{"version":"1.0.0","segments":[{"startTime":0.54,"endTime":1.62,"body":"- Hello everyone."},{"startTime":1.62,"endTime":5.757,"body":"Today's talk is on \"The Maths"},{"startTime":1.62,"endTime":5.757,"body":"of Coins and Currencies.\""},{"startTime":6.93,"endTime":9.27,"body":"And maths can help us"},{"startTime":6.93,"endTime":9.27,"body":"with lots of questions"},{"startTime":9.27,"endTime":10.95,"body":"about coins and currencies."},{"startTime":10.95,"endTime":13.593,"body":"How do you detect a counterfeit coin?"},{"startTime":14.79,"endTime":16.62,"body":"We know that bank notes are rectangular,"},{"startTime":16.62,"endTime":18.66,"body":"why aren't coins rectangular?"},{"startTime":18.66,"endTime":20.25,"body":"What's so great about circles,"},{"startTime":20.25,"endTime":22.41,"body":"if they are indeed the best solution?"},{"startTime":22.41,"endTime":25.74,"body":"And finally, why is the money"},{"startTime":22.41,"endTime":25.74,"body":"in Harry Potter's pockets"},{"startTime":25.74,"endTime":27.45,"body":"so implausible?"},{"startTime":27.45,"endTime":29.01,"body":"We'll get onto all of those things,"},{"startTime":29.01,"endTime":32.46,"body":"but let's begin with the fakers."},{"startTime":32.46,"endTime":35.64,"body":"There's a very old mathematical problem"},{"startTime":35.64,"endTime":40.64,"body":"about detecting a fake, and"},{"startTime":35.64,"endTime":40.64,"body":"it's the story of Hiero's crown."},{"startTime":41.67,"endTime":45.42,"body":"This was recounted by the"},{"startTime":41.67,"endTime":45.42,"body":"Roman architect Vitruvius,"},{"startTime":45.42,"endTime":48.06,"body":"in his book, \"De Architectura,\""},{"startTime":45.42,"endTime":48.06,"body":"which had lots in it,"},{"startTime":48.06,"endTime":49.95,"body":"apart from architecture as well."},{"startTime":49.95,"endTime":52.47,"body":"And one of the things was this story about"},{"startTime":52.47,"endTime":55.5,"body":"King Hiero II of Syracuse."},{"startTime":55.5,"endTime":57.33,"body":"And he had commissioned a goldsmith"},{"startTime":57.33,"endTime":61.59,"body":"to make a votive crown out of gold,"},{"startTime":61.59,"endTime":64.56,"body":"he gave him a certain amount"},{"startTime":61.59,"endTime":64.56,"body":"of gold to do this with,"},{"startTime":64.56,"endTime":67.83,"body":"and the goldsmith presented"},{"startTime":64.56,"endTime":67.83,"body":"him with this beautiful thing."},{"startTime":67.83,"endTime":69.72,"body":"But Hiero had heard a rumor"},{"startTime":69.72,"endTime":72.15,"body":"that the goldsmith may have cheated him,"},{"startTime":72.15,"endTime":75.24,"body":"that he may have kept some"},{"startTime":72.15,"endTime":75.24,"body":"of the gold for himself"},{"startTime":75.24,"endTime":78.24,"body":"and replaced that amount with silver."},{"startTime":78.24,"endTime":81.57,"body":"So Hiero wanted to know if this was true."},{"startTime":81.57,"endTime":84.87,"body":"So he asked nearby"},{"startTime":81.57,"endTime":84.87,"body":"mathematician Archimedes,"},{"startTime":84.87,"endTime":87.33,"body":"who lived in Syracuse very conveniently,"},{"startTime":87.33,"endTime":88.83,"body":"to help him with this."},{"startTime":88.83,"endTime":90.12,"body":"How can we work out"},{"startTime":90.12,"endTime":93.15,"body":"if this thing that's meant to be pure gold"},{"startTime":93.15,"endTime":96.54,"body":"actually has been mixed with"},{"startTime":93.15,"endTime":96.54,"body":"silver or something else?"},{"startTime":96.54,"endTime":100.47,"body":"And the problem is, you"},{"startTime":96.54,"endTime":100.47,"body":"could check quite easily,"},{"startTime":100.47,"endTime":103.2,"body":"because gold and silver"},{"startTime":100.47,"endTime":103.2,"body":"have different densities,"},{"startTime":103.2,"endTime":105.06,"body":"they take up different volumes of space,"},{"startTime":105.06,"endTime":108.9,"body":"the same mass of silver"},{"startTime":105.06,"endTime":108.9,"body":"would take up more space"},{"startTime":108.9,"endTime":110.82,"body":"than the same mass of gold."},{"startTime":110.82,"endTime":114.3,"body":"But it's very difficult to"},{"startTime":110.82,"endTime":114.3,"body":"measure the volume of some,"},{"startTime":114.3,"endTime":117.99,"body":"you know, fancy artifact like"},{"startTime":114.3,"endTime":117.99,"body":"a crown, without damaging it."},{"startTime":117.99,"endTime":120.39,"body":"So the problem that"},{"startTime":117.99,"endTime":120.39,"body":"Archimedes is faced with,"},{"startTime":120.39,"endTime":123.9,"body":"is how can I test this thing"},{"startTime":123.9,"endTime":126.45,"body":"to see if it's pure gold or not?"},{"startTime":126.45,"endTime":129.69,"body":"How can I measure the volume"},{"startTime":126.45,"endTime":129.69,"body":"of this very delicate object,"},{"startTime":129.69,"endTime":130.8,"body":"without damaging it?"},{"startTime":130.8,"endTime":132.42,"body":"So we all know the legend, right?"},{"startTime":132.42,"endTime":135.3,"body":"Archimedes is sitting there in the bath"},{"startTime":135.3,"endTime":137.07,"body":"having a think about this."},{"startTime":137.07,"endTime":139.68,"body":"There's the crown over"},{"startTime":137.07,"endTime":139.68,"body":"there in this old picture."},{"startTime":139.68,"endTime":141.18,"body":"And he's just sort of musing,"},{"startTime":141.18,"endTime":143.04,"body":"like all the best"},{"startTime":141.18,"endTime":143.04,"body":"theories are come up with"},{"startTime":143.04,"endTime":144.66,"body":"in the bath, right?"},{"startTime":144.66,"endTime":146.31,"body":"He suddenly has this great idea."},{"startTime":146.31,"endTime":149.43,"body":"He realizes that when"},{"startTime":146.31,"endTime":149.43,"body":"he gets into the bath,"},{"startTime":149.43,"endTime":151.71,"body":"the amount of water that's displaced"},{"startTime":151.71,"endTime":154.29,"body":"is exactly equal to his volume."},{"startTime":154.29,"endTime":157.11,"body":"So this, he could do this"},{"startTime":154.29,"endTime":157.11,"body":"with the crown as well."},{"startTime":157.11,"endTime":160.053,"body":"So eureka, runs through"},{"startTime":157.11,"endTime":160.053,"body":"the streets, great story."},{"startTime":161.67,"endTime":163.41,"body":"There's a postscript to this though,"},{"startTime":163.41,"endTime":165.12,"body":"comes quite a while later."},{"startTime":165.12,"endTime":166.95,"body":"Over a thousand years later,"},{"startTime":166.95,"endTime":171.36,"body":"Galileo wrote a little"},{"startTime":166.95,"endTime":171.36,"body":"treaties about this story,"},{"startTime":171.36,"endTime":174.927,"body":"and he said, Vitruvius, he thought,"},{"startTime":174.927,"endTime":177.27,"body":"had got the story a bit wrong."},{"startTime":177.27,"endTime":180.9,"body":"He perhaps had heard that"},{"startTime":177.27,"endTime":180.9,"body":"Archimedes had solved this problem"},{"startTime":180.9,"endTime":182.61,"body":"using something to do with water,"},{"startTime":182.61,"endTime":184.83,"body":"and had made the assumption"},{"startTime":182.61,"endTime":184.83,"body":"that he'd solved it"},{"startTime":184.83,"endTime":187.41,"body":"basically by chucking a crown in a bath."},{"startTime":187.41,"endTime":190.38,"body":"That didn't seem very elegant to Galileo."},{"startTime":190.38,"endTime":192.3,"body":"He thought it would be too inaccurate"},{"startTime":192.3,"endTime":193.29,"body":"and it wouldn't quite work."},{"startTime":193.29,"endTime":195.12,"body":"So Galileo wrote this little treaties"},{"startTime":195.12,"endTime":198.09,"body":"on what he thought"},{"startTime":195.12,"endTime":198.09,"body":"Archimedes had really done."},{"startTime":198.09,"endTime":200.55,"body":"I mean, we'll never know, but"},{"startTime":198.09,"endTime":200.55,"body":"I really like Galileo's idea,"},{"startTime":200.55,"endTime":202.14,"body":"so I'm going to share it with you."},{"startTime":202.14,"endTime":203.463,"body":"So there's Galileo."},{"startTime":204.33,"endTime":208.56,"body":"The basic underlying"},{"startTime":204.33,"endTime":208.56,"body":"solution that Vitruvius said"},{"startTime":208.56,"endTime":210.09,"body":"that Archimedes had done,"},{"startTime":210.09,"endTime":213.36,"body":"was based on comparing or knowing that"},{"startTime":213.36,"endTime":215.79,"body":"the densities of gold"},{"startTime":213.36,"endTime":215.79,"body":"and silver are different."},{"startTime":215.79,"endTime":220.29,"body":"And, so using modern units for ease,"},{"startTime":220.29,"endTime":223.68,"body":"we can see what's happening,"},{"startTime":220.29,"endTime":223.68,"body":"the density of gold is high."},{"startTime":223.68,"endTime":228.03,"body":"It's sort of 19.32 grams"},{"startTime":223.68,"endTime":228.03,"body":"per centimeter cubed."},{"startTime":228.03,"endTime":229.41,"body":"The density of silver is much less,"},{"startTime":229.41,"endTime":232.5,"body":"about 10 1/2 grams per centimeter cubed."},{"startTime":232.5,"endTime":234.99,"body":"So density is mass"},{"startTime":232.5,"endTime":234.99,"body":"divided by volume, right?"},{"startTime":234.99,"endTime":236.64,"body":"So you can see from that, from the units."},{"startTime":236.64,"endTime":240.99,"body":"So then the volume would be,"},{"startTime":236.64,"endTime":240.99,"body":"just flip that upside down,"},{"startTime":240.99,"endTime":242.73,"body":"mass divided by density."},{"startTime":242.73,"endTime":247.08,"body":"So that tells you something"},{"startTime":242.73,"endTime":247.08,"body":"that's very dense will have,"},{"startTime":247.08,"endTime":249.36,"body":"for a given mass, a smaller volume."},{"startTime":249.36,"endTime":253.11,"body":"So gold will take up"},{"startTime":249.36,"endTime":253.11,"body":"less space per kilogram"},{"startTime":253.11,"endTime":254.16,"body":"than silver will."},{"startTime":254.16,"endTime":257.13,"body":"And you can work out exactly"},{"startTime":254.16,"endTime":257.13,"body":"how much in modern units."},{"startTime":257.13,"endTime":260.16,"body":"So a kilogram of gold, you"},{"startTime":257.13,"endTime":260.16,"body":"can just put the numbers in,"},{"startTime":260.16,"endTime":262.62,"body":"a kilogram of gold will"},{"startTime":260.16,"endTime":262.62,"body":"take up a volume of"},{"startTime":262.62,"endTime":266.25,"body":"51.76-ish cubic centimeters,"},{"startTime":266.25,"endTime":271.233,"body":"whereas the same mass of silver"},{"startTime":266.25,"endTime":271.233,"body":"takes up much more space."},{"startTime":272.28,"endTime":273.51,"body":"So that's all fine."},{"startTime":273.51,"endTime":276.213,"body":"So you could compare the"},{"startTime":273.51,"endTime":276.213,"body":"volumes and have a look."},{"startTime":277.11,"endTime":279.51,"body":"The issue is that of"},{"startTime":277.11,"endTime":279.51,"body":"course we're not saying"},{"startTime":279.51,"endTime":282.6,"body":"that the goldsmith has replaced"},{"startTime":279.51,"endTime":282.6,"body":"all of the gold with silver."},{"startTime":282.6,"endTime":284.16,"body":"That would be easy to"},{"startTime":282.6,"endTime":284.16,"body":"see with the naked eye,"},{"startTime":284.16,"endTime":285.41,"body":"the color would be wrong."},{"startTime":286.32,"endTime":288.51,"body":"So what's probably"},{"startTime":286.32,"endTime":288.51,"body":"likely to have happened,"},{"startTime":288.51,"endTime":290.16,"body":"if there has been cheating,"},{"startTime":290.16,"endTime":293.25,"body":"is that a bit of the gold has"},{"startTime":290.16,"endTime":293.25,"body":"been replaced, but not lots."},{"startTime":293.25,"endTime":294.96,"body":"So maybe let's say about 10%,"},{"startTime":294.96,"endTime":297.24,"body":"just to have something to work with."},{"startTime":297.24,"endTime":300.96,"body":"If this crown, let's say"},{"startTime":297.24,"endTime":300.96,"body":"it weighs a kilogram,"},{"startTime":300.96,"endTime":302.25,"body":"is supposed to weigh a kilogram."},{"startTime":302.25,"endTime":305.13,"body":"If actually 10% of it is silver, not gold,"},{"startTime":305.13,"endTime":306.45,"body":"so it still weighs a kilogram,"},{"startTime":306.45,"endTime":308.49,"body":"but it has a slightly"},{"startTime":306.45,"endTime":308.49,"body":"different composition,"},{"startTime":308.49,"endTime":310.086,"body":"then that's 900 grams of gold,"},{"startTime":310.086,"endTime":312.112,"body":"and you can work out the volume of that."},{"startTime":312.112,"endTime":312.945,"body":"And a hundred grams of silver,"},{"startTime":312.945,"endTime":314.82,"body":"and you can work out the volume of that,"},{"startTime":314.82,"endTime":316.59,"body":"based on those densities."},{"startTime":316.59,"endTime":321.39,"body":"And so the total amount that"},{"startTime":316.59,"endTime":321.39,"body":"you would arrive at for volume"},{"startTime":321.39,"endTime":323.61,"body":"is about 56 centimeters cubed."},{"startTime":323.61,"endTime":328.08,"body":"That's only just over four"},{"startTime":323.61,"endTime":328.08,"body":"cubic centimeters discrepancy."},{"startTime":328.08,"endTime":331.38,"body":"So how are you actually"},{"startTime":328.08,"endTime":331.38,"body":"going to measure that?"},{"startTime":331.38,"endTime":335.07,"body":"The crown in a bucket method,"},{"startTime":331.38,"endTime":335.07,"body":"which is sort of the crude,"},{"startTime":335.07,"endTime":336.69,"body":"the crude thing you could do."},{"startTime":336.69,"endTime":338.28,"body":"There's your vessel, you take a, you know,"},{"startTime":338.28,"endTime":339.84,"body":"something that's big"},{"startTime":338.28,"endTime":339.84,"body":"enough to hold a crown."},{"startTime":339.84,"endTime":341.13,"body":"So how big is a crown?"},{"startTime":341.13,"endTime":343.32,"body":"I mean, it's going to have"},{"startTime":341.13,"endTime":343.32,"body":"to be sort of at least like"},{"startTime":343.32,"endTime":346.82,"body":"20 centimeters wide,"},{"startTime":343.32,"endTime":346.82,"body":"let's say this vessel."},{"startTime":346.82,"endTime":350.673,"body":"So you get some kind"},{"startTime":346.82,"endTime":350.673,"body":"of, let's say a bucket,"},{"startTime":351.51,"endTime":352.637,"body":"you're going to put your crown"},{"startTime":351.51,"endTime":352.637,"body":"in, you fill it with water,"},{"startTime":352.637,"endTime":354.51,"body":"or you put some water in it."},{"startTime":354.51,"endTime":358.2,"body":"Then you want to know how"},{"startTime":354.51,"endTime":358.2,"body":"much volume the crown takes up"},{"startTime":358.2,"endTime":360.75,"body":"compared to how much volume"},{"startTime":360.75,"endTime":363.48,"body":"the equivalent weight of"},{"startTime":360.75,"endTime":363.48,"body":"pure gold will take up."},{"startTime":363.48,"endTime":367.5,"body":"So you take some gold, the"},{"startTime":363.48,"endTime":367.5,"body":"water level will rise a bit,"},{"startTime":367.5,"endTime":370.23,"body":"you can mark that off,"},{"startTime":367.5,"endTime":370.23,"body":"then you replace it,"},{"startTime":370.23,"endTime":372.63,"body":"take out the gold and put in the crown,"},{"startTime":372.63,"endTime":375.0,"body":"and the water level, if it's different,"},{"startTime":375.0,"endTime":378.03,"body":"then you'll be able to see"},{"startTime":375.0,"endTime":378.03,"body":"that there's been a problem."},{"startTime":378.03,"endTime":382.41,"body":"But the issue is, it's not"},{"startTime":378.03,"endTime":382.41,"body":"going to be very much different."},{"startTime":382.41,"endTime":384.57,"body":"And let's think about how"},{"startTime":382.41,"endTime":384.57,"body":"different it's going to be."},{"startTime":384.57,"endTime":387.09,"body":"We've got this, let's"},{"startTime":384.57,"endTime":387.09,"body":"say it has a diameter"},{"startTime":387.09,"endTime":388.08,"body":"of 20 centimeters,"},{"startTime":388.08,"endTime":390.51,"body":"because then the radius is 10 centimeters,"},{"startTime":390.51,"endTime":393.15,"body":"and we can all do pi"},{"startTime":390.51,"endTime":393.15,"body":"r squared in our heads"},{"startTime":393.15,"endTime":396.858,"body":"and get the surface area of this."},{"startTime":396.858,"endTime":399.6,"body":"The surface area of the water"},{"startTime":396.858,"endTime":399.6,"body":"will be a hundred pi, right,"},{"startTime":399.6,"endTime":402.513,"body":"314 centimeters squared, roughly."},{"startTime":404.34,"endTime":408.06,"body":"So that's the surface area"},{"startTime":404.34,"endTime":408.06,"body":"of the water at the top."},{"startTime":408.06,"endTime":411.66,"body":"And then if our difference"},{"startTime":408.06,"endTime":411.66,"body":"that we're looking for"},{"startTime":411.66,"endTime":412.77,"body":"between the pure gold"},{"startTime":412.77,"endTime":415.8,"body":"and the potentially slightly dodgy crown,"},{"startTime":415.8,"endTime":417.72,"body":"the difference is only four"},{"startTime":415.8,"endTime":417.72,"body":"point, whatever it was,"},{"startTime":417.72,"endTime":421.53,"body":"three six centimeters cubed,"},{"startTime":417.72,"endTime":421.53,"body":"we've got to spread that out"},{"startTime":421.53,"endTime":423.78,"body":"over the whole surface of the water."},{"startTime":423.78,"endTime":425.28,"body":"So when you do that,"},{"startTime":425.28,"endTime":428.31,"body":"you find that the actual"},{"startTime":425.28,"endTime":428.31,"body":"difference in the water level"},{"startTime":428.31,"endTime":430.77,"body":"is tiny, tiny."},{"startTime":430.77,"endTime":433.05,"body":"So that, you know, would"},{"startTime":430.77,"endTime":433.05,"body":"be completely outweighed by"},{"startTime":433.05,"endTime":434.948,"body":"losing a few drops here and there."},{"startTime":434.948,"endTime":437.25,"body":"You're not going to be"},{"startTime":434.948,"endTime":437.25,"body":"able to detect that."},{"startTime":437.25,"endTime":440.25,"body":"Now this is kind of a"},{"startTime":437.25,"endTime":440.25,"body":"false problem really,"},{"startTime":440.25,"endTime":443.28,"body":"because this isn't a good way to measure"},{"startTime":443.28,"endTime":444.45,"body":"the difference in volume."},{"startTime":444.45,"endTime":446.97,"body":"If you just did it like"},{"startTime":444.45,"endTime":446.97,"body":"this, crown in a bucket,"},{"startTime":446.97,"endTime":450.03,"body":"that's, you know, this isn't"},{"startTime":446.97,"endTime":450.03,"body":"the Archimedes that is famous."},{"startTime":450.03,"endTime":451.05,"body":"He wouldn't have done it like this,"},{"startTime":451.05,"endTime":452.64,"body":"even if he was using this method."},{"startTime":452.64,"endTime":454.89,"body":"So there is a way to do this better."},{"startTime":454.89,"endTime":459.09,"body":"A better approach to this"},{"startTime":454.89,"endTime":459.09,"body":"would be to use something,"},{"startTime":459.09,"endTime":460.5,"body":"and this is sort of the kind of technology"},{"startTime":460.5,"endTime":463.47,"body":"you might have used with the"},{"startTime":460.5,"endTime":463.47,"body":"water clocks of the time."},{"startTime":463.47,"endTime":466.08,"body":"So you have your vessel,"},{"startTime":463.47,"endTime":466.08,"body":"but you have a little hole,"},{"startTime":466.08,"endTime":470.13,"body":"a little opening, some of the way up,"},{"startTime":470.13,"endTime":471.72,"body":"and you fill it up to that level,"},{"startTime":471.72,"endTime":474.15,"body":"and then you put your lump of gold in,"},{"startTime":474.15,"endTime":476.55,"body":"your one kilogram of gold, and, you know,"},{"startTime":476.55,"endTime":478.59,"body":"the water level will"},{"startTime":476.55,"endTime":478.59,"body":"rise and will overflow."},{"startTime":478.59,"endTime":479.94,"body":"Don't worry about that."},{"startTime":479.94,"endTime":482.01,"body":"Then before you put the crown in,"},{"startTime":482.01,"endTime":485.22,"body":"you take out the gold and"},{"startTime":482.01,"endTime":485.22,"body":"replace with the crown."},{"startTime":485.22,"endTime":487.2,"body":"What you're going to do"},{"startTime":485.22,"endTime":487.2,"body":"is put a very narrow tube,"},{"startTime":487.2,"endTime":489.963,"body":"like a test tube, at the opening there."},{"startTime":491.04,"endTime":492.87,"body":"And because it's very narrow,"},{"startTime":492.87,"endTime":495.39,"body":"then it'll be much easier"},{"startTime":492.87,"endTime":495.39,"body":"to detect what's going on."},{"startTime":495.39,"endTime":500.143,"body":"So you do that, and you put"},{"startTime":495.39,"endTime":500.143,"body":"the crown in, and there we go."},{"startTime":501.09,"endTime":505.62,"body":"And so some extra bit of"},{"startTime":501.09,"endTime":505.62,"body":"water will now be displaced."},{"startTime":505.62,"endTime":509.46,"body":"4.36 centimeters cubed of"},{"startTime":505.62,"endTime":509.46,"body":"water will be displaced."},{"startTime":509.46,"endTime":514.383,"body":"But because now we're putting"},{"startTime":509.46,"endTime":514.383,"body":"it into a narrow tube,"},{"startTime":515.31,"endTime":517.86,"body":"if it was a centimeter"},{"startTime":515.31,"endTime":517.86,"body":"wide, you would get,"},{"startTime":517.86,"endTime":521.46,"body":"I mean 4.36 centimeters is"},{"startTime":517.86,"endTime":521.46,"body":"clearly enough to detect,"},{"startTime":521.46,"endTime":524.37,"body":"and that would've been"},{"startTime":521.46,"endTime":524.37,"body":"possible at the time to do,"},{"startTime":524.37,"endTime":526.65,"body":"you could get that kind of accuracy."},{"startTime":526.65,"endTime":530.73,"body":"So that would be a way to deal with this."},{"startTime":530.73,"endTime":534.434,"body":"But I still like Galileo's"},{"startTime":530.73,"endTime":534.434,"body":"suggestion because it,"},{"startTime":534.434,"endTime":535.35,"body":"and we'll come onto that now,"},{"startTime":535.35,"endTime":539.67,"body":"because it incorporates more"},{"startTime":535.35,"endTime":539.67,"body":"things that Archimedes knew."},{"startTime":539.67,"endTime":541.53,"body":"So he knew about the volume,"},{"startTime":541.53,"endTime":545.46,"body":"his eureka moment was about"},{"startTime":541.53,"endTime":545.46,"body":"the volume being displaced"},{"startTime":545.46,"endTime":547.86,"body":"is equal to the volume of the object"},{"startTime":547.86,"endTime":549.66,"body":"that you're putting into the water."},{"startTime":549.66,"endTime":551.58,"body":"But he also said things about buoyancy,"},{"startTime":551.58,"endTime":553.29,"body":"and he said things about levers."},{"startTime":553.29,"endTime":557.19,"body":"So Galileo's suggestion"},{"startTime":553.29,"endTime":557.19,"body":"involves all of those things."},{"startTime":557.19,"endTime":558.84,"body":"So it's pretty cool."},{"startTime":558.84,"endTime":560.01,"body":"So what Galileo says is,"},{"startTime":560.01,"endTime":562.59,"body":"and we'll do a little thought experiment."},{"startTime":562.59,"endTime":566.28,"body":"Imagine we've got some scales,"},{"startTime":566.28,"endTime":568.533,"body":"sort of balance the scales."},{"startTime":569.52,"endTime":574.47,"body":"And we've put, just to start us off,"},{"startTime":574.47,"endTime":576.54,"body":"we're going to weigh"},{"startTime":574.47,"endTime":576.54,"body":"equal amounts of gold,"},{"startTime":576.54,"endTime":578.22,"body":"equal masses of gold and silver."},{"startTime":578.22,"endTime":581.37,"body":"So say you have a kilogram of"},{"startTime":578.22,"endTime":581.37,"body":"gold and a kilogram of silver,"},{"startTime":581.37,"endTime":584.52,"body":"and you know that they're equal"},{"startTime":581.37,"endTime":584.52,"body":"'cause the scales balance."},{"startTime":584.52,"endTime":586.41,"body":"So now, thought experiment."},{"startTime":586.41,"endTime":590.91,"body":"Imagine if we take this whole"},{"startTime":586.41,"endTime":590.91,"body":"setup and put it underwater,"},{"startTime":590.91,"endTime":592.86,"body":"we submerge it."},{"startTime":592.86,"endTime":593.86,"body":"What's going to happen?"},{"startTime":594.72,"endTime":597.69,"body":"So I've put it underwater,"},{"startTime":594.72,"endTime":597.69,"body":"I haven't let go yet."},{"startTime":597.69,"endTime":599.97,"body":"Let's think about what's going to happen."},{"startTime":599.97,"endTime":603.69,"body":"So now that the scales are underwater,"},{"startTime":603.69,"endTime":606.42,"body":"imagine if they had nothing in the pans,"},{"startTime":606.42,"endTime":608.37,"body":"then you've got the same amount of water"},{"startTime":608.37,"endTime":610.35,"body":"pressing down on each side."},{"startTime":610.35,"endTime":612.57,"body":"So they're going to balance."},{"startTime":612.57,"endTime":615.84,"body":"Then you put the gold"},{"startTime":612.57,"endTime":615.84,"body":"in the left hand pan,"},{"startTime":615.84,"endTime":620.84,"body":"and you put say a kilogram"},{"startTime":615.84,"endTime":620.84,"body":"of gold in the left hand pan."},{"startTime":620.97,"endTime":622.53,"body":"Now, that is pressing down,"},{"startTime":622.53,"endTime":625.62,"body":"a kilogram is pressing"},{"startTime":622.53,"endTime":625.62,"body":"down on the left hand pan."},{"startTime":625.62,"endTime":630.62,"body":"But, but remember, we have"},{"startTime":625.62,"endTime":630.62,"body":"displaced its volume of water."},{"startTime":632.4,"endTime":636.33,"body":"So we've got a kilogram"},{"startTime":632.4,"endTime":636.33,"body":"of gold, that has volume,"},{"startTime":636.33,"endTime":638.91,"body":"we worked out earlier,"},{"startTime":636.33,"endTime":638.91,"body":"51 point whatever it is,"},{"startTime":638.91,"endTime":640.47,"body":"76 centimeters cubed."},{"startTime":640.47,"endTime":642.36,"body":"That amount of water will be displaced,"},{"startTime":642.36,"endTime":643.68,"body":"pushed out of the way,"},{"startTime":643.68,"endTime":647.31,"body":"and will therefore not be"},{"startTime":643.68,"endTime":647.31,"body":"pressing down on the scales."},{"startTime":647.31,"endTime":652.26,"body":"So how much does 51.76"},{"startTime":647.31,"endTime":652.26,"body":"centimeters cubed of water weigh?"},{"startTime":652.26,"endTime":654.27,"body":"Well luckily, it's very easy because"},{"startTime":654.27,"endTime":657.81,"body":"one centimeter cubed of water"},{"startTime":654.27,"endTime":657.81,"body":"by definition weighs a gram."},{"startTime":657.81,"endTime":660.72,"body":"So that's just, replace"},{"startTime":657.81,"endTime":660.72,"body":"centimeters cubed by grams"},{"startTime":660.72,"endTime":661.89,"body":"and that's how much it is."},{"startTime":661.89,"endTime":665.31,"body":"So the apparent weight of the gold now,"},{"startTime":665.31,"endTime":666.87,"body":"because we've pushed"},{"startTime":665.31,"endTime":666.87,"body":"some water out of the way"},{"startTime":666.87,"endTime":669.03,"body":"and so that's not pushing down the scales,"},{"startTime":669.03,"endTime":674.03,"body":"the gold will appear to weigh"},{"startTime":669.03,"endTime":674.03,"body":"948.24 grams underwater."},{"startTime":675.63,"endTime":678.57,"body":"Whereas the silver, now silver,"},{"startTime":678.57,"endTime":681.27,"body":"a kilogram of silver takes up more space."},{"startTime":681.27,"endTime":685.05,"body":"It's more like 95 centimeters cubed,"},{"startTime":685.05,"endTime":687.24,"body":"and so its apparent weight will only be"},{"startTime":687.24,"endTime":690.99,"body":"904 and a bit grams underwater."},{"startTime":690.99,"endTime":694.86,"body":"So this is really cool"},{"startTime":690.99,"endTime":694.86,"body":"that the less dense thing"},{"startTime":694.86,"endTime":697.53,"body":"will appear to weigh less"},{"startTime":694.86,"endTime":697.53,"body":"when you put it underwater."},{"startTime":697.53,"endTime":700.17,"body":"So actually the scales won't balance."},{"startTime":700.17,"endTime":704.43,"body":"The gold, the true gold, will"},{"startTime":700.17,"endTime":704.43,"body":"look like it weighs more,"},{"startTime":704.43,"endTime":706.56,"body":"because of this buoyancy effect."},{"startTime":706.56,"endTime":708.303,"body":"And this is a really clever"},{"startTime":706.56,"endTime":708.303,"body":"and sort of an instant thing"},{"startTime":708.303,"endTime":711.09,"body":"that you could say, okay, look, yeah."},{"startTime":711.09,"endTime":713.85,"body":"So if you've got your crown,"},{"startTime":711.09,"endTime":713.85,"body":"which has a bit of silver in,"},{"startTime":713.85,"endTime":716.46,"body":"it's going to appear"},{"startTime":713.85,"endTime":716.46,"body":"underwater to weigh less."},{"startTime":716.46,"endTime":718.29,"body":"So that's a very nice thing."},{"startTime":718.29,"endTime":720.39,"body":"I don't have practicalities to"},{"startTime":718.29,"endTime":720.39,"body":"really do, but I don't care,"},{"startTime":720.39,"endTime":722.1,"body":"I really liked that solution."},{"startTime":722.1,"endTime":724.74,"body":"And Galileo actually"},{"startTime":722.1,"endTime":724.74,"body":"went on to explain how,"},{"startTime":724.74,"endTime":729.74,"body":"by moving how far away the pan"},{"startTime":724.74,"endTime":729.74,"body":"is from the central fulcrum,"},{"startTime":730.17,"endTime":732.12,"body":"and that's the levers bit, you can,"},{"startTime":732.12,"endTime":733.92,"body":"if you make them balance underwater,"},{"startTime":733.92,"endTime":736.02,"body":"you can then work out actually"},{"startTime":736.02,"endTime":740.16,"body":"how much silver has been"},{"startTime":736.02,"endTime":740.16,"body":"added into the crown."},{"startTime":740.16,"endTime":743.76,"body":"So this is a really cool"},{"startTime":740.16,"endTime":743.76,"body":"solution, I really like it,"},{"startTime":743.76,"endTime":746.52,"body":"but it also, it struck me suddenly,"},{"startTime":746.52,"endTime":748.2,"body":"while thinking about this,"},{"startTime":748.2,"endTime":751.26,"body":"that actually air doesn't weigh nothing."},{"startTime":751.26,"endTime":754.62,"body":"Air, I mean of course it's"},{"startTime":751.26,"endTime":754.62,"body":"much less dense than water,"},{"startTime":754.62,"endTime":758.4,"body":"but even air weighs a little bit."},{"startTime":758.4,"endTime":762.09,"body":"And so actually, if you've got"},{"startTime":758.4,"endTime":762.09,"body":"a pair of scales like this,"},{"startTime":762.09,"endTime":765.42,"body":"and you want to truly compare"},{"startTime":765.42,"endTime":766.83,"body":"the different masses of things,"},{"startTime":766.83,"endTime":769.77,"body":"you'd have to do it in a"},{"startTime":766.83,"endTime":769.77,"body":"vacuum to get the right answer."},{"startTime":769.77,"endTime":772.68,"body":"Because actually, if you"},{"startTime":769.77,"endTime":772.68,"body":"think about, you know,"},{"startTime":772.68,"endTime":777.06,"body":"the old puzzle, the old trick"},{"startTime":772.68,"endTime":777.06,"body":"question that we ask kids,"},{"startTime":777.06,"endTime":779.79,"body":"which weighs more, a ton of"},{"startTime":777.06,"endTime":779.79,"body":"bricks or a ton of feathers."},{"startTime":779.79,"endTime":782.55,"body":"And of course, you know,"},{"startTime":779.79,"endTime":782.55,"body":"the unwary person will say,"},{"startTime":782.55,"endTime":784.89,"body":"oh well it's the bricks,"},{"startTime":782.55,"endTime":784.89,"body":"'cause bricks are really heavy."},{"startTime":784.89,"endTime":786.72,"body":"And you say ah-ha-ha,"},{"startTime":784.89,"endTime":786.72,"body":"and you laugh at them."},{"startTime":786.72,"endTime":789.96,"body":"But no, but no, because if you weigh them,"},{"startTime":789.96,"endTime":792.57,"body":"if you actually have genuinely a real,"},{"startTime":792.57,"endTime":794.64,"body":"genuinely the same mass of these things"},{"startTime":794.64,"endTime":796.95,"body":"and you've weighed them in a"},{"startTime":794.64,"endTime":796.95,"body":"vacuum and you're really sure,"},{"startTime":796.95,"endTime":800.28,"body":"and then you weigh them in the open air,"},{"startTime":800.28,"endTime":801.69,"body":"I hope the feathers don't fly away,"},{"startTime":801.69,"endTime":803.94,"body":"but weigh them in the open air,"},{"startTime":803.94,"endTime":806.64,"body":"then the feathers take"},{"startTime":803.94,"endTime":806.64,"body":"up so much more space"},{"startTime":806.64,"endTime":808.98,"body":"than the bricks, that they"},{"startTime":806.64,"endTime":808.98,"body":"would displace a lot of air,"},{"startTime":808.98,"endTime":810.81,"body":"and so actually"},{"startTime":810.81,"endTime":813.06,"body":"maybe the bricks do weigh"},{"startTime":810.81,"endTime":813.06,"body":"more than the feathers."},{"startTime":813.06,"endTime":816.18,"body":"So if you ever accidentally"},{"startTime":813.06,"endTime":816.18,"body":"fall for that question,"},{"startTime":816.18,"endTime":817.71,"body":"you can now come back and say yes,"},{"startTime":817.71,"endTime":819.54,"body":"but actually they would"},{"startTime":817.71,"endTime":819.54,"body":"appear to weigh more"},{"startTime":819.54,"endTime":820.373,"body":"than the feathers."},{"startTime":820.373,"endTime":824.13,"body":"So there we are, it even is true for air."},{"startTime":824.13,"endTime":826.47,"body":"We don't need to worry"},{"startTime":824.13,"endTime":826.47,"body":"about that too much."},{"startTime":826.47,"endTime":829.89,"body":"But it's a nice little"},{"startTime":826.47,"endTime":829.89,"body":"solution to the question."},{"startTime":829.89,"endTime":833.223,"body":"So that's one way of"},{"startTime":829.89,"endTime":833.223,"body":"catching the counterfeiters."},{"startTime":834.39,"endTime":839.22,"body":"I believe history tells"},{"startTime":834.39,"endTime":839.22,"body":"that actually the goldsmith"},{"startTime":839.22,"endTime":840.51,"body":"had been cheating."},{"startTime":840.51,"endTime":843.06,"body":"I do not know what his fate was."},{"startTime":843.06,"endTime":845.34,"body":"It probably wasn't a good fate."},{"startTime":845.34,"endTime":849.453,"body":"So yes, do not cheat the"},{"startTime":845.34,"endTime":849.453,"body":"king, if you're a goldsmith."},{"startTime":850.29,"endTime":853.53,"body":"Now if you are thinking about coins,"},{"startTime":853.53,"endTime":856.02,"body":"it could be a little easier to check,"},{"startTime":856.02,"endTime":858.66,"body":"or to show that you've"},{"startTime":856.02,"endTime":858.66,"body":"got a counterfeit coin,"},{"startTime":858.66,"endTime":861.78,"body":"because, of course, a"},{"startTime":858.66,"endTime":861.78,"body":"counterfeit coin has to look like"},{"startTime":861.78,"endTime":863.12,"body":"the real thing."},{"startTime":863.12,"endTime":865.8,"body":"So it would have to be"},{"startTime":863.12,"endTime":865.8,"body":"the same size and shape,"},{"startTime":865.8,"endTime":869.34,"body":"and therefore the volume"},{"startTime":865.8,"endTime":869.34,"body":"should be the same"},{"startTime":869.34,"endTime":870.78,"body":"as the real coin."},{"startTime":870.78,"endTime":872.37,"body":"So then you can just weigh them"},{"startTime":872.37,"endTime":875.76,"body":"and the counterfeit is likely"},{"startTime":872.37,"endTime":875.76,"body":"to have a different weight."},{"startTime":875.76,"endTime":878.46,"body":"So that's one way of finding counterfeits."},{"startTime":878.46,"endTime":880.74,"body":"There are various other"},{"startTime":878.46,"endTime":880.74,"body":"methods that have been used"},{"startTime":880.74,"endTime":882.15,"body":"over the centuries."},{"startTime":882.15,"endTime":884.1,"body":"You can measure things,"},{"startTime":884.1,"endTime":888.33,"body":"and to stop people"},{"startTime":884.1,"endTime":888.33,"body":"snipping bits off coins,"},{"startTime":888.33,"endTime":889.96,"body":"we have things like milling"},{"startTime":888.33,"endTime":889.96,"body":"round the edge of coins."},{"startTime":889.96,"endTime":892.65,"body":"We've put some sort of markings or words"},{"startTime":892.65,"endTime":893.52,"body":"round the edge of the coin,"},{"startTime":893.52,"endTime":895.11,"body":"that stops people clipping coins,"},{"startTime":895.11,"endTime":898.41,"body":"which used to be a real problem"},{"startTime":895.11,"endTime":898.41,"body":"in Tudor times, for example."},{"startTime":898.41,"endTime":900.3,"body":"So you can have a look"},{"startTime":898.41,"endTime":900.3,"body":"at the coin and check"},{"startTime":900.3,"endTime":902.4,"body":"has it been tampered with in that way."},{"startTime":902.4,"endTime":904.59,"body":"And then if you've got"},{"startTime":902.4,"endTime":904.59,"body":"a whole batch of coins,"},{"startTime":904.59,"endTime":905.88,"body":"or you want to analyze, you know,"},{"startTime":905.88,"endTime":908.52,"body":"a foreign currency or something like this,"},{"startTime":908.52,"endTime":909.57,"body":"you can do an assay."},{"startTime":909.57,"endTime":913.62,"body":"So you can actually sort of chemically,"},{"startTime":913.62,"endTime":916.53,"body":"first you melt it, you"},{"startTime":913.62,"endTime":916.53,"body":"melt away the impurities,"},{"startTime":916.53,"endTime":918.99,"body":"then perhaps you're left with"},{"startTime":916.53,"endTime":918.99,"body":"a mix of gold and silver,"},{"startTime":918.99,"endTime":922.74,"body":"and then you can dissolve the"},{"startTime":918.99,"endTime":922.74,"body":"silver away with nitric acid."},{"startTime":922.74,"endTime":924.57,"body":"That leaves you with just the gold."},{"startTime":924.57,"endTime":925.89,"body":"So then you can weigh that"},{"startTime":925.89,"endTime":928.98,"body":"and then you can sort of"},{"startTime":925.89,"endTime":928.98,"body":"work back, reverse engineer,"},{"startTime":928.98,"endTime":930.9,"body":"this is how much silver"},{"startTime":928.98,"endTime":930.9,"body":"there must've been,"},{"startTime":930.9,"endTime":934.08,"body":"this is how much impurities"},{"startTime":930.9,"endTime":934.08,"body":"there must've been all along."},{"startTime":934.08,"endTime":936.993,"body":"So there are various"},{"startTime":934.08,"endTime":936.993,"body":"ways of analyzing coins,"},{"startTime":938.01,"endTime":940.98,"body":"to check whether they're"},{"startTime":938.01,"endTime":940.98,"body":"counterfeit or not,"},{"startTime":940.98,"endTime":942.873,"body":"or what their composition is."},{"startTime":944.94,"endTime":947.16,"body":"But, so that's sort of testing to see"},{"startTime":947.16,"endTime":949.2,"body":"if people are trying to cheat you."},{"startTime":949.2,"endTime":952.56,"body":"But kind of the other, well"},{"startTime":949.2,"endTime":952.56,"body":"the other side of the coin is,"},{"startTime":952.56,"endTime":956.25,"body":"if you are trying to genuinely make coins."},{"startTime":956.25,"endTime":958.83,"body":"If you are, you know, master of the mint,"},{"startTime":958.83,"endTime":961.23,"body":"and you have been instructed,"},{"startTime":961.23,"endTime":965.22,"body":"we want coins of this"},{"startTime":961.23,"endTime":965.22,"body":"particular grade of silver,"},{"startTime":965.22,"endTime":968.88,"body":"this particular level of"},{"startTime":965.22,"endTime":968.88,"body":"purity, how do you do that?"},{"startTime":968.88,"endTime":971.67,"body":"And that's what I want to turn to next."},{"startTime":971.67,"endTime":976.67,"body":"And we're going to meet"},{"startTime":971.67,"endTime":976.67,"body":"this chap, Leonardo of Pisa,"},{"startTime":976.74,"endTime":980.73,"body":"who wrote a very influential"},{"startTime":976.74,"endTime":980.73,"body":"book in the early 13th century,"},{"startTime":980.73,"endTime":982.608,"body":"which had many things in it."},{"startTime":982.608,"endTime":984.57,"body":"And one of the things it had in it"},{"startTime":984.57,"endTime":987.66,"body":"is a sequence of numbers that we know now,"},{"startTime":987.66,"endTime":989.64,"body":"not as the Leonardo of Pisa sequence,"},{"startTime":989.64,"endTime":992.88,"body":"but by this guy's nickname,"},{"startTime":989.64,"endTime":992.88,"body":"which is Fibonacci."},{"startTime":992.88,"endTime":995.73,"body":"And so the Fibonacci"},{"startTime":992.88,"endTime":995.73,"body":"Sequence, as it's called,"},{"startTime":995.73,"endTime":998.76,"body":"he didn't discover it, but"},{"startTime":995.73,"endTime":998.76,"body":"he kind of popularized it"},{"startTime":998.76,"endTime":999.593,"body":"in the West."},{"startTime":999.593,"endTime":1001.25,"body":"It'd been known in India"},{"startTime":999.593,"endTime":1001.25,"body":"and other countries"},{"startTime":1001.25,"endTime":1002.27,"body":"for hundreds of years."},{"startTime":1002.27,"endTime":1003.98,"body":"That's one of the things"},{"startTime":1002.27,"endTime":1003.98,"body":"that's in his book."},{"startTime":1003.98,"endTime":1006.753,"body":"But another thing that's in his book is"},{"startTime":1006.753,"endTime":1008.75,"body":"a really, really useful trick"},{"startTime":1008.75,"endTime":1013.75,"body":"for how to blend different"},{"startTime":1008.75,"endTime":1013.75,"body":"qualities of metal that you have,"},{"startTime":1015.41,"endTime":1018.47,"body":"to produce a desired quality."},{"startTime":1018.47,"endTime":1020.3,"body":"And this is a real genuine problem"},{"startTime":1020.3,"endTime":1025.3,"body":"that you would be faced"},{"startTime":1020.3,"endTime":1025.3,"body":"with, if your monarch says,"},{"startTime":1025.67,"endTime":1029.45,"body":"I decree that silver coins have to be"},{"startTime":1029.45,"endTime":1033.183,"body":"precisely this level of,"},{"startTime":1029.45,"endTime":1033.183,"body":"this amount of silver,"},{"startTime":1033.183,"endTime":1036.2,"body":"and this purity, and you have"},{"startTime":1033.183,"endTime":1036.2,"body":"different kinds of levels"},{"startTime":1036.2,"endTime":1039.68,"body":"of quality of your bullion,"},{"startTime":1039.68,"endTime":1041.33,"body":"you've got to work out"},{"startTime":1039.68,"endTime":1041.33,"body":"how to actually produce"},{"startTime":1041.33,"endTime":1042.89,"body":"the desire amount."},{"startTime":1042.89,"endTime":1044.9,"body":"This is a tricky mathematical challenge."},{"startTime":1044.9,"endTime":1048.2,"body":"So just an example of"},{"startTime":1044.9,"endTime":1048.2,"body":"this that Fibonacci gave."},{"startTime":1048.2,"endTime":1050.723,"body":"Imagine you've got two"},{"startTime":1048.2,"endTime":1050.723,"body":"kinds of silver bullion,"},{"startTime":1052.13,"endTime":1056.09,"body":"and one is too impure and one is too pure,"},{"startTime":1056.09,"endTime":1057.5,"body":"it's very good quality."},{"startTime":1057.5,"endTime":1061.94,"body":"So your bad bullion has"},{"startTime":1057.5,"endTime":1061.94,"body":"fineness, let's say, four."},{"startTime":1061.94,"endTime":1063.92,"body":"We don't need to worry"},{"startTime":1061.94,"endTime":1063.92,"body":"about what the units"},{"startTime":1063.92,"endTime":1065.21,"body":"the fineness is measured in."},{"startTime":1065.21,"endTime":1068.21,"body":"It might be ounces of"},{"startTime":1065.21,"endTime":1068.21,"body":"silver per pound of metal,"},{"startTime":1068.21,"endTime":1069.44,"body":"or something like that."},{"startTime":1069.44,"endTime":1072.56,"body":"But the fineness is four, by some measure."},{"startTime":1072.56,"endTime":1074.45,"body":"This is sort of the purity level."},{"startTime":1074.45,"endTime":1078.473,"body":"And then you've got some good"},{"startTime":1074.45,"endTime":1078.473,"body":"bullion with fineness nine."},{"startTime":1079.31,"endTime":1081.71,"body":"And your desired fineness,"},{"startTime":1081.71,"endTime":1083.54,"body":"the decree says that your coins must have"},{"startTime":1083.54,"endTime":1085.04,"body":"fineness of seven."},{"startTime":1085.04,"endTime":1087.02,"body":"So how do you mix these?"},{"startTime":1087.02,"endTime":1089.99,"body":"In what proportions should you"},{"startTime":1087.02,"endTime":1089.99,"body":"mix these different bullions"},{"startTime":1089.99,"endTime":1093.11,"body":"to get the desired number of seven?"},{"startTime":1093.11,"endTime":1096.26,"body":"And what Fibonacci says is that,"},{"startTime":1096.26,"endTime":1098.72,"body":"so this process is called"},{"startTime":1096.26,"endTime":1098.72,"body":"allegation, by the way,"},{"startTime":1098.72,"endTime":1100.43,"body":"this process of deciding"},{"startTime":1100.43,"endTime":1102.14,"body":"how we're going to mix"},{"startTime":1100.43,"endTime":1102.14,"body":"quantities together."},{"startTime":1102.14,"endTime":1104.99,"body":"So his allegation"},{"startTime":1102.14,"endTime":1104.99,"body":"solution is the following."},{"startTime":1104.99,"endTime":1109.99,"body":"He says, okay, so the bad"},{"startTime":1104.99,"endTime":1109.99,"body":"bullion, the impure one,"},{"startTime":1111.44,"endTime":1115.19,"body":"that has fineness of four,"},{"startTime":1111.44,"endTime":1115.19,"body":"that's three worse than you want."},{"startTime":1115.19,"endTime":1117.71,"body":"And the good bullion has fineness of nine,"},{"startTime":1117.71,"endTime":1119.9,"body":"and that's two better than you want."},{"startTime":1119.9,"endTime":1123.08,"body":"So you've got that"},{"startTime":1119.9,"endTime":1123.08,"body":"three to two proportion."},{"startTime":1123.08,"endTime":1125.84,"body":"And what Fibonacci says is"},{"startTime":1123.08,"endTime":1125.84,"body":"you just swap that over,"},{"startTime":1125.84,"endTime":1129.14,"body":"and mix them in the opposite ratio."},{"startTime":1129.14,"endTime":1132.2,"body":"So he says, therefore you mix in the ratio"},{"startTime":1132.2,"endTime":1134.78,"body":"two bad to three good."},{"startTime":1134.78,"endTime":1137.66,"body":"And if you sort of think"},{"startTime":1134.78,"endTime":1137.66,"body":"intuitively that makes sense."},{"startTime":1137.66,"endTime":1139.43,"body":"I'm not going to write"},{"startTime":1137.66,"endTime":1139.43,"body":"down a detailed proof,"},{"startTime":1139.43,"endTime":1140.96,"body":"although I have put one in the transcript"},{"startTime":1140.96,"endTime":1142.79,"body":"which you can pick up afterwards,"},{"startTime":1142.79,"endTime":1144.56,"body":"or download if you're"},{"startTime":1142.79,"endTime":1144.56,"body":"watching this online."},{"startTime":1144.56,"endTime":1147.11,"body":"I have sort of done the"},{"startTime":1144.56,"endTime":1147.11,"body":"working out in the transcript,"},{"startTime":1147.11,"endTime":1148.85,"body":"if you're interested to see that."},{"startTime":1148.85,"endTime":1151.73,"body":"But intuitively, we can see"},{"startTime":1148.85,"endTime":1151.73,"body":"that it kind of makes sense"},{"startTime":1151.73,"endTime":1156.26,"body":"to do this because the bad"},{"startTime":1151.73,"endTime":1156.26,"body":"one is kind of three bad."},{"startTime":1156.26,"endTime":1158.39,"body":"So three times two is six, right?"},{"startTime":1158.39,"endTime":1163.343,"body":"And the good one is three,"},{"startTime":1158.39,"endTime":1163.343,"body":"is two good plus two,"},{"startTime":1165.92,"endTime":1167.3,"body":"and you have three lots of that."},{"startTime":1167.3,"endTime":1170.0,"body":"So you've got a two"},{"startTime":1167.3,"endTime":1170.0,"body":"times three is six there."},{"startTime":1170.0,"endTime":1173.12,"body":"And those six plus six and"},{"startTime":1170.0,"endTime":1173.12,"body":"minus six kind of cancel out."},{"startTime":1173.12,"endTime":1175.19,"body":"So intuitively it makes sense that you,"},{"startTime":1175.19,"endTime":1178.46,"body":"this feels plausible,"},{"startTime":1175.19,"endTime":1178.46,"body":"and indeed it does work."},{"startTime":1178.46,"endTime":1181.37,"body":"So this is a very simple trick."},{"startTime":1181.37,"endTime":1183.8,"body":"A bit of algebra proves that it works."},{"startTime":1183.8,"endTime":1186.65,"body":"But what Fibonacci does so well,"},{"startTime":1186.65,"endTime":1188.72,"body":"and if you want to read about this,"},{"startTime":1188.72,"endTime":1192.56,"body":"Fibonacci and the history"},{"startTime":1188.72,"endTime":1192.56,"body":"of this whole development,"},{"startTime":1192.56,"endTime":1195.68,"body":"the development of this whole"},{"startTime":1192.56,"endTime":1195.68,"body":"kind of process or techniques"},{"startTime":1195.68,"endTime":1197.45,"body":"for solving this kind of problem,"},{"startTime":1197.45,"endTime":1199.55,"body":"you should check out a"},{"startTime":1197.45,"endTime":1199.55,"body":"paper that I've linked to"},{"startTime":1199.55,"endTime":1201.02,"body":"in the transcript by Norman Biggs,"},{"startTime":1201.02,"endTime":1203.423,"body":"who was a former Gresham"},{"startTime":1201.02,"endTime":1203.423,"body":"professor of geometry."},{"startTime":1204.41,"endTime":1205.79,"body":"You can generalize this idea."},{"startTime":1205.79,"endTime":1208.4,"body":"So you can have lots of"},{"startTime":1205.79,"endTime":1208.4,"body":"different kinds of bullion"},{"startTime":1208.4,"endTime":1209.39,"body":"that you can mix."},{"startTime":1209.39,"endTime":1211.31,"body":"And what you do is,"},{"startTime":1209.39,"endTime":1211.31,"body":"according to Fibonacci,"},{"startTime":1211.31,"endTime":1212.63,"body":"you can pair them off."},{"startTime":1212.63,"endTime":1214.34,"body":"You take a good one and a bad one."},{"startTime":1214.34,"endTime":1217.25,"body":"You work out this process"},{"startTime":1214.34,"endTime":1217.25,"body":"for all those pairs,"},{"startTime":1217.25,"endTime":1220.04,"body":"and then you can mix"},{"startTime":1217.25,"endTime":1220.04,"body":"them in any way you like."},{"startTime":1220.04,"endTime":1223.16,"body":"So you can combine the"},{"startTime":1220.04,"endTime":1223.16,"body":"solutions in any way you like."},{"startTime":1223.16,"endTime":1226.07,"body":"What's called a linear"},{"startTime":1223.16,"endTime":1226.07,"body":"combination of the solutions."},{"startTime":1226.07,"endTime":1229.31,"body":"So you can generalize it to"},{"startTime":1226.07,"endTime":1229.31,"body":"more than two kinds of bullion,"},{"startTime":1229.31,"endTime":1232.61,"body":"but you can also apply"},{"startTime":1229.31,"endTime":1232.61,"body":"this kind of reasoning"},{"startTime":1232.61,"endTime":1235.97,"body":"in other situations beyond coins."},{"startTime":1235.97,"endTime":1239.243,"body":"And that's another thing"},{"startTime":1235.97,"endTime":1239.243,"body":"Fibonacci's really great at."},{"startTime":1240.14,"endTime":1242.0,"body":"For example, you can apply it,"},{"startTime":1242.0,"endTime":1243.95,"body":"say you are a winemaker"},{"startTime":1243.95,"endTime":1246.56,"body":"and you've got lots of different wines"},{"startTime":1246.56,"endTime":1248.45,"body":"of different kind of unit costs,"},{"startTime":1248.45,"endTime":1251.66,"body":"and you want to blend to produce"},{"startTime":1248.45,"endTime":1251.66,"body":"a wine that you will charge"},{"startTime":1251.66,"endTime":1253.13,"body":"a certain amount of money for."},{"startTime":1253.13,"endTime":1254.93,"body":"How do you do that?"},{"startTime":1254.93,"endTime":1257.06,"body":"Or if you have a certain budget"},{"startTime":1257.06,"endTime":1259.46,"body":"and you want to buy"},{"startTime":1257.06,"endTime":1259.46,"body":"different amounts of things,"},{"startTime":1260.78,"endTime":1262.16,"body":"what are the possibilities for doing that?"},{"startTime":1262.16,"endTime":1263.87,"body":"If each thing has a different price,"},{"startTime":1263.87,"endTime":1266.06,"body":"how many of each thing should you buy?"},{"startTime":1266.06,"endTime":1268.46,"body":"And Fibonacci included, he had"},{"startTime":1266.06,"endTime":1268.46,"body":"lots of puzzles in his books."},{"startTime":1268.46,"endTime":1271.46,"body":"He liked a puzzle, he liked a"},{"startTime":1268.46,"endTime":1271.46,"body":"sequence, he liked a pattern."},{"startTime":1271.46,"endTime":1273.74,"body":"He's got this lovely puzzle in his book,"},{"startTime":1273.74,"endTime":1276.32,"body":"the puzzle of the birds, which, oh yeah,"},{"startTime":1276.32,"endTime":1277.82,"body":"we all like birds, right?"},{"startTime":1277.82,"endTime":1280.37,"body":"So this is a really cute puzzle."},{"startTime":1280.37,"endTime":1284.27,"body":"It seems to have nothing to"},{"startTime":1280.37,"endTime":1284.27,"body":"do with blending bullions"},{"startTime":1284.27,"endTime":1286.1,"body":"to make coins, right?"},{"startTime":1286.1,"endTime":1289.49,"body":"But you can apply exactly"},{"startTime":1286.1,"endTime":1289.49,"body":"the same kind of reasoning"},{"startTime":1289.49,"endTime":1290.323,"body":"to solve it."},{"startTime":1290.323,"endTime":1295.06,"body":"So the puzzle says, I bought"},{"startTime":1290.323,"endTime":1295.06,"body":"30 birds of three kinds,"},{"startTime":1296.39,"endTime":1301.22,"body":"so it's sparrows, pigeons, and"},{"startTime":1296.39,"endTime":1301.22,"body":"partridges, for 30 pennies."},{"startTime":1301.22,"endTime":1303.41,"body":"Well denari, of course,"},{"startTime":1301.22,"endTime":1303.41,"body":"it said in the original."},{"startTime":1303.41,"endTime":1304.94,"body":"But for 30 pennies."},{"startTime":1304.94,"endTime":1307.37,"body":"So you get the price of each one."},{"startTime":1307.37,"endTime":1309.02,"body":"Per sparrow, half a penny."},{"startTime":1309.02,"endTime":1311.258,"body":"Per pigeon, 2p, and so on."},{"startTime":1311.258,"endTime":1316.1,"body":"And the question is, what, how"},{"startTime":1311.258,"endTime":1316.1,"body":"many of each bird did I get?"},{"startTime":1316.1,"endTime":1320.603,"body":"So 30 birds for 30 pence, how"},{"startTime":1316.1,"endTime":1320.603,"body":"many of each bird did I get?"},{"startTime":1321.491,"endTime":1323.45,"body":"And the answer to this,"},{"startTime":1323.45,"endTime":1325.52,"body":"the solution to this that Fibonacci gave,"},{"startTime":1325.52,"endTime":1329.54,"body":"is he converted it to"},{"startTime":1325.52,"endTime":1329.54,"body":"an allegation question."},{"startTime":1329.54,"endTime":1331.58,"body":"So what's the, you"},{"startTime":1329.54,"endTime":1331.58,"body":"know, what's my bullion,"},{"startTime":1331.58,"endTime":1333.23,"body":"what's my fineness here?"},{"startTime":1333.23,"endTime":1337.25,"body":"Well if we think, the final"},{"startTime":1333.23,"endTime":1337.25,"body":"outcome is that I've got"},{"startTime":1337.25,"endTime":1338.99,"body":"30 birds for 30p."},{"startTime":1338.99,"endTime":1341.66,"body":"So the sort of price per bird,"},{"startTime":1341.66,"endTime":1344.783,"body":"the fineness is one penny per bird."},{"startTime":1345.65,"endTime":1349.01,"body":"So what we're aiming"},{"startTime":1345.65,"endTime":1349.01,"body":"at is to produce that."},{"startTime":1349.01,"endTime":1352.1,"body":"And what we do is to"},{"startTime":1349.01,"endTime":1352.1,"body":"pair them off, as I said,"},{"startTime":1352.1,"endTime":1355.34,"body":"so you take, you have to"},{"startTime":1352.1,"endTime":1355.34,"body":"pair something that's worse,"},{"startTime":1355.34,"endTime":1358.94,"body":"worse fineness, so cheaper"},{"startTime":1355.34,"endTime":1358.94,"body":"than one penny per bird,"},{"startTime":1358.94,"endTime":1361.19,"body":"which in this case is just the sparrows,"},{"startTime":1361.19,"endTime":1364.49,"body":"with something that's better,"},{"startTime":1361.19,"endTime":1364.49,"body":"more than one penny per bird."},{"startTime":1364.49,"endTime":1367.07,"body":"And then you work out the"},{"startTime":1364.49,"endTime":1367.07,"body":"proportions that you must have"},{"startTime":1367.07,"endTime":1368.84,"body":"just for that pair."},{"startTime":1368.84,"endTime":1372.08,"body":"And then you combine them as you like."},{"startTime":1372.08,"endTime":1374.84,"body":"So, for example, right, sparrows,"},{"startTime":1374.84,"endTime":1377.87,"body":"we want to end up with a fineness of one,"},{"startTime":1377.87,"endTime":1379.34,"body":"one penny per bird."},{"startTime":1379.34,"endTime":1383.06,"body":"Sparrows are worse than that,"},{"startTime":1379.34,"endTime":1383.06,"body":"they're half worse than that."},{"startTime":1383.06,"endTime":1386.36,"body":"And pigeons are one better"},{"startTime":1383.06,"endTime":1386.36,"body":"than that cause they cost 2p,"},{"startTime":1386.36,"endTime":1388.61,"body":"that's 1p above what we're aiming at."},{"startTime":1388.61,"endTime":1391.64,"body":"So sparrows are kind of badness of half,"},{"startTime":1391.64,"endTime":1394.52,"body":"pigeons are goodness one."},{"startTime":1394.52,"endTime":1396.56,"body":"So that ratio, a half to one."},{"startTime":1396.56,"endTime":1397.393,"body":"What's the rule?"},{"startTime":1397.393,"endTime":1398.3,"body":"We flip it."},{"startTime":1398.3,"endTime":1403.067,"body":"So for every sparrow,"},{"startTime":1398.3,"endTime":1403.067,"body":"we buy half a pigeon."},{"startTime":1403.067,"endTime":1405.2,"body":"And you can't buy half a pigeon,"},{"startTime":1405.2,"endTime":1406.73,"body":"wouldn't be very nice to the pigeon,"},{"startTime":1406.73,"endTime":1407.78,"body":"so let's just double that up."},{"startTime":1407.78,"endTime":1410.15,"body":"For every two sparrows,"},{"startTime":1407.78,"endTime":1410.15,"body":"you buy one pigeon."},{"startTime":1410.15,"endTime":1412.49,"body":"So that's a total of three birds."},{"startTime":1412.49,"endTime":1415.13,"body":"Two sparrows cost you total of a penny."},{"startTime":1415.13,"endTime":1417.02,"body":"The pigeon costs you 2p."},{"startTime":1417.02,"endTime":1420.11,"body":"So you've spent 3p,"},{"startTime":1417.02,"endTime":1420.11,"body":"you've got three birds."},{"startTime":1420.11,"endTime":1424.1,"body":"So just that pairing, two"},{"startTime":1420.11,"endTime":1424.1,"body":"to one sparrows to pigeons,"},{"startTime":1424.1,"endTime":1427.43,"body":"that will give you a fineness of one."},{"startTime":1427.43,"endTime":1431.3,"body":"So you can have as many"},{"startTime":1427.43,"endTime":1431.3,"body":"of those as you like"},{"startTime":1431.3,"endTime":1434.81,"body":"in your final solution, and"},{"startTime":1431.3,"endTime":1434.81,"body":"you'll get a fineness of one."},{"startTime":1434.81,"endTime":1437.6,"body":"You can do the same thing with"},{"startTime":1434.81,"endTime":1437.6,"body":"the sparrows and partridges,"},{"startTime":1437.6,"endTime":1441.02,"body":"sparrows have badness half,"},{"startTime":1441.02,"endTime":1444.11,"body":"partridges have goodness"},{"startTime":1441.02,"endTime":1444.11,"body":"two more than you want."},{"startTime":1444.11,"endTime":1446.99,"body":"So you have that two to a half ratio."},{"startTime":1446.99,"endTime":1449.78,"body":"So you should buy two sparrows"},{"startTime":1446.99,"endTime":1449.78,"body":"for every half a partridge,"},{"startTime":1449.78,"endTime":1452.36,"body":"or four sparrows for every partridge."},{"startTime":1452.36,"endTime":1455.18,"body":"And that gives you five birds there,"},{"startTime":1455.18,"endTime":1457.04,"body":"four sparrows and a partridge."},{"startTime":1457.04,"endTime":1459.47,"body":"And you can check that"},{"startTime":1457.04,"endTime":1459.47,"body":"it would cost you 5p."},{"startTime":1459.47,"endTime":1463.64,"body":"So then, any combination"},{"startTime":1459.47,"endTime":1463.64,"body":"of these threes and fives"},{"startTime":1463.64,"endTime":1466.88,"body":"is going to give you a price of"},{"startTime":1466.88,"endTime":1469.22,"body":"one penny per bird on average."},{"startTime":1469.22,"endTime":1473.06,"body":"So the only remaining bit is to work out"},{"startTime":1473.06,"endTime":1475.55,"body":"the ways that you can combine"},{"startTime":1473.06,"endTime":1475.55,"body":"these threes and fives"},{"startTime":1475.55,"endTime":1477.56,"body":"to come up with a total of 30."},{"startTime":1477.56,"endTime":1478.58,"body":"And if you think about it for a bit,"},{"startTime":1478.58,"endTime":1481.102,"body":"you can see there's actually"},{"startTime":1478.58,"endTime":1481.102,"body":"only one way to do it"},{"startTime":1481.102,"endTime":1482.952,"body":"that really involves all three birds."},{"startTime":1484.01,"endTime":1487.49,"body":"And that is, if you have five lots of the"},{"startTime":1487.49,"endTime":1490.82,"body":"sparrow pigeon combo, so five pigeons,"},{"startTime":1490.82,"endTime":1493.49,"body":"and three lots of the"},{"startTime":1490.82,"endTime":1493.49,"body":"sparrow partridge combo,"},{"startTime":1493.49,"endTime":1496.4,"body":"so five partridges, and then"},{"startTime":1493.49,"endTime":1496.4,"body":"you just work out, you know,"},{"startTime":1496.4,"endTime":1498.59,"body":"from that how many sparrows are required."},{"startTime":1498.59,"endTime":1502.79,"body":"And if you check, 22"},{"startTime":1498.59,"endTime":1502.79,"body":"sparrows, that costs you 11p,"},{"startTime":1502.79,"endTime":1505.61,"body":"five pigeons will cost you 10p,"},{"startTime":1505.61,"endTime":1507.74,"body":"three partridges cost you 9p."},{"startTime":1507.74,"endTime":1509.36,"body":"If you add that up you get 30p."},{"startTime":1509.36,"endTime":1511.28,"body":"So it's a nice little solution,"},{"startTime":1511.28,"endTime":1512.51,"body":"and it's one of those examples"},{"startTime":1512.51,"endTime":1514.67,"body":"where you've got a mathematical technique,"},{"startTime":1514.67,"endTime":1518.09,"body":"that if you have a kind of"},{"startTime":1514.67,"endTime":1518.09,"body":"a little bit of inspiration,"},{"startTime":1518.09,"endTime":1519.47,"body":"you can apply in what looks like"},{"startTime":1519.47,"endTime":1521.39,"body":"a completely different area."},{"startTime":1521.39,"endTime":1523.73,"body":"And this idea, this allegation thing,"},{"startTime":1523.73,"endTime":1527.3,"body":"and the techniques for"},{"startTime":1523.73,"endTime":1527.3,"body":"solving problems like this,"},{"startTime":1527.3,"endTime":1529.91,"body":"have come to be a mathematical subject"},{"startTime":1529.91,"endTime":1530.87,"body":"really in their own right."},{"startTime":1530.87,"endTime":1532.62,"body":"It's now called linear programming."},{"startTime":1533.54,"endTime":1536.6,"body":"And so this is, again, it's"},{"startTime":1533.54,"endTime":1536.6,"body":"come from working with money"},{"startTime":1536.6,"endTime":1537.92,"body":"and how to produce correct amounts,"},{"startTime":1537.92,"endTime":1542.39,"body":"but it has lots of applications"},{"startTime":1537.92,"endTime":1542.39,"body":"beyond bird puzzles"},{"startTime":1542.39,"endTime":1543.56,"body":"to real life."},{"startTime":1543.56,"endTime":1546.083,"body":"And so it's really kind"},{"startTime":1543.56,"endTime":1546.083,"body":"of interesting there."},{"startTime":1546.98,"endTime":1550.73,"body":"So we now know, we're master of the mint,"},{"startTime":1550.73,"endTime":1555.65,"body":"we know how to make the"},{"startTime":1550.73,"endTime":1555.65,"body":"desired alloys for our coins."},{"startTime":1555.65,"endTime":1557.33,"body":"We know what to make our coins of"},{"startTime":1557.33,"endTime":1560.06,"body":"and how to produce the desired purity."},{"startTime":1560.06,"endTime":1563.81,"body":"So the next question for us is"},{"startTime":1563.81,"endTime":1565.613,"body":"what shape should our coins be?"},{"startTime":1567.26,"endTime":1572.26,"body":"Most coins are round, circular."},{"startTime":1572.27,"endTime":1574.55,"body":"Not all but most."},{"startTime":1574.55,"endTime":1577.37,"body":"Now one reason for this"},{"startTime":1574.55,"endTime":1577.37,"body":"is kind of, you know,"},{"startTime":1577.37,"endTime":1581.87,"body":"the organic development of using"},{"startTime":1577.37,"endTime":1581.87,"body":"bits of metal for currency."},{"startTime":1581.87,"endTime":1584.9,"body":"Originally you might have just"},{"startTime":1581.87,"endTime":1584.9,"body":"like a little lump of metal,"},{"startTime":1584.9,"endTime":1587.57,"body":"and like a pebble, it might"},{"startTime":1584.9,"endTime":1587.57,"body":"get smoothed down or whatever."},{"startTime":1587.57,"endTime":1591.32,"body":"But then at some point"},{"startTime":1587.57,"endTime":1591.32,"body":"people started stamping them"},{"startTime":1591.32,"endTime":1594.23,"body":"with pictures of emperors"},{"startTime":1591.32,"endTime":1594.23,"body":"or something like this,"},{"startTime":1594.23,"endTime":1596.84,"body":"or amounts or writing."},{"startTime":1596.84,"endTime":1598.07,"body":"So then if you take a sort of"},{"startTime":1598.07,"endTime":1601.34,"body":"vaguely round ball of"},{"startTime":1598.07,"endTime":1601.34,"body":"something and you squash it,"},{"startTime":1601.34,"endTime":1603.53,"body":"by impressing it with some kind of image,"},{"startTime":1603.53,"endTime":1606.743,"body":"then it will become a flat"},{"startTime":1603.53,"endTime":1606.743,"body":"disc, and essentially a circle."},{"startTime":1607.64,"endTime":1609.32,"body":"Some coins have holes in them."},{"startTime":1609.32,"endTime":1611.15,"body":"It kind of makes it easier"},{"startTime":1609.32,"endTime":1611.15,"body":"for threading them onto"},{"startTime":1611.15,"endTime":1613.673,"body":"strings or necklaces for"},{"startTime":1611.15,"endTime":1613.673,"body":"carrying them around."},{"startTime":1614.69,"endTime":1617.57,"body":"Not all coins are circular,"},{"startTime":1614.69,"endTime":1617.57,"body":"but quite a lot are."},{"startTime":1617.57,"endTime":1619.76,"body":"There's a more modern"},{"startTime":1617.57,"endTime":1619.76,"body":"reason why the circle is"},{"startTime":1619.76,"endTime":1622.91,"body":"a good solution for the shape of a coin."},{"startTime":1622.91,"endTime":1624.92,"body":"A good solution, not the only one."},{"startTime":1624.92,"endTime":1626.66,"body":"And that's vending machines."},{"startTime":1626.66,"endTime":1629.12,"body":"Because when you, often"},{"startTime":1626.66,"endTime":1629.12,"body":"vending machines will have,"},{"startTime":1629.12,"endTime":1632.24,"body":"they'll have a single slot"},{"startTime":1629.12,"endTime":1632.24,"body":"where you put your money in,"},{"startTime":1632.24,"endTime":1634.82,"body":"and then the machine has to somehow decide"},{"startTime":1634.82,"endTime":1637.49,"body":"what's the value of that"},{"startTime":1634.82,"endTime":1637.49,"body":"coin, what coin is it,"},{"startTime":1637.49,"endTime":1639.8,"body":"so that it can then give"},{"startTime":1637.49,"endTime":1639.8,"body":"you change or whatever,"},{"startTime":1639.8,"endTime":1641.72,"body":"and know if you've paid enough."},{"startTime":1641.72,"endTime":1642.89,"body":"So how does it do that?"},{"startTime":1642.89,"endTime":1647.89,"body":"Well, the most often, the most"},{"startTime":1642.89,"endTime":1647.89,"body":"commonly usual way to tell"},{"startTime":1648.38,"endTime":1652.28,"body":"what value of coin you've put"},{"startTime":1648.38,"endTime":1652.28,"body":"in, is to measure its width."},{"startTime":1652.28,"endTime":1656.3,"body":"So if you've got a coin that"},{"startTime":1652.28,"endTime":1656.3,"body":"has very varying widths,"},{"startTime":1656.3,"endTime":1657.26,"body":"depending on, you know,"},{"startTime":1657.26,"endTime":1659.27,"body":"where it is as you're moving it along,"},{"startTime":1659.27,"endTime":1662.358,"body":"like if you had a long thin"},{"startTime":1659.27,"endTime":1662.358,"body":"rectangle or something,"},{"startTime":1662.358,"endTime":1664.94,"body":"if you have a coin whose"},{"startTime":1662.358,"endTime":1664.94,"body":"width keeps varying,"},{"startTime":1664.94,"endTime":1666.86,"body":"that can make it very"},{"startTime":1664.94,"endTime":1666.86,"body":"difficult for a machine"},{"startTime":1666.86,"endTime":1669.92,"body":"to work out actually,"},{"startTime":1666.86,"endTime":1669.92,"body":"what is the coin I've got?"},{"startTime":1669.92,"endTime":1672.47,"body":"And of course, circles,"},{"startTime":1669.92,"endTime":1672.47,"body":"you know, archetypically,"},{"startTime":1672.47,"endTime":1674.81,"body":"they are the things that"},{"startTime":1672.47,"endTime":1674.81,"body":"have a constant width."},{"startTime":1674.81,"endTime":1675.92,"body":"The width of a circle,"},{"startTime":1675.92,"endTime":1679.55,"body":"if you just kind of put"},{"startTime":1675.92,"endTime":1679.55,"body":"it between two lines,"},{"startTime":1679.55,"endTime":1680.87,"body":"it's going to be the diameter,"},{"startTime":1680.87,"endTime":1682.88,"body":"and the diameter of the"},{"startTime":1680.87,"endTime":1682.88,"body":"circle is always the same."},{"startTime":1682.88,"endTime":1685.76,"body":"Wherever you measure it,"},{"startTime":1682.88,"endTime":1685.76,"body":"it's twice the radius."},{"startTime":1685.76,"endTime":1689.0,"body":"So circles have this property"},{"startTime":1685.76,"endTime":1689.0,"body":"of having a constant width,"},{"startTime":1689.0,"endTime":1691.43,"body":"but they are not the only"},{"startTime":1689.0,"endTime":1691.43,"body":"shape with that property."},{"startTime":1691.43,"endTime":1693.74,"body":"I'm going to show you another example."},{"startTime":1693.74,"endTime":1694.7,"body":"Not this."},{"startTime":1694.7,"endTime":1697.58,"body":"Equilateral triangles very much"},{"startTime":1694.7,"endTime":1697.58,"body":"do not have constant width."},{"startTime":1697.58,"endTime":1700.4,"body":"But we can modify this triangle."},{"startTime":1700.4,"endTime":1703.55,"body":"So what I'm going to do is"},{"startTime":1700.4,"endTime":1703.55,"body":"I'm going to replace each edge"},{"startTime":1703.55,"endTime":1706.61,"body":"by the arc of a circle, like that."},{"startTime":1706.61,"endTime":1710.48,"body":"So this circle, it has"},{"startTime":1706.61,"endTime":1710.48,"body":"center, the top vertex there,"},{"startTime":1710.48,"endTime":1713.087,"body":"and it has radius, whatever"},{"startTime":1710.48,"endTime":1713.087,"body":"the side of the triangle is."},{"startTime":1713.087,"endTime":1715.46,"body":"And so you just make an"},{"startTime":1713.087,"endTime":1715.46,"body":"arc of a circle there,"},{"startTime":1715.46,"endTime":1719.36,"body":"and you do that for all of the edges."},{"startTime":1719.36,"endTime":1724.13,"body":"So you get this sort of"},{"startTime":1719.36,"endTime":1724.13,"body":"slightly inflated triangle."},{"startTime":1724.13,"endTime":1727.25,"body":"And this triangle, well let's"},{"startTime":1724.13,"endTime":1727.25,"body":"have a look at it, this,"},{"startTime":1727.25,"endTime":1728.96,"body":"whatever it's called."},{"startTime":1728.96,"endTime":1733.1,"body":"Let's have a look at what"},{"startTime":1728.96,"endTime":1733.1,"body":"the width of this thing is,"},{"startTime":1733.1,"endTime":1733.933,"body":"when we measure it."},{"startTime":1733.933,"endTime":1737.36,"body":"So we're going to roll it along"},{"startTime":1733.933,"endTime":1737.36,"body":"the bottom blue line there"},{"startTime":1737.36,"endTime":1738.71,"body":"and see what happens."},{"startTime":1738.71,"endTime":1741.98,"body":"And I'm saying that we'll"},{"startTime":1738.71,"endTime":1741.98,"body":"call the side length s,"},{"startTime":1741.98,"endTime":1745.49,"body":"and then the edges of the"},{"startTime":1741.98,"endTime":1745.49,"body":"triangle were replaced by"},{"startTime":1745.49,"endTime":1748.37,"body":"these new edges, which"},{"startTime":1745.49,"endTime":1748.37,"body":"are arcs of circles."},{"startTime":1748.37,"endTime":1751.55,"body":"And the radius of these circles is also s."},{"startTime":1751.55,"endTime":1754.1,"body":"So to begin with, the top blue line"},{"startTime":1754.1,"endTime":1755.3,"body":"and the bottom blue line,"},{"startTime":1755.3,"endTime":1757.01,"body":"they just have an edge of"},{"startTime":1755.3,"endTime":1757.01,"body":"the triangle between them."},{"startTime":1757.01,"endTime":1759.62,"body":"So the distance between them is s."},{"startTime":1759.62,"endTime":1763.058,"body":"Then we set the thing rolling,"},{"startTime":1759.62,"endTime":1763.058,"body":"and it starts rolling along,"},{"startTime":1763.058,"endTime":1766.07,"body":"and at the top you've still"},{"startTime":1763.058,"endTime":1766.07,"body":"got the vertex of the triangle."},{"startTime":1766.07,"endTime":1769.25,"body":"And at the bottom you're rolling"},{"startTime":1766.07,"endTime":1769.25,"body":"along this arc of a circle."},{"startTime":1769.25,"endTime":1771.35,"body":"So at all points along that arc,"},{"startTime":1771.35,"endTime":1773.6,"body":"the distance from top to"},{"startTime":1771.35,"endTime":1773.6,"body":"bottom is still going to be s."},{"startTime":1773.6,"endTime":1775.55,"body":"That's the radius of the circle."},{"startTime":1775.55,"endTime":1777.59,"body":"So then we reach this point again."},{"startTime":1777.59,"endTime":1780.23,"body":"So the end of that edge,"},{"startTime":1777.59,"endTime":1780.23,"body":"the end of that arc,"},{"startTime":1780.23,"endTime":1782.36,"body":"and again, we've got a vertex at the top,"},{"startTime":1782.36,"endTime":1785.63,"body":"vertex at the bottom, distance"},{"startTime":1782.36,"endTime":1785.63,"body":"between them is still s."},{"startTime":1785.63,"endTime":1787.52,"body":"And we keep going, and now, you know,"},{"startTime":1787.52,"endTime":1790.07,"body":"now we've got an arc at the"},{"startTime":1787.52,"endTime":1790.07,"body":"top that's rolling along."},{"startTime":1790.07,"endTime":1791.63,"body":"Again, the distance is s."},{"startTime":1791.63,"endTime":1794.63,"body":"And you can keep doing this,"},{"startTime":1791.63,"endTime":1794.63,"body":"and you find it's always,"},{"startTime":1794.63,"endTime":1797.3,"body":"it's always going to have this width s."},{"startTime":1797.3,"endTime":1799.193,"body":"This is a shape of constant width."},{"startTime":1800.21,"endTime":1802.43,"body":"And so you could roll this along,"},{"startTime":1802.43,"endTime":1804.56,"body":"and it really does fit nicely between"},{"startTime":1804.56,"endTime":1806.0,"body":"these two parallel lines"},{"startTime":1806.0,"endTime":1807.59,"body":"that do not have to move up and down."},{"startTime":1807.59,"endTime":1810.29,"body":"So this could potentially"},{"startTime":1807.59,"endTime":1810.29,"body":"be a shape for a coin maybe."},{"startTime":1811.55,"endTime":1815.54,"body":"Shapes like this, where you take a polygon"},{"startTime":1815.54,"endTime":1818.9,"body":"and you replace edges"},{"startTime":1815.54,"endTime":1818.9,"body":"with arcs of circles,"},{"startTime":1818.9,"endTime":1822.65,"body":"are called Reuleaux"},{"startTime":1818.9,"endTime":1822.65,"body":"polygons after this guy,"},{"startTime":1822.65,"endTime":1824.63,"body":"Franz Reuleaux, a German"},{"startTime":1822.65,"endTime":1824.63,"body":"mechanical engineer,"},{"startTime":1824.63,"endTime":1826.85,"body":"who investigated these shapes."},{"startTime":1826.85,"endTime":1828.98,"body":"This trick works as"},{"startTime":1826.85,"endTime":1828.98,"body":"long as your polygon has"},{"startTime":1828.98,"endTime":1830.96,"body":"an odd number of sides."},{"startTime":1830.96,"endTime":1833.51,"body":"And the reason is, if you"},{"startTime":1830.96,"endTime":1833.51,"body":"had an even number of sides,"},{"startTime":1833.51,"endTime":1836.365,"body":"like a square, you don't,"},{"startTime":1836.365,"endTime":1839.48,"body":"you always have vertices"},{"startTime":1836.365,"endTime":1839.48,"body":"opposite vertices in a square,"},{"startTime":1839.48,"endTime":1840.313,"body":"or a hexagon, or something like that."},{"startTime":1840.313,"endTime":1841.88,"body":"If you have an even number of sides"},{"startTime":1841.88,"endTime":1843.83,"body":"you have vertices, opposite vertices,"},{"startTime":1843.83,"endTime":1845.84,"body":"and that distance is always going to be"},{"startTime":1845.84,"endTime":1847.94,"body":"greater than anything you could achieve"},{"startTime":1847.94,"endTime":1849.65,"body":"by curving the sides a little bit."},{"startTime":1849.65,"endTime":1853.16,"body":"So this works for polygons"},{"startTime":1849.65,"endTime":1853.16,"body":"with an odd number of sides,"},{"startTime":1853.16,"endTime":1855.35,"body":"these Reuleaux polygons can be produced,"},{"startTime":1855.35,"endTime":1859.64,"body":"and you may well find you"},{"startTime":1855.35,"endTime":1859.64,"body":"have one in your pocket,"},{"startTime":1859.64,"endTime":1864.64,"body":"because lots of currencies have"},{"startTime":1859.64,"endTime":1864.64,"body":"Reuleaux polygons as coins."},{"startTime":1865.16,"endTime":1868.79,"body":"In Britain we have the"},{"startTime":1865.16,"endTime":1868.79,"body":"50p and the 20p coin."},{"startTime":1868.79,"endTime":1871.04,"body":"They are seven-sided Reuleaux polygons."},{"startTime":1871.04,"endTime":1872.483,"body":"Shapes of constant width."},{"startTime":1873.62,"endTime":1877.16,"body":"So you see these little"},{"startTime":1873.62,"endTime":1877.16,"body":"curved edges, not just,"},{"startTime":1877.16,"endTime":1879.26,"body":"you know, flat heptagons."},{"startTime":1879.26,"endTime":1882.287,"body":"They have these curved edges,"},{"startTime":1879.26,"endTime":1882.287,"body":"they're Reuleaux heptagons."},{"startTime":1883.64,"endTime":1888.64,"body":"Canada has got an 11-sided"},{"startTime":1883.64,"endTime":1888.64,"body":"Reuleaux polygon for its,"},{"startTime":1889.01,"endTime":1892.52,"body":"they call it the loonie, 'cause"},{"startTime":1889.01,"endTime":1892.52,"body":"of the bird that's shown."},{"startTime":1892.52,"endTime":1895.55,"body":"The Canadian dollar is an"},{"startTime":1892.52,"endTime":1895.55,"body":"11-sided Reuleaux polygon,"},{"startTime":1895.55,"endTime":1897.95,"body":"shape of constant width."},{"startTime":1897.95,"endTime":1901.46,"body":"Perhaps my favorite one though"},{"startTime":1897.95,"endTime":1901.46,"body":"is the Bermuda Triangle."},{"startTime":1901.46,"endTime":1902.66,"body":"Look at that."},{"startTime":1902.66,"endTime":1905.42,"body":"What a great, great name, great polygon."},{"startTime":1905.42,"endTime":1908.78,"body":"I mean it's just, they have"},{"startTime":1905.42,"endTime":1908.78,"body":"won the competition there."},{"startTime":1908.78,"endTime":1911.9,"body":"Fantastic, fantastic choice."},{"startTime":1911.9,"endTime":1914.6,"body":"So that you do get these things as coins."},{"startTime":1914.6,"endTime":1917.96,"body":"You also see Reuleaux triangles"},{"startTime":1914.6,"endTime":1917.96,"body":"around and about the place."},{"startTime":1917.96,"endTime":1920.0,"body":"So if you don't have one in your pocket,"},{"startTime":1920.0,"endTime":1923.24,"body":"you may have one at home,"},{"startTime":1920.0,"endTime":1923.24,"body":"because a lot of guitar picks"},{"startTime":1923.24,"endTime":1928.24,"body":"or plectrums, plectra, those"},{"startTime":1923.24,"endTime":1928.24,"body":"are often Reuleaux triangles."},{"startTime":1929.6,"endTime":1931.04,"body":"Not always, but often."},{"startTime":1931.04,"endTime":1933.56,"body":"And the reason, I think, I mean you get,"},{"startTime":1933.56,"endTime":1934.73,"body":"you've got a mixture there."},{"startTime":1934.73,"endTime":1937.4,"body":"You've got your pointed,"},{"startTime":1934.73,"endTime":1937.4,"body":"you've got three points"},{"startTime":1937.4,"endTime":1940.1,"body":"and they give you sort"},{"startTime":1937.4,"endTime":1940.1,"body":"of a sharp, bright sound,"},{"startTime":1940.1,"endTime":1944.27,"body":"or you could use the"},{"startTime":1940.1,"endTime":1944.27,"body":"more gentle curved parts"},{"startTime":1944.27,"endTime":1946.73,"body":"if you want a softer, warmer sound,"},{"startTime":1946.73,"endTime":1948.5,"body":"and you could sort of move"},{"startTime":1946.73,"endTime":1948.5,"body":"it around in your hand"},{"startTime":1948.5,"endTime":1951.14,"body":"and it has constant"},{"startTime":1948.5,"endTime":1951.14,"body":"width, so it's nice to do."},{"startTime":1951.14,"endTime":1953.36,"body":"So that's one use."},{"startTime":1953.36,"endTime":1955.61,"body":"Perhaps, how am I going to"},{"startTime":1953.36,"endTime":1955.61,"body":"say it's more important?"},{"startTime":1955.61,"endTime":1958.373,"body":"Well, it's a different"},{"startTime":1955.61,"endTime":1958.373,"body":"use, and a really nice one,"},{"startTime":1958.373,"endTime":1960.17,"body":"'cause it involves triangles"},{"startTime":1960.17,"endTime":1962.54,"body":"which have something"},{"startTime":1960.17,"endTime":1962.54,"body":"in common with circles,"},{"startTime":1962.54,"endTime":1965.51,"body":"but they're being used to make squares."},{"startTime":1965.51,"endTime":1966.92,"body":"Oh, just, this is amazing."},{"startTime":1966.92,"endTime":1969.443,"body":"So this thing is a drill bit,"},{"startTime":1970.4,"endTime":1972.59,"body":"and it can drill a square hole."},{"startTime":1972.59,"endTime":1977.59,"body":"You can rotate a Reuleaux"},{"startTime":1972.59,"endTime":1977.59,"body":"triangle around a little circle,"},{"startTime":1978.02,"endTime":1981.65,"body":"and it will trace out"},{"startTime":1978.02,"endTime":1981.65,"body":"almost a whole square."},{"startTime":1981.65,"endTime":1984.68,"body":"You do lose a tiny little"},{"startTime":1981.65,"endTime":1984.68,"body":"bit at the corners,"},{"startTime":1984.68,"endTime":1986.81,"body":"you get slightly curved corners."},{"startTime":1986.81,"endTime":1989.66,"body":"But this is a really, really"},{"startTime":1986.81,"endTime":1989.66,"body":"good approximation to a square."},{"startTime":1989.66,"endTime":1990.86,"body":"This isn't just theoretical."},{"startTime":1990.86,"endTime":1994.88,"body":"This thing was patented"},{"startTime":1990.86,"endTime":1994.88,"body":"by Harry Watts in 1916."},{"startTime":1994.88,"endTime":1997.58,"body":"The Harry Watts drill bit,"},{"startTime":1994.88,"endTime":1997.58,"body":"still available, still on sale."},{"startTime":1997.58,"endTime":2001.36,"body":"And it can produce very nearly"},{"startTime":2001.36,"endTime":2004.72,"body":"over 98% of the area of"},{"startTime":2001.36,"endTime":2004.72,"body":"the square is covered."},{"startTime":2004.72,"endTime":2006.76,"body":"So that's a, I just love"},{"startTime":2004.72,"endTime":2006.76,"body":"these kind of things"},{"startTime":2006.76,"endTime":2010.39,"body":"that link triangles, circles,"},{"startTime":2006.76,"endTime":2010.39,"body":"squares, in unexpected ways."},{"startTime":2010.39,"endTime":2011.223,"body":"So that's another place"},{"startTime":2011.223,"endTime":2013.78,"body":"you might have seen Reuleaux triangles."},{"startTime":2013.78,"endTime":2015.313,"body":"Why use them for coins?"},{"startTime":2017.05,"endTime":2020.5,"body":"Well, answer one, which"},{"startTime":2017.05,"endTime":2020.5,"body":"is the correct answer,"},{"startTime":2020.5,"endTime":2024.4,"body":"is they're super cool and"},{"startTime":2020.5,"endTime":2024.4,"body":"we love them, and why not?"},{"startTime":2024.4,"endTime":2025.57,"body":"But there may be another,"},{"startTime":2025.57,"endTime":2027.4,"body":"if you really want a practical answer,"},{"startTime":2027.4,"endTime":2029.953,"body":"there may be something about areas."},{"startTime":2030.88,"endTime":2033.88,"body":"If we think about shapes"},{"startTime":2030.88,"endTime":2033.88,"body":"of constant width."},{"startTime":2033.88,"endTime":2035.47,"body":"So circles are one example,"},{"startTime":2035.47,"endTime":2037.03,"body":"and these Reuleaux polygons are another."},{"startTime":2037.03,"endTime":2040.99,"body":"There are others, but we won't"},{"startTime":2037.03,"endTime":2040.99,"body":"go into detail about that."},{"startTime":2040.99,"endTime":2043.99,"body":"But there's a theorem called"},{"startTime":2040.99,"endTime":2043.99,"body":"Barbier's theorem that says"},{"startTime":2043.99,"endTime":2047.35,"body":"if you have a shape of constant"},{"startTime":2043.99,"endTime":2047.35,"body":"width and that width is s,"},{"startTime":2047.35,"endTime":2050.98,"body":"then its perimeter is always pi times s,"},{"startTime":2050.98,"endTime":2052.63,"body":"whatever the shape is."},{"startTime":2052.63,"endTime":2054.43,"body":"So we can see that's"},{"startTime":2052.63,"endTime":2054.43,"body":"true for circles, right?"},{"startTime":2054.43,"endTime":2057.43,"body":"Because a circle whose width"},{"startTime":2054.43,"endTime":2057.43,"body":"is s, that's its diameter."},{"startTime":2057.43,"endTime":2060.43,"body":"We know the circumference of"},{"startTime":2057.43,"endTime":2060.43,"body":"a circle whose diameter is d,"},{"startTime":2060.43,"endTime":2061.57,"body":"is pi times d."},{"startTime":2061.57,"endTime":2063.73,"body":"So if it's s then it'd be pi times s."},{"startTime":2063.73,"endTime":2066.43,"body":"So we know already"},{"startTime":2063.73,"endTime":2066.43,"body":"that's true for circles."},{"startTime":2066.43,"endTime":2068.53,"body":"It turns out to be true"},{"startTime":2066.43,"endTime":2068.53,"body":"of all of these shapes"},{"startTime":2068.53,"endTime":2069.73,"body":"of constant width."},{"startTime":2069.73,"endTime":2070.81,"body":"So that's nice."},{"startTime":2070.81,"endTime":2074.593,"body":"But, the area of these things,"},{"startTime":2075.85,"endTime":2077.29,"body":"that gets a bit more complicated."},{"startTime":2077.29,"endTime":2081.013,"body":"So I'm just going to do a"},{"startTime":2077.29,"endTime":2081.013,"body":"quick calculation of the area"},{"startTime":2081.013,"endTime":2083.77,"body":"of a Reuleaux triangle whose width is s,"},{"startTime":2083.77,"endTime":2085.63,"body":"and the way we can do"},{"startTime":2083.77,"endTime":2085.63,"body":"it with pizza slices,"},{"startTime":2085.63,"endTime":2087.91,"body":"so that's always an advantage."},{"startTime":2087.91,"endTime":2090.01,"body":"So here's the first pizza slice."},{"startTime":2090.01,"endTime":2092.41,"body":"You take that little,"},{"startTime":2090.01,"endTime":2092.41,"body":"it's a sixth of a circle,"},{"startTime":2092.41,"endTime":2094.54,"body":"'cause the angle there is 60 degrees,"},{"startTime":2094.54,"endTime":2098.14,"body":"it's got that equilateral"},{"startTime":2094.54,"endTime":2098.14,"body":"triangle and then the curved bit."},{"startTime":2098.14,"endTime":2101.47,"body":"So you get one of those,"},{"startTime":2098.14,"endTime":2101.47,"body":"and then add another one,"},{"startTime":2101.47,"endTime":2104.17,"body":"and another one, that's"},{"startTime":2101.47,"endTime":2104.17,"body":"three pizza slices."},{"startTime":2104.17,"endTime":2106.0,"body":"So that's kind of half the circle."},{"startTime":2106.0,"endTime":2108.16,"body":"But of course we've done"},{"startTime":2106.0,"endTime":2108.16,"body":"a bit of overcounting."},{"startTime":2108.16,"endTime":2109.66,"body":"We've got a lot of overlap here."},{"startTime":2109.66,"endTime":2111.13,"body":"We've counted that equilateral triangle"},{"startTime":2111.13,"endTime":2112.75,"body":"in the middle three times."},{"startTime":2112.75,"endTime":2115.3,"body":"So we should take that away twice."},{"startTime":2115.3,"endTime":2119.71,"body":"So what we'll end up with is"},{"startTime":2115.3,"endTime":2119.71,"body":"three of those pizza slices,"},{"startTime":2119.71,"endTime":2123.16,"body":"but then take away two"},{"startTime":2119.71,"endTime":2123.16,"body":"equilateral triangles."},{"startTime":2123.16,"endTime":2126.67,"body":"So you can work out the,"},{"startTime":2123.16,"endTime":2126.67,"body":"do a bit of algebra,"},{"startTime":2126.67,"endTime":2128.2,"body":"you're doing it now in your heads,"},{"startTime":2128.2,"endTime":2130.06,"body":"the area of an equilateral triangle,"},{"startTime":2130.06,"endTime":2131.08,"body":"little bit more complicated"},{"startTime":2131.08,"endTime":2132.1,"body":"'cause you have to find the height"},{"startTime":2132.1,"endTime":2134.86,"body":"using Pythagoras' theorem, but you do that"},{"startTime":2134.86,"endTime":2138.363,"body":"and you work it all through, and you get,"},{"startTime":2138.363,"endTime":2139.81,"body":"this is the answer."},{"startTime":2139.81,"endTime":2141.76,"body":"So in terms of s, s is the width."},{"startTime":2141.76,"endTime":2144.37,"body":"You get this pi minus"},{"startTime":2141.76,"endTime":2144.37,"body":"the square root of three"},{"startTime":2144.37,"endTime":2149.37,"body":"times half s squared, which"},{"startTime":2144.37,"endTime":2149.37,"body":"is about .7 times s squared."},{"startTime":2151.57,"endTime":2155.35,"body":"If you work out the same"},{"startTime":2151.57,"endTime":2155.35,"body":"calculation for a circle,"},{"startTime":2155.35,"endTime":2159.43,"body":"so a circle with width s,"},{"startTime":2155.35,"endTime":2159.43,"body":"and s is the diameter then,"},{"startTime":2159.43,"endTime":2161.17,"body":"so the radius is half s,"},{"startTime":2161.17,"endTime":2163.69,"body":"so you do sort of pi r squared on it,"},{"startTime":2163.69,"endTime":2168.69,"body":"and you find that it has area"},{"startTime":2163.69,"endTime":2168.69,"body":"much more, .785-ish s squared."},{"startTime":2170.2,"endTime":2174.07,"body":"So a circle has a really"},{"startTime":2170.2,"endTime":2174.07,"body":"big area for its perimeter."},{"startTime":2174.07,"endTime":2175.72,"body":"The Reuleaux triangle much smaller,"},{"startTime":2175.72,"endTime":2177.967,"body":"and it can be shown that actually"},{"startTime":2175.72,"endTime":2177.967,"body":"it has the best possible,"},{"startTime":2177.967,"endTime":2181.27,"body":"the least possible area,"},{"startTime":2177.967,"endTime":2181.27,"body":"is the Reuleaux triangle"},{"startTime":2181.27,"endTime":2183.97,"body":"of all these shapes of constant width."},{"startTime":2183.97,"endTime":2186.52,"body":"So, you know, if you're"},{"startTime":2183.97,"endTime":2186.52,"body":"worried about how much metal"},{"startTime":2186.52,"endTime":2188.14,"body":"you're going to use for your coin,"},{"startTime":2188.14,"endTime":2190.94,"body":"maybe the Reuleaux triangle"},{"startTime":2188.14,"endTime":2190.94,"body":"is the right answer for you."},{"startTime":2193.3,"endTime":2194.32,"body":"Okay?"},{"startTime":2194.32,"endTime":2198.49,"body":"So, of course you're sitting here,"},{"startTime":2198.49,"endTime":2200.59,"body":"every person in the audience"},{"startTime":2200.59,"endTime":2204.103,"body":"who has used British coins of late,"},{"startTime":2205.3,"endTime":2207.46,"body":"knows, I'm standing here saying,"},{"startTime":2207.46,"endTime":2209.47,"body":"oh these things can only be done with"},{"startTime":2209.47,"endTime":2211.42,"body":"odd numbers of sides, right?"},{"startTime":2211.42,"endTime":2212.92,"body":"And you are sitting there thinking,"},{"startTime":2212.92,"endTime":2215.44,"body":"but pound coins have 12 sides,"},{"startTime":2215.44,"endTime":2217.24,"body":"which has been true since 2017."},{"startTime":2217.24,"endTime":2218.073,"body":"Yes they do."},{"startTime":2218.073,"endTime":2219.52,"body":"They do."},{"startTime":2219.52,"endTime":2221.38,"body":"So how can that be the case?"},{"startTime":2221.38,"endTime":2226.06,"body":"Well, if you are a more mature person,"},{"startTime":2226.06,"endTime":2230.68,"body":"you may remember that we once"},{"startTime":2226.06,"endTime":2230.68,"body":"had other coins with 12 sides,"},{"startTime":2230.68,"endTime":2235.0,"body":"the thrupenny bits used to have 12 sides,"},{"startTime":2235.0,"endTime":2238.433,"body":"and they were just, you"},{"startTime":2235.0,"endTime":2238.433,"body":"know, genuinely a polygon,"},{"startTime":2238.433,"endTime":2241.63,"body":"a dodecagon with straight edges."},{"startTime":2241.63,"endTime":2244.6,"body":"When it was mooted,"},{"startTime":2244.6,"endTime":2245.843,"body":"when the mint was thinking of bringing in"},{"startTime":2245.843,"endTime":2249.25,"body":"a 12-sided coin again, they"},{"startTime":2245.843,"endTime":2249.25,"body":"did some consulting with,"},{"startTime":2249.25,"endTime":2253.33,"body":"you know, the manufacturers,"},{"startTime":2249.25,"endTime":2253.33,"body":"about how would this work."},{"startTime":2253.33,"endTime":2254.92,"body":"Would this be all right?"},{"startTime":2254.92,"endTime":2257.29,"body":"And the manufacturers said"},{"startTime":2254.92,"endTime":2257.29,"body":"no, it would not be all right,"},{"startTime":2257.29,"endTime":2259.75,"body":"because the difference"},{"startTime":2257.29,"endTime":2259.75,"body":"between the maximum width"},{"startTime":2259.75,"endTime":2261.67,"body":"and the minimum width is too big,"},{"startTime":2261.67,"endTime":2263.53,"body":"and our machines would not"},{"startTime":2261.67,"endTime":2263.53,"body":"be able to cope with it."},{"startTime":2263.53,"endTime":2268.45,"body":"So they made the edges a bit"},{"startTime":2263.53,"endTime":2268.45,"body":"curved, of the new pound coin,"},{"startTime":2268.45,"endTime":2272.05,"body":"when it came in, and they did it so that"},{"startTime":2272.05,"endTime":2275.08,"body":"that difference between the"},{"startTime":2272.05,"endTime":2275.08,"body":"minimum and maximum width"},{"startTime":2275.08,"endTime":2278.77,"body":"was small enough that the"},{"startTime":2275.08,"endTime":2278.77,"body":"machines could cope with it."},{"startTime":2278.77,"endTime":2281.38,"body":"So it's not a Reuleaux polygon"},{"startTime":2281.38,"endTime":2284.56,"body":"and it can't be a Reuleaux"},{"startTime":2281.38,"endTime":2284.56,"body":"polygon because it has 12 sides,"},{"startTime":2284.56,"endTime":2287.86,"body":"but it was a near enough solution"},{"startTime":2287.86,"endTime":2291.34,"body":"that was okay for the vending machines."},{"startTime":2291.34,"endTime":2292.78,"body":"So it sort of worked."},{"startTime":2292.78,"endTime":2294.583,"body":"It doesn't mean I have to like it."},{"startTime":2295.753,"endTime":2296.586,"body":"I don't like it."},{"startTime":2296.586,"endTime":2299.65,"body":"So yeah, I think the Canadians"},{"startTime":2296.586,"endTime":2299.65,"body":"have done better than us in,"},{"startTime":2299.65,"endTime":2302.65,"body":"just have 11 sides, or"},{"startTime":2299.65,"endTime":2302.65,"body":"13, I wouldn't mind 13."},{"startTime":2302.65,"endTime":2304.72,"body":"12, no, okay."},{"startTime":2304.72,"endTime":2306.1,"body":"Unsatisfactory."},{"startTime":2306.1,"endTime":2308.75,"body":"But there we go, this is,"},{"startTime":2306.1,"endTime":2308.75,"body":"we do have a 12-sided coin."},{"startTime":2310.0,"endTime":2315.0,"body":"So we've got metals to make our coins."},{"startTime":2315.76,"endTime":2319.27,"body":"We've got some good candidates"},{"startTime":2315.76,"endTime":2319.27,"body":"for shapes for our coins."},{"startTime":2319.27,"endTime":2320.833,"body":"Circles, Reuleaux polygons."},{"startTime":2321.88,"endTime":2324.91,"body":"The final piece of the puzzle"},{"startTime":2321.88,"endTime":2324.91,"body":"that I want to talk about"},{"startTime":2324.91,"endTime":2329.91,"body":"is what actual values"},{"startTime":2324.91,"endTime":2329.91,"body":"should we give our coins?"},{"startTime":2330.43,"endTime":2331.78,"body":"What coins should we have?"},{"startTime":2331.78,"endTime":2334.99,"body":"So in Britain we have 1ps,"},{"startTime":2331.78,"endTime":2334.99,"body":"2ps, and so on, like this."},{"startTime":2334.99,"endTime":2336.85,"body":"What should we choose?"},{"startTime":2336.85,"endTime":2338.8,"body":"And this is the great debate."},{"startTime":2338.8,"endTime":2340.57,"body":"Denominations."},{"startTime":2340.57,"endTime":2344.8,"body":"Now, a few hundred years ago,"},{"startTime":2344.8,"endTime":2348.91,"body":"almost every currency in"},{"startTime":2344.8,"endTime":2348.91,"body":"the world was non-decimal,"},{"startTime":2348.91,"endTime":2350.323,"body":"was not a decimal currency."},{"startTime":2351.16,"endTime":2353.56,"body":"Now, almost every country in the world"},{"startTime":2353.56,"endTime":2354.943,"body":"has a decimal currency."},{"startTime":2356.5,"endTime":2358.78,"body":"Humans are the same as we were."},{"startTime":2358.78,"endTime":2360.37,"body":"So something, what's changed, right?"},{"startTime":2360.37,"endTime":2365.2,"body":"Why was non-decimal good 300 years ago,"},{"startTime":2365.2,"endTime":2367.36,"body":"and now decimal is good?"},{"startTime":2367.36,"endTime":2368.77,"body":"What has changed in that time?"},{"startTime":2368.77,"endTime":2372.133,"body":"Why did we change and what's going on?"},{"startTime":2373.54,"endTime":2376.54,"body":"We are the same people"},{"startTime":2373.54,"endTime":2376.54,"body":"using money, after all."},{"startTime":2376.54,"endTime":2380.587,"body":"So what, if you think what"},{"startTime":2376.54,"endTime":2380.587,"body":"you want out of coins,"},{"startTime":2380.587,"endTime":2383.29,"body":"and coin values, denominations."},{"startTime":2383.29,"endTime":2384.64,"body":"There are two things you might want."},{"startTime":2384.64,"endTime":2385.48,"body":"The first thing,"},{"startTime":2385.48,"endTime":2387.34,"body":"you want it to be kind"},{"startTime":2385.48,"endTime":2387.34,"body":"of easy to work with,"},{"startTime":2387.34,"endTime":2389.62,"body":"easy to calculate with."},{"startTime":2389.62,"endTime":2393.7,"body":"But you also would like to be"},{"startTime":2389.62,"endTime":2393.7,"body":"able to kind of divide money,"},{"startTime":2393.7,"endTime":2397.21,"body":"amounts of money up into fractions,"},{"startTime":2397.21,"endTime":2398.5,"body":"to commonly used fractions."},{"startTime":2398.5,"endTime":2400.39,"body":"If you and your friend do something"},{"startTime":2400.39,"endTime":2402.01,"body":"and get paid an amount of money,"},{"startTime":2402.01,"endTime":2404.05,"body":"you want to be able to share"},{"startTime":2402.01,"endTime":2404.05,"body":"it equally between you,"},{"startTime":2404.05,"endTime":2405.7,"body":"if you have any sense of justice."},{"startTime":2405.7,"endTime":2406.99,"body":"So you want to be able to have, you know,"},{"startTime":2406.99,"endTime":2410.11,"body":"half of amounts of money,"},{"startTime":2410.11,"endTime":2412.6,"body":"and maybe thirds and quarters and things."},{"startTime":2412.6,"endTime":2416.47,"body":"So it's quite nice to be"},{"startTime":2412.6,"endTime":2416.47,"body":"able to divide up money"},{"startTime":2416.47,"endTime":2419.59,"body":"into small fractions like this."},{"startTime":2419.59,"endTime":2421.93,"body":"So the smallest first few fractions"},{"startTime":2421.93,"endTime":2423.79,"body":"are a half, a third, and a quarter, right?"},{"startTime":2423.79,"endTime":2425.92,"body":"If you want to be able"},{"startTime":2423.79,"endTime":2425.92,"body":"to have a unit of money"},{"startTime":2425.92,"endTime":2428.68,"body":"that you can divide into"},{"startTime":2425.92,"endTime":2428.68,"body":"all of those fractions,"},{"startTime":2428.68,"endTime":2431.831,"body":"then you need it to be"},{"startTime":2428.68,"endTime":2431.831,"body":"divisible by 12, right?"},{"startTime":2431.831,"endTime":2434.26,"body":"'Cause if it's divisible"},{"startTime":2431.831,"endTime":2434.26,"body":"by three and by four,"},{"startTime":2434.26,"endTime":2435.61,"body":"you get two for free."},{"startTime":2435.61,"endTime":2437.47,"body":"So it must be divisible by 12."},{"startTime":2437.47,"endTime":2440.74,"body":"And so that is one really good reason"},{"startTime":2440.74,"endTime":2442.27,"body":"to have shillings, right?"},{"startTime":2442.27,"endTime":2446.71,"body":"So in the old, old, olden days,"},{"startTime":2442.27,"endTime":2446.71,"body":"we used to have shillings,"},{"startTime":2446.71,"endTime":2451.15,"body":"and they, one shilling had 12 old pennies,"},{"startTime":2451.15,"endTime":2454.57,"body":"which we write d, 12d."},{"startTime":2454.57,"endTime":2456.49,"body":"And a really good thing about them"},{"startTime":2456.49,"endTime":2461.094,"body":"is that you can then divide"},{"startTime":2456.49,"endTime":2461.094,"body":"them up into these parts in,"},{"startTime":2461.094,"endTime":2462.46,"body":"you know, an equal way."},{"startTime":2462.46,"endTime":2466.06,"body":"So I'll show you a few old"},{"startTime":2462.46,"endTime":2466.06,"body":"coins, I love old coins."},{"startTime":2466.06,"endTime":2469.84,"body":"We've got a sixpence up"},{"startTime":2466.06,"endTime":2469.84,"body":"there, that's half a shilling,"},{"startTime":2469.84,"endTime":2471.88,"body":"'cause a shilling is 12 pence."},{"startTime":2471.88,"endTime":2473.92,"body":"Then on the bottom here, this"},{"startTime":2471.88,"endTime":2473.92,"body":"sort of gold colored one,"},{"startTime":2473.92,"endTime":2476.67,"body":"that's a thrupenny bit, that's"},{"startTime":2473.92,"endTime":2476.67,"body":"a quarter of a shilling."},{"startTime":2478.21,"endTime":2480.67,"body":"You might not recognize the top one there."},{"startTime":2480.67,"endTime":2483.34,"body":"That little silver one,"},{"startTime":2480.67,"endTime":2483.34,"body":"the round silver one,"},{"startTime":2483.34,"endTime":2484.99,"body":"that's a groat."},{"startTime":2484.99,"endTime":2487.15,"body":"Four pence, that's a third of a shilling."},{"startTime":2487.15,"endTime":2490.09,"body":"So we've got a half, a third, a quarter,"},{"startTime":2490.09,"endTime":2492.52,"body":"and of course you can have one"},{"startTime":2490.09,"endTime":2492.52,"body":"penny, a 12th of a shilling."},{"startTime":2492.52,"endTime":2494.65,"body":"There was tuppence as well."},{"startTime":2494.65,"endTime":2497.38,"body":"So there were lots of"},{"startTime":2494.65,"endTime":2497.38,"body":"subdivisions of a shilling."},{"startTime":2497.38,"endTime":2498.91,"body":"You could go further even."},{"startTime":2498.91,"endTime":2503.59,"body":"So the cutest coin that we"},{"startTime":2498.91,"endTime":2503.59,"body":"have had is the farthing."},{"startTime":2503.59,"endTime":2504.61,"body":"It's got a wren on it"},{"startTime":2504.61,"endTime":2507.07,"body":"because the wren is the"},{"startTime":2504.61,"endTime":2507.07,"body":"smallest British bird, right,"},{"startTime":2507.07,"endTime":2509.11,"body":"and the farthing was the"},{"startTime":2507.07,"endTime":2509.11,"body":"smallest British coin."},{"startTime":2509.11,"endTime":2510.94,"body":"That was a quarter of a penny."},{"startTime":2510.94,"endTime":2513.52,"body":"So farthing comes from fourthling, right?"},{"startTime":2513.52,"endTime":2515.38,"body":"A fourth quarter."},{"startTime":2515.38,"endTime":2520.0,"body":"So, because, as well, there"},{"startTime":2515.38,"endTime":2520.0,"body":"are 20 shillings in a pound,"},{"startTime":2520.0,"endTime":2521.38,"body":"if you think how much can I divide,"},{"startTime":2521.38,"endTime":2523.78,"body":"how many different ways"},{"startTime":2521.38,"endTime":2523.78,"body":"can I divide a pound up,"},{"startTime":2524.89,"endTime":2528.1,"body":"it's divisible by a half, a"},{"startTime":2524.89,"endTime":2528.1,"body":"third, a quarter, a fifth,"},{"startTime":2528.1,"endTime":2530.683,"body":"a sixth, loads of things"},{"startTime":2528.1,"endTime":2530.683,"body":"you can break it up into."},{"startTime":2531.64,"endTime":2535.99,"body":"That seems really convenient"},{"startTime":2531.64,"endTime":2535.99,"body":"and good, so why did we switch?"},{"startTime":2535.99,"endTime":2537.91,"body":"Why go decimal?"},{"startTime":2537.91,"endTime":2540.79,"body":"And the reason is 'cause we're,"},{"startTime":2540.79,"endTime":2542.62,"body":"'cause of the way we count, right?"},{"startTime":2542.62,"endTime":2547.21,"body":"We have 10 fingers, and because of that,"},{"startTime":2547.21,"endTime":2550.57,"body":"most cultures have developed at some point"},{"startTime":2550.57,"endTime":2553.24,"body":"a decimal counting system,"},{"startTime":2553.24,"endTime":2556.307,"body":"where we have tens and hundreds"},{"startTime":2553.24,"endTime":2556.307,"body":"and thousands and so on."},{"startTime":2556.307,"endTime":2559.36,"body":"And this is how we write down"},{"startTime":2556.307,"endTime":2559.36,"body":"our calculations nowadays,"},{"startTime":2559.36,"endTime":2561.34,"body":"and it's how we do our arithmetic."},{"startTime":2561.34,"endTime":2565.99,"body":"So the problem with the"},{"startTime":2561.34,"endTime":2565.99,"body":"pre-decimal currencies"},{"startTime":2565.99,"endTime":2570.25,"body":"is that it becomes quite"},{"startTime":2565.99,"endTime":2570.25,"body":"difficult to do calculations"},{"startTime":2570.25,"endTime":2572.98,"body":"using the kind of techniques we've learned"},{"startTime":2572.98,"endTime":2574.63,"body":"for decimal calculations."},{"startTime":2574.63,"endTime":2577.99,"body":"So just as an example, in the, you know,"},{"startTime":2577.99,"endTime":2581.41,"body":"the old pre-decimal system,"},{"startTime":2577.99,"endTime":2581.41,"body":"a pound is 20 shillings,"},{"startTime":2581.41,"endTime":2583.75,"body":"and it's 240 pence."},{"startTime":2583.75,"endTime":2586.33,"body":"So suppose you have some"},{"startTime":2583.75,"endTime":2586.33,"body":"little maths problem"},{"startTime":2586.33,"endTime":2587.32,"body":"that says you've got,"},{"startTime":2587.32,"endTime":2589.84,"body":"how much will it cost to buy 40 items"},{"startTime":2589.84,"endTime":2592.8,"body":"at one shilling and seven pence each?"},{"startTime":2592.8,"endTime":2595.93,"body":"So you think, okay, first I"},{"startTime":2592.8,"endTime":2595.93,"body":"have to do that seven pence,"},{"startTime":2595.93,"endTime":2599.56,"body":"so multiply that by 40, that's 280 pence."},{"startTime":2599.56,"endTime":2603.29,"body":"But then I have to divide by"},{"startTime":2599.56,"endTime":2603.29,"body":"12 to get how many shillings."},{"startTime":2603.29,"endTime":2606.25,"body":"Well okay, so do that and"},{"startTime":2603.29,"endTime":2606.25,"body":"I get 23, remainder eight,"},{"startTime":2606.25,"endTime":2608.5,"body":"but then, oh no, it's 20"},{"startTime":2606.25,"endTime":2608.5,"body":"shillings and a pound."},{"startTime":2608.5,"endTime":2609.333,"body":"So I've got to remember that."},{"startTime":2609.333,"endTime":2611.95,"body":"So it's one pound three"},{"startTime":2609.333,"endTime":2611.95,"body":"shillings and eight pence."},{"startTime":2611.95,"endTime":2613.72,"body":"And then, oh I forgot"},{"startTime":2611.95,"endTime":2613.72,"body":"about the one shilling"},{"startTime":2613.72,"endTime":2614.62,"body":"at the beginning."},{"startTime":2614.62,"endTime":2616.6,"body":"40 shillings, okay that's"},{"startTime":2614.62,"endTime":2616.6,"body":"two pounds, so I add that,"},{"startTime":2616.6,"endTime":2620.02,"body":"da-da-da, and you know,"},{"startTime":2616.6,"endTime":2620.02,"body":"you're there for a while."},{"startTime":2620.02,"endTime":2621.46,"body":"Now of course people could practice this"},{"startTime":2621.46,"endTime":2625.18,"body":"and get better at it, but"},{"startTime":2621.46,"endTime":2625.18,"body":"estimates people have made"},{"startTime":2625.18,"endTime":2627.34,"body":"has said that actually it would take kids"},{"startTime":2627.34,"endTime":2629.8,"body":"almost an extra six months in school"},{"startTime":2629.8,"endTime":2634.03,"body":"to be able to learn how to do"},{"startTime":2629.8,"endTime":2634.03,"body":"all this stuff and convert,"},{"startTime":2634.03,"endTime":2635.89,"body":"you know, it's not just"},{"startTime":2634.03,"endTime":2635.89,"body":"that it's all base 12,"},{"startTime":2635.89,"endTime":2638.53,"body":"it's a mix of base 12 and base 20."},{"startTime":2638.53,"endTime":2642.13,"body":"And don't even get me started"},{"startTime":2638.53,"endTime":2642.13,"body":"on imperial units of weight,"},{"startTime":2642.13,"endTime":2644.86,"body":"like ounces in a pound"},{"startTime":2642.13,"endTime":2644.86,"body":"and pounds in a stone."},{"startTime":2644.86,"endTime":2646.33,"body":"You know, it's really tough"},{"startTime":2646.33,"endTime":2647.98,"body":"to be able to get adept at these things."},{"startTime":2647.98,"endTime":2650.95,"body":"Whereas, if you think what's"},{"startTime":2647.98,"endTime":2650.95,"body":"one shilling and seven pence"},{"startTime":2650.95,"endTime":2653.427,"body":"in new money, well a shilling is like 5p,"},{"startTime":2653.427,"endTime":2655.45,"body":"'cause it's a 20th of a pound."},{"startTime":2655.45,"endTime":2658.06,"body":"Seven pence is about three new pence."},{"startTime":2658.06,"endTime":2660.4,"body":"So basically around 8p."},{"startTime":2660.4,"endTime":2665.35,"body":"So now, 40 items at 8p,"},{"startTime":2660.4,"endTime":2665.35,"body":"three pound 20, done."},{"startTime":2665.35,"endTime":2668.23,"body":"So it is much quicker to"},{"startTime":2665.35,"endTime":2668.23,"body":"use the decimal calculation,"},{"startTime":2668.23,"endTime":2670.13,"body":"because that's what we learned to use."},{"startTime":2671.2,"endTime":2676.2,"body":"Okay, so people have been"},{"startTime":2671.2,"endTime":2676.2,"body":"saying okay, this system,"},{"startTime":2676.24,"endTime":2677.65,"body":"we should switch to decimal."},{"startTime":2677.65,"endTime":2679.36,"body":"Before I talk about who had said that"},{"startTime":2679.36,"endTime":2680.53,"body":"and how long ago they said it,"},{"startTime":2680.53,"endTime":2682.54,"body":"you can be guessing while we wait,"},{"startTime":2682.54,"endTime":2684.37,"body":"I just want to mention"},{"startTime":2682.54,"endTime":2684.37,"body":"the worst currency ever"},{"startTime":2684.37,"endTime":2687.343,"body":"in the history of real"},{"startTime":2684.37,"endTime":2687.343,"body":"or imagined realities."},{"startTime":2688.9,"endTime":2691.75,"body":"Poor old Harry Potter, right?"},{"startTime":2691.75,"endTime":2694.36,"body":"Those wizards, I mean"},{"startTime":2691.75,"endTime":2694.36,"body":"they had to invent magic"},{"startTime":2694.36,"endTime":2697.06,"body":"purely to be able to pay,"},{"startTime":2694.36,"endTime":2697.06,"body":"I reckon, for things,"},{"startTime":2697.06,"endTime":2700.3,"body":"because they've got,"},{"startTime":2697.06,"endTime":2700.3,"body":"I dunno what the real,"},{"startTime":2700.3,"endTime":2701.23,"body":"the currency's supposed to look like,"},{"startTime":2701.23,"endTime":2705.58,"body":"but they've got Bronze-Knuts,"},{"startTime":2701.23,"endTime":2705.58,"body":"there are 29 of them"},{"startTime":2705.58,"endTime":2708.792,"body":"in a Silver-Sickle, 29."},{"startTime":2708.792,"endTime":2712.72,"body":"And then there are 17"},{"startTime":2708.792,"endTime":2712.72,"body":"Silver-Sickles in a Golden-Galleon."},{"startTime":2712.72,"endTime":2716.11,"body":"So if you need to give"},{"startTime":2712.72,"endTime":2716.11,"body":"your friend half a Galleon,"},{"startTime":2716.11,"endTime":2720.94,"body":"you can't, no, I'm sorry."},{"startTime":2720.94,"endTime":2723.61,"body":"This is, you know, I can"},{"startTime":2720.94,"endTime":2723.61,"body":"believe in all the rest,"},{"startTime":2723.61,"endTime":2725.71,"body":"but the currency is just dreadful."},{"startTime":2725.71,"endTime":2728.14,"body":"I'm glad that dragon destroyed the bank."},{"startTime":2728.14,"endTime":2730.056,"body":"It's never going to work."},{"startTime":2730.056,"endTime":2733.09,"body":"So that has all the worst"},{"startTime":2730.056,"endTime":2733.09,"body":"things of non-decimal coinage,"},{"startTime":2733.09,"endTime":2734.89,"body":"impossible to calculate with."},{"startTime":2734.89,"endTime":2737.32,"body":"And it also doesn't have"},{"startTime":2734.89,"endTime":2737.32,"body":"any of the benefits,"},{"startTime":2737.32,"endTime":2738.31,"body":"because you can't divide things"},{"startTime":2738.31,"endTime":2739.69,"body":"by half or third or anything."},{"startTime":2739.69,"endTime":2743.77,"body":"17 and 29 are prime,"},{"startTime":2739.69,"endTime":2743.77,"body":"like nothing to be done."},{"startTime":2743.77,"endTime":2744.603,"body":"Let's move on."},{"startTime":2745.6,"endTime":2748.81,"body":"Who do you think was a very early person"},{"startTime":2748.81,"endTime":2751.36,"body":"to suggest that we switched"},{"startTime":2748.81,"endTime":2751.36,"body":"to decimal currency,"},{"startTime":2751.36,"endTime":2752.193,"body":"and when was that?"},{"startTime":2752.193,"endTime":2753.7,"body":"Was it in 1920?"},{"startTime":2753.7,"endTime":2755.56,"body":"Was it in 1850?"},{"startTime":2755.56,"endTime":2758.38,"body":"Was it perhaps in 1696?"},{"startTime":2758.38,"endTime":2761.14,"body":"Christopher Wren, already back then,"},{"startTime":2761.14,"endTime":2764.8,"body":"former Gresham professor of"},{"startTime":2761.14,"endTime":2764.8,"body":"astronomy, as we probably know,"},{"startTime":2764.8,"endTime":2767.383,"body":"was suggesting we switch"},{"startTime":2764.8,"endTime":2767.383,"body":"to a decimal currency."},{"startTime":2768.28,"endTime":2769.66,"body":"And he even wasn't the first."},{"startTime":2769.66,"endTime":2771.449,"body":"People had started saying this"},{"startTime":2771.449,"endTime":2772.69,"body":"sort of in the early 17th century,"},{"startTime":2772.69,"endTime":2775.36,"body":"and from then on people"},{"startTime":2772.69,"endTime":2775.36,"body":"had started saying this."},{"startTime":2775.36,"endTime":2778.27,"body":"And I think it was because"},{"startTime":2775.36,"endTime":2778.27,"body":"more and more people"},{"startTime":2778.27,"endTime":2779.2,"body":"were becoming literate,"},{"startTime":2779.2,"endTime":2781.81,"body":"they were being able to"},{"startTime":2779.2,"endTime":2781.81,"body":"write down calculations."},{"startTime":2781.81,"endTime":2782.98,"body":"It wasn't all just like, you know,"},{"startTime":2782.98,"endTime":2784.33,"body":"I'll share with you and you share with me."},{"startTime":2784.33,"endTime":2785.47,"body":"People were writing things down,"},{"startTime":2785.47,"endTime":2788.68,"body":"they were doing actual proper"},{"startTime":2785.47,"endTime":2788.68,"body":"accounting more and more."},{"startTime":2788.68,"endTime":2792.13,"body":"And so then you really come"},{"startTime":2788.68,"endTime":2792.13,"body":"up against this wall of"},{"startTime":2792.13,"endTime":2795.31,"body":"we're using hundreds,"},{"startTime":2792.13,"endTime":2795.31,"body":"tens, hundreds, thousands"},{"startTime":2795.31,"endTime":2796.93,"body":"in our calculations,"},{"startTime":2796.93,"endTime":2799.5,"body":"but we're using these weird"},{"startTime":2796.93,"endTime":2799.5,"body":"twelves and all sorts of things"},{"startTime":2799.5,"endTime":2800.333,"body":"in our coins."},{"startTime":2800.333,"endTime":2803.17,"body":"So he said we should have a silver noble"},{"startTime":2803.17,"endTime":2806.41,"body":"divided into 10 primes,"},{"startTime":2803.17,"endTime":2806.41,"body":"and a hundred seconds,"},{"startTime":2806.41,"endTime":2808.33,"body":"and this centesimal division"},{"startTime":2808.33,"endTime":2810.28,"body":"will be very proper for accounts."},{"startTime":2810.28,"endTime":2813.07,"body":"So he suggested that in 1696."},{"startTime":2813.07,"endTime":2814.42,"body":"It did take a while"},{"startTime":2814.42,"endTime":2817.123,"body":"for Britain to adopt a"},{"startTime":2814.42,"endTime":2817.123,"body":"decimal currency after that."},{"startTime":2817.96,"endTime":2820.72,"body":"I think the first country"},{"startTime":2817.96,"endTime":2820.72,"body":"to adopt a decimal currency,"},{"startTime":2820.72,"endTime":2824.56,"body":"as far as I can tell, was Russia in 1704."},{"startTime":2824.56,"endTime":2826.873,"body":"The Russian Rouble became 100 kopeks."},{"startTime":2828.34,"endTime":2830.83,"body":"The U.S. dollar, in 1792 now,"},{"startTime":2830.83,"endTime":2832.81,"body":"of course, one of the main objections"},{"startTime":2832.81,"endTime":2836.11,"body":"to switching to a decimal system"},{"startTime":2832.81,"endTime":2836.11,"body":"is that people said, look,"},{"startTime":2836.11,"endTime":2837.25,"body":"people just won't go for it."},{"startTime":2837.25,"endTime":2838.57,"body":"They're used to what they're used to,"},{"startTime":2838.57,"endTime":2840.79,"body":"they won't want to change,"},{"startTime":2838.57,"endTime":2840.79,"body":"it's too complicated,"},{"startTime":2840.79,"endTime":2842.2,"body":"you'll never be able to do it."},{"startTime":2842.2,"endTime":2845.47,"body":"But if you're creating a new"},{"startTime":2842.2,"endTime":2845.47,"body":"currency in a new country,"},{"startTime":2845.47,"endTime":2847.33,"body":"you don't have that problem."},{"startTime":2847.33,"endTime":2848.62,"body":"You just say, well, yeah,"},{"startTime":2848.62,"endTime":2850.33,"body":"let's have this wonderful new thing,"},{"startTime":2850.33,"endTime":2852.88,"body":"have a hundred cents, a"},{"startTime":2850.33,"endTime":2852.88,"body":"hundred cents in a dollar"},{"startTime":2852.88,"endTime":2854.05,"body":"right from the off."},{"startTime":2854.05,"endTime":2856.51,"body":"So that happened in 1792."},{"startTime":2856.51,"endTime":2857.74,"body":"Britain."},{"startTime":2857.74,"endTime":2858.76,"body":"Oh, Britain."},{"startTime":2858.76,"endTime":2862.03,"body":"So we talked about it a"},{"startTime":2858.76,"endTime":2862.03,"body":"lot for hundreds of years."},{"startTime":2862.03,"endTime":2865.15,"body":"In the Victorian period,"},{"startTime":2865.15,"endTime":2867.07,"body":"there was really an awful"},{"startTime":2865.15,"endTime":2867.07,"body":"lot of talk about it."},{"startTime":2867.07,"endTime":2868.69,"body":"In parliament there were lots of debates,"},{"startTime":2868.69,"endTime":2870.07,"body":"there were royal commissions."},{"startTime":2870.07,"endTime":2871.96,"body":"They even started to bring in coins"},{"startTime":2871.96,"endTime":2875.05,"body":"that would sort of help"},{"startTime":2871.96,"endTime":2875.05,"body":"ease us into decimalization."},{"startTime":2875.05,"endTime":2879.01,"body":"In 1849, right, they brought"},{"startTime":2875.05,"endTime":2879.01,"body":"in a coin that was going to be,"},{"startTime":2879.01,"endTime":2881.41,"body":"this is sort of a step along the route."},{"startTime":2881.41,"endTime":2883.96,"body":"And I don't know if you"},{"startTime":2881.41,"endTime":2883.96,"body":"can see, but on this coin,"},{"startTime":2883.96,"endTime":2886.9,"body":"so it says one decade at the"},{"startTime":2883.96,"endTime":2886.9,"body":"top, and at the bottom it says"},{"startTime":2886.9,"endTime":2888.67,"body":"one-tenth of a pound."},{"startTime":2888.67,"endTime":2891.7,"body":"So they're sort of explicitly"},{"startTime":2888.67,"endTime":2891.7,"body":"saying that's two shillings,"},{"startTime":2891.7,"endTime":2892.78,"body":"that's a florin."},{"startTime":2892.78,"endTime":2894.49,"body":"But they didn't say two shillings,"},{"startTime":2894.49,"endTime":2896.32,"body":"they said one-tenth of a pound."},{"startTime":2896.32,"endTime":2901.32,"body":"This is kind of, let's edge"},{"startTime":2896.32,"endTime":2901.32,"body":"people towards decimalization."},{"startTime":2901.96,"endTime":2904.063,"body":"Still took many, many more years,"},{"startTime":2904.93,"endTime":2907.36,"body":"took another 120 years after that,"},{"startTime":2907.36,"endTime":2909.07,"body":"after the one-tenth of a pound,"},{"startTime":2909.07,"endTime":2914.07,"body":"to actually go decimal in 1971"},{"startTime":2909.07,"endTime":2914.07,"body":"on D-Day, in 15th February."},{"startTime":2914.32,"endTime":2917.02,"body":"But the, you know, it took people a while"},{"startTime":2917.02,"endTime":2917.853,"body":"to get the hang of it,"},{"startTime":2917.853,"endTime":2920.74,"body":"but now our school"},{"startTime":2917.853,"endTime":2920.74,"body":"children have a better time"},{"startTime":2920.74,"endTime":2922.29,"body":"with trying to do calculations."},{"startTime":2924.16,"endTime":2925.93,"body":"So, okay."},{"startTime":2925.93,"endTime":2930.49,"body":"Maybe we accept that"},{"startTime":2925.93,"endTime":2930.49,"body":"decimalization is a good idea"},{"startTime":2930.49,"endTime":2933.52,"body":"just to help us do our"},{"startTime":2930.49,"endTime":2933.52,"body":"calculations and our arithmetic."},{"startTime":2933.52,"endTime":2936.01,"body":"We still haven't decided"},{"startTime":2933.52,"endTime":2936.01,"body":"what coins to have."},{"startTime":2936.01,"endTime":2937.27,"body":"So the next question is,"},{"startTime":2937.27,"endTime":2939.82,"body":"what denominations within"},{"startTime":2937.27,"endTime":2939.82,"body":"the decimal system."},{"startTime":2939.82,"endTime":2942.43,"body":"We should have things that go"},{"startTime":2939.82,"endTime":2942.43,"body":"into a pound exactly, right."},{"startTime":2942.43,"endTime":2944.41,"body":"So you don't want to have a 3p coin,"},{"startTime":2944.41,"endTime":2946.36,"body":"because you can't get a whole"},{"startTime":2944.41,"endTime":2946.36,"body":"number of those in a pound"},{"startTime":2946.36,"endTime":2948.16,"body":"if you have a hundred pennies."},{"startTime":2948.16,"endTime":2950.14,"body":"Similarly with cents and dollars."},{"startTime":2950.14,"endTime":2952.81,"body":"So there's a mathematician"},{"startTime":2950.14,"endTime":2952.81,"body":"called Adam Townsend"},{"startTime":2952.81,"endTime":2954.16,"body":"at the University of Durham,"},{"startTime":2954.16,"endTime":2959.16,"body":"suggested that one way of"},{"startTime":2954.16,"endTime":2959.16,"body":"determining kind of the efficiency"},{"startTime":2960.04,"endTime":2961.78,"body":"or, you know, best possible choices"},{"startTime":2961.78,"endTime":2963.94,"body":"of different coins to have,"},{"startTime":2963.94,"endTime":2967.84,"body":"is to look at the average number of coins"},{"startTime":2967.84,"endTime":2969.94,"body":"you would get in change."},{"startTime":2969.94,"endTime":2974.77,"body":"So, you know, if you get 40p"},{"startTime":2969.94,"endTime":2974.77,"body":"change, then you're going to get,"},{"startTime":2974.77,"endTime":2977.92,"body":"in Britain you'd get two 20p coins."},{"startTime":2977.92,"endTime":2981.28,"body":"In America, if you're"},{"startTime":2977.92,"endTime":2981.28,"body":"having 40 cents change,"},{"startTime":2981.28,"endTime":2984.417,"body":"you might get a quarter and"},{"startTime":2981.28,"endTime":2984.417,"body":"a nickel and a dime, right?"},{"startTime":2984.417,"endTime":2986.32,"body":"'Cause they don't have a 20 cent coin,"},{"startTime":2986.32,"endTime":2988.69,"body":"they have a quarter which is 25 cents."},{"startTime":2988.69,"endTime":2992.02,"body":"So if you look, if you"},{"startTime":2988.69,"endTime":2992.02,"body":"sort of average out over"},{"startTime":2992.02,"endTime":2994.45,"body":"all possible amounts of"},{"startTime":2992.02,"endTime":2994.45,"body":"change you could get,"},{"startTime":2994.45,"endTime":2998.71,"body":"up to some limit, then you can see,"},{"startTime":2998.71,"endTime":3000.78,"body":"this might be an"},{"startTime":2998.71,"endTime":3000.78,"body":"indication of which system"},{"startTime":3000.78,"endTime":3002.733,"body":"is the most efficient."},{"startTime":3003.66,"endTime":3006.81,"body":"So he'd written a couple"},{"startTime":3003.66,"endTime":3006.81,"body":"of articles about this,"},{"startTime":3006.81,"endTime":3010.74,"body":"I've put links in the"},{"startTime":3006.81,"endTime":3010.74,"body":"transcript again to those."},{"startTime":3010.74,"endTime":3013.44,"body":"And he'd also come up with a"},{"startTime":3010.74,"endTime":3013.44,"body":"bit of code that you can try,"},{"startTime":3013.44,"endTime":3016.35,"body":"and I modified it for, slightly,"},{"startTime":3016.35,"endTime":3017.43,"body":"to do something slightly different,"},{"startTime":3017.43,"endTime":3019.35,"body":"and I put the code I"},{"startTime":3017.43,"endTime":3019.35,"body":"used in the transcript"},{"startTime":3019.35,"endTime":3020.45,"body":"if you want to try it."},{"startTime":3021.3,"endTime":3024.18,"body":"But the basic idea is that"},{"startTime":3021.3,"endTime":3024.18,"body":"you do this and you work out"},{"startTime":3024.18,"endTime":3027.0,"body":"the most efficient change"},{"startTime":3024.18,"endTime":3027.0,"body":"you could get for each value"},{"startTime":3027.0,"endTime":3031.113,"body":"up to, so I went up to 99p or 99 cents,"},{"startTime":3032.34,"endTime":3034.65,"body":"on the grounds that then you're"},{"startTime":3032.34,"endTime":3034.65,"body":"comparing like with like."},{"startTime":3034.65,"endTime":3036.93,"body":"But Dr. Townsend actually went up to"},{"startTime":3036.93,"endTime":3039.93,"body":"the smallest bills, bank"},{"startTime":3036.93,"endTime":3039.93,"body":"note that you can get."},{"startTime":3039.93,"endTime":3042.06,"body":"So in America that's the dollar bill,"},{"startTime":3042.06,"endTime":3044.419,"body":"but in Britain it's a five pound note."},{"startTime":3044.419,"endTime":3046.17,"body":"So you can, you know,"},{"startTime":3044.419,"endTime":3046.17,"body":"there's no right answer here."},{"startTime":3046.17,"endTime":3047.4,"body":"We're just looking at different ways"},{"startTime":3047.4,"endTime":3049.92,"body":"we might measure the"},{"startTime":3047.4,"endTime":3049.92,"body":"efficiency of these things."},{"startTime":3049.92,"endTime":3051.78,"body":"So if we look at the coins in the U.K."},{"startTime":3051.78,"endTime":3054.423,"body":"that are less than a pound,"},{"startTime":3051.78,"endTime":3054.423,"body":"you've got lots of them."},{"startTime":3055.29,"endTime":3059.37,"body":"Six coins, we have 1p,"},{"startTime":3055.29,"endTime":3059.37,"body":"2p, 5p, 10p, 20p, 50p."},{"startTime":3059.37,"endTime":3060.63,"body":"That's a lot of coins."},{"startTime":3060.63,"endTime":3062.67,"body":"In America they only have four coins."},{"startTime":3062.67,"endTime":3065.223,"body":"The cent, the nickel,"},{"startTime":3062.67,"endTime":3065.223,"body":"the dime, the quarter."},{"startTime":3066.06,"endTime":3068.58,"body":"So you might expect that"},{"startTime":3066.06,"endTime":3068.58,"body":"you're going to need"},{"startTime":3068.58,"endTime":3070.23,"body":"potentially more coins."},{"startTime":3070.23,"endTime":3072.27,"body":"You might get more coins"},{"startTime":3070.23,"endTime":3072.27,"body":"in change in the U.S.,"},{"startTime":3072.27,"endTime":3074.262,"body":"'cause they have fewer different ones."},{"startTime":3074.262,"endTime":3075.095,"body":"And that, indeed, is what happens."},{"startTime":3075.095,"endTime":3080.095,"body":"So in the U.K. you get"},{"startTime":3075.095,"endTime":3080.095,"body":"a mean of 3.43 coins,"},{"startTime":3080.22,"endTime":3081.9,"body":"and a median of three."},{"startTime":3081.9,"endTime":3083.16,"body":"So kind of about half the time"},{"startTime":3083.16,"endTime":3084.51,"body":"you're getting three or fewer,"},{"startTime":3084.51,"endTime":3086.16,"body":"about half the time you get three or more."},{"startTime":3086.16,"endTime":3088.11,"body":"In the U.S. that's a lot more."},{"startTime":3088.11,"endTime":3089.16,"body":"A median of five."},{"startTime":3089.16,"endTime":3091.08,"body":"So you get more things"},{"startTime":3089.16,"endTime":3091.08,"body":"in change in the U.S."},{"startTime":3091.08,"endTime":3095.34,"body":"So we might say, maybe our"},{"startTime":3091.08,"endTime":3095.34,"body":"system's a bit more efficient,"},{"startTime":3095.34,"endTime":3096.173,"body":"but on the other hand,"},{"startTime":3096.173,"endTime":3097.59,"body":"we do have more different kinds of coins,"},{"startTime":3097.59,"endTime":3099.84,"body":"and maybe that has disadvantages."},{"startTime":3099.84,"endTime":3103.86,"body":"You could also think we have"},{"startTime":3099.84,"endTime":3103.86,"body":"a 20p, they have a quarter."},{"startTime":3103.86,"endTime":3106.26,"body":"Is one of those better than the other?"},{"startTime":3106.26,"endTime":3107.697,"body":"You can test that,"},{"startTime":3107.697,"endTime":3111.57,"body":"and actually doesn't"},{"startTime":3107.697,"endTime":3111.57,"body":"make too much difference."},{"startTime":3111.57,"endTime":3114.6,"body":"So in Britain, if you"},{"startTime":3111.57,"endTime":3114.6,"body":"swap 20ps for quarters,"},{"startTime":3114.6,"endTime":3115.77,"body":"absolutely no effect."},{"startTime":3115.77,"endTime":3116.603,"body":"It would have no effect,"},{"startTime":3116.603,"endTime":3117.84,"body":"of course everyone would get very cross,"},{"startTime":3117.84,"endTime":3119.0,"body":"so we probably won't do it."},{"startTime":3119.0,"endTime":3121.74,"body":"In the U.S., things would"},{"startTime":3119.0,"endTime":3121.74,"body":"actually get slightly worse."},{"startTime":3121.74,"endTime":3125.981,"body":"If you tried to introduce"},{"startTime":3121.74,"endTime":3125.981,"body":"a 20p instead of a quarter,"},{"startTime":3125.981,"endTime":3129.21,"body":"or 20 cent coin, because they"},{"startTime":3125.981,"endTime":3129.21,"body":"have so few coins already,"},{"startTime":3129.21,"endTime":3130.89,"body":"that actually would make"},{"startTime":3129.21,"endTime":3130.89,"body":"things a little bit worse."},{"startTime":3130.89,"endTime":3132.87,"body":"So within their system,"},{"startTime":3132.87,"endTime":3135.363,"body":"that's probably the best coin to have."},{"startTime":3136.53,"endTime":3139.38,"body":"I encourage you to try"},{"startTime":3136.53,"endTime":3139.38,"body":"this and think of your own,"},{"startTime":3139.38,"endTime":3141.24,"body":"what do you think is the"},{"startTime":3139.38,"endTime":3141.24,"body":"best thing to measure,"},{"startTime":3141.24,"endTime":3142.413,"body":"and have a go at it."},{"startTime":3143.58,"endTime":3144.9,"body":"One thing to mention,"},{"startTime":3144.9,"endTime":3148.2,"body":"and this is something that"},{"startTime":3144.9,"endTime":3148.2,"body":"Adam Townsend pointed out"},{"startTime":3148.2,"endTime":3149.22,"body":"in his article,"},{"startTime":3149.22,"endTime":3152.28,"body":"that self-checkout machines"},{"startTime":3149.22,"endTime":3152.28,"body":"are the work of the devil."},{"startTime":3152.28,"endTime":3154.2,"body":"He didn't put it like that."},{"startTime":3154.2,"endTime":3156.96,"body":"He said, so these machines are kind of,"},{"startTime":3156.96,"endTime":3161.31,"body":"usually come from global"},{"startTime":3156.96,"endTime":3161.31,"body":"designs, and they,"},{"startTime":3161.31,"endTime":3163.29,"body":"because we have quite a"},{"startTime":3161.31,"endTime":3163.29,"body":"lot of coins in Britain,"},{"startTime":3163.29,"endTime":3165.81,"body":"we have those six coins"},{"startTime":3163.29,"endTime":3165.81,"body":"that are below a pound,"},{"startTime":3165.81,"endTime":3167.54,"body":"but we also have a one"},{"startTime":3165.81,"endTime":3167.54,"body":"pound and a two pound coin."},{"startTime":3167.54,"endTime":3169.893,"body":"We have eight coins, which is quite a lot."},{"startTime":3170.85,"endTime":3172.948,"body":"Self checkout machines usually only give"},{"startTime":3172.948,"endTime":3175.4,"body":"up to six different coins in change."},{"startTime":3175.4,"endTime":3176.94,"body":"So you can put in any coin,"},{"startTime":3176.94,"endTime":3179.937,"body":"but they'll only give you"},{"startTime":3176.94,"endTime":3179.937,"body":"out six different coins."},{"startTime":3179.937,"endTime":3183.768,"body":"And the ones that were chosen"},{"startTime":3179.937,"endTime":3183.768,"body":"are usually 1p, 2p, 5p,"},{"startTime":3183.768,"endTime":3188.768,"body":"not a 10p, 20p, not 50p, and"},{"startTime":3183.768,"endTime":3188.768,"body":"then a pound and two pound."},{"startTime":3189.36,"endTime":3192.72,"body":"Now if you only have those"},{"startTime":3189.36,"endTime":3192.72,"body":"coins available to you,"},{"startTime":3192.72,"endTime":3194.85,"body":"then for getting amounts"},{"startTime":3192.72,"endTime":3194.85,"body":"of change up to 99p,"},{"startTime":3194.85,"endTime":3198.45,"body":"as I checked before, it's suddenly awful."},{"startTime":3198.45,"endTime":3202.2,"body":"You suddenly get an average"},{"startTime":3198.45,"endTime":3202.2,"body":"of five coins in your change."},{"startTime":3202.2,"endTime":3203.85,"body":"If you ever curse one of those machines"},{"startTime":3203.85,"endTime":3205.83,"body":"for giving you a load of"},{"startTime":3203.85,"endTime":3205.83,"body":"shrapnel, now you know,"},{"startTime":3205.83,"endTime":3207.69,"body":"you're not just sort of being grumpy,"},{"startTime":3207.69,"endTime":3210.63,"body":"they really do give you too much change."},{"startTime":3210.63,"endTime":3213.547,"body":"They're not as efficient"},{"startTime":3210.63,"endTime":3213.547,"body":"as a human would be."},{"startTime":3213.547,"endTime":3215.91,"body":"They give you loads of"},{"startTime":3213.547,"endTime":3215.91,"body":"5ps, usually, typically,"},{"startTime":3215.91,"endTime":3217.81,"body":"why have I got a whole handful of 5ps?"},{"startTime":3219.57,"endTime":3221.31,"body":"This isn't even the right choice."},{"startTime":3221.31,"endTime":3224.25,"body":"If you have to limit"},{"startTime":3221.31,"endTime":3224.25,"body":"yourself to six coins,"},{"startTime":3224.25,"endTime":3225.63,"body":"then if you change,"},{"startTime":3225.63,"endTime":3228.99,"body":"if instead of giving"},{"startTime":3225.63,"endTime":3228.99,"body":"pound coins you give 50ps,"},{"startTime":3228.99,"endTime":3231.09,"body":"then actually can bring that right down,"},{"startTime":3231.09,"endTime":3234.42,"body":"almost back to the what"},{"startTime":3231.09,"endTime":3234.42,"body":"a human would give you."},{"startTime":3234.42,"endTime":3238.41,"body":"So it's kind of interesting"},{"startTime":3234.42,"endTime":3238.41,"body":"this affects our daily lives,"},{"startTime":3238.41,"endTime":3240.96,"body":"but we don't necessarily"},{"startTime":3238.41,"endTime":3240.96,"body":"realize what's happening."},{"startTime":3240.96,"endTime":3243.93,"body":"And actually all supermarkets"},{"startTime":3240.96,"endTime":3243.93,"body":"should immediately change"},{"startTime":3243.93,"endTime":3246.93,"body":"to this second system,"},{"startTime":3243.93,"endTime":3246.93,"body":"right away, starting today."},{"startTime":3246.93,"endTime":3249.663,"body":"That's my order, as a Gresham"},{"startTime":3246.93,"endTime":3249.663,"body":"professor of geometry."},{"startTime":3251.07,"endTime":3252.72,"body":"The final thing I want to mention."},{"startTime":3252.72,"endTime":3254.77,"body":"We've talked about having a lot of coins."},{"startTime":3255.84,"endTime":3258.3,"body":"The 1p coin, has it had its day?"},{"startTime":3258.3,"endTime":3261.06,"body":"Is it time to ditch the 1p coin?"},{"startTime":3261.06,"endTime":3263.67,"body":"1p doesn't buy you a lot these days."},{"startTime":3263.67,"endTime":3266.19,"body":"Inflation is quite high."},{"startTime":3266.19,"endTime":3270.12,"body":"How can we decide when"},{"startTime":3266.19,"endTime":3270.12,"body":"it's time to let it go?"},{"startTime":3270.12,"endTime":3272.64,"body":"Well, one way is to compare"},{"startTime":3270.12,"endTime":3272.64,"body":"when other small coins"},{"startTime":3272.64,"endTime":3274.86,"body":"have been discontinued."},{"startTime":3274.86,"endTime":3275.97,"body":"And the Bank of England has this"},{"startTime":3275.97,"endTime":3278.157,"body":"amazing inflation calculator"},{"startTime":3275.97,"endTime":3278.157,"body":"that will tell you,"},{"startTime":3278.157,"endTime":3280.231,"body":"you can put in an amount of money,"},{"startTime":3280.231,"endTime":3282.3,"body":"and it tells you what it would"},{"startTime":3280.231,"endTime":3282.3,"body":"be worth today for any year,"},{"startTime":3282.3,"endTime":3285.15,"body":"going back to 1209."},{"startTime":3285.15,"endTime":3286.74,"body":"So, wow!"},{"startTime":3286.74,"endTime":3289.38,"body":"So you can try this out, or"},{"startTime":3286.74,"endTime":3289.38,"body":"you could tell your kids,"},{"startTime":3289.38,"endTime":3291.45,"body":"you know, how lucky they are"},{"startTime":3291.45,"endTime":3294.18,"body":"and how they should be grateful"},{"startTime":3291.45,"endTime":3294.18,"body":"for anything you've given,"},{"startTime":3294.18,"endTime":3296.64,"body":"because, you know, the amount"},{"startTime":3294.18,"endTime":3296.64,"body":"you had when you were a kid."},{"startTime":3296.64,"endTime":3299.793,"body":"I do this to my children"},{"startTime":3296.64,"endTime":3299.793,"body":"and they love it so much."},{"startTime":3301.08,"endTime":3304.563,"body":"Halfpenny coin discontinued, 1984."},{"startTime":3306.349,"endTime":3307.8,"body":"People didn't like it."},{"startTime":3307.8,"endTime":3309.81,"body":"They didn't care about it as a coin."},{"startTime":3309.81,"endTime":3311.49,"body":"Very important issues arose."},{"startTime":3311.49,"endTime":3314.01,"body":"So this is an irate letter to \"The Times.\""},{"startTime":3314.01,"endTime":3317.077,"body":"You cannot, how dare"},{"startTime":3314.01,"endTime":3317.077,"body":"you get rid of the 1/2p?"},{"startTime":3317.077,"endTime":3320.01,"body":"\"It's indispensable for"},{"startTime":3317.077,"endTime":3320.01,"body":"leveling off pendulum clocks.\""},{"startTime":3320.01,"endTime":3323.85,"body":"So, you know, in touch with"},{"startTime":3320.01,"endTime":3323.85,"body":"the common person as ever."},{"startTime":3323.85,"endTime":3326.07,"body":"In spite of this urgent need,"},{"startTime":3326.07,"endTime":3329.61,"body":"I hope they kept a few in"},{"startTime":3326.07,"endTime":3329.61,"body":"reserve, they were discontinued."},{"startTime":3329.61,"endTime":3331.05,"body":"Now looking at inflation,"},{"startTime":3331.05,"endTime":3334.68,"body":"one pound in 1984 could"},{"startTime":3331.05,"endTime":3334.68,"body":"buy you the same as"},{"startTime":3334.68,"endTime":3336.84,"body":"two pounds 63 today."},{"startTime":3336.84,"endTime":3340.8,"body":"So that would mean that a"},{"startTime":3336.84,"endTime":3340.8,"body":"1/2p, when it was discontinued,"},{"startTime":3340.8,"endTime":3345.06,"body":"would have the same value as"},{"startTime":3340.8,"endTime":3345.06,"body":"1.3p today, more than a penny."},{"startTime":3345.06,"endTime":3346.23,"body":"Interesting."},{"startTime":3346.23,"endTime":3348.03,"body":"Let's think about the other"},{"startTime":3346.23,"endTime":3348.03,"body":"coin that was discontinued,"},{"startTime":3348.03,"endTime":3350.763,"body":"the lovely farthing, discontinued in 1960."},{"startTime":3351.78,"endTime":3353.34,"body":"One pound in 1960"},{"startTime":3353.34,"endTime":3356.31,"body":"would be worth 16 pounds and 18p today."},{"startTime":3356.31,"endTime":3358.29,"body":"That little asterisk is to remind me"},{"startTime":3358.29,"endTime":3360.03,"body":"that the inflation calculator goes to"},{"startTime":3360.03,"endTime":3362.46,"body":"the end of the last full calendar year."},{"startTime":3362.46,"endTime":3365.55,"body":"So I'm talking now in October, 2022,"},{"startTime":3365.55,"endTime":3367.35,"body":"this is up to the end of 2021."},{"startTime":3367.35,"endTime":3369.65,"body":"So things will only have"},{"startTime":3367.35,"endTime":3369.65,"body":"got worse since then."},{"startTime":3370.92,"endTime":3372.96,"body":"Let's say it's worth"},{"startTime":3370.92,"endTime":3372.96,"body":"about 40 pounds today."},{"startTime":3372.96,"endTime":3376.68,"body":"So a farthing, back in 1960,"},{"startTime":3376.68,"endTime":3378.84,"body":"that's equivalent to 1.7p today."},{"startTime":3378.84,"endTime":3381.87,"body":"So actually, not only should"},{"startTime":3378.84,"endTime":3381.87,"body":"we be getting rid of the penny,"},{"startTime":3381.87,"endTime":3384.42,"body":"we might have to get rid of"},{"startTime":3381.87,"endTime":3384.42,"body":"the 2p as well, and just,"},{"startTime":3384.42,"endTime":3387.6,"body":"you know, get rid of"},{"startTime":3384.42,"endTime":3387.6,"body":"those coppers altogether."},{"startTime":3387.6,"endTime":3390.48,"body":"We'd miss them, but maybe"},{"startTime":3387.6,"endTime":3390.48,"body":"we would like our purses"},{"startTime":3390.48,"endTime":3391.83,"body":"being a bit lighter."},{"startTime":3391.83,"endTime":3393.03,"body":"We could manage without them."},{"startTime":3393.03,"endTime":3394.77,"body":"You might think but how are"},{"startTime":3393.03,"endTime":3394.77,"body":"we going to pay for things"},{"startTime":3394.77,"endTime":3398.67,"body":"that have, you know, whose"},{"startTime":3394.77,"endTime":3398.67,"body":"values are 99p or something."},{"startTime":3398.67,"endTime":3399.503,"body":"That's okay."},{"startTime":3399.503,"endTime":3402.9,"body":"I mean, Canada actually got"},{"startTime":3399.503,"endTime":3402.9,"body":"rid of their one cent coin"},{"startTime":3402.9,"endTime":3405.15,"body":"a long time ago in 2013."},{"startTime":3405.15,"endTime":3408.42,"body":"Their smallest coin now is five cents."},{"startTime":3408.42,"endTime":3410.94,"body":"And what happens is you can"},{"startTime":3408.42,"endTime":3410.94,"body":"still have things priced"},{"startTime":3410.94,"endTime":3412.26,"body":"at any amount you want."},{"startTime":3412.26,"endTime":3414.3,"body":"You can still deal electronically"},{"startTime":3414.3,"endTime":3416.49,"body":"in whole numbers of pennies or cents,"},{"startTime":3416.49,"endTime":3418.08,"body":"but when you get change in a shop,"},{"startTime":3418.08,"endTime":3419.97,"body":"it's just rounded to the nearest five."},{"startTime":3419.97,"endTime":3423.21,"body":"And that's, you know, no"},{"startTime":3419.97,"endTime":3423.21,"body":"one worries about that."},{"startTime":3423.21,"endTime":3427.08,"body":"So we might, it might be time"},{"startTime":3423.21,"endTime":3427.08,"body":"to say goodbye to the penny."},{"startTime":3427.08,"endTime":3430.35,"body":"But, you know, one day we"},{"startTime":3427.08,"endTime":3430.35,"body":"may not use coins at all."},{"startTime":3430.35,"endTime":3433.11,"body":"I think that day is happily quite far off."},{"startTime":3433.11,"endTime":3434.79,"body":"Coins are so interesting and fascinating,"},{"startTime":3434.79,"endTime":3438.06,"body":"and they're bits of history"},{"startTime":3434.79,"endTime":3438.06,"body":"and mathematics, as we've seen,"},{"startTime":3438.06,"endTime":3439.32,"body":"in our pockets."},{"startTime":3439.32,"endTime":3442.56,"body":"So I hope you've enjoyed this"},{"startTime":3439.32,"endTime":3442.56,"body":"mathematical guide to coins."},{"startTime":3442.56,"endTime":3446.19,"body":"In my next lecture in the"},{"startTime":3442.56,"endTime":3446.19,"body":"Maths and Money series,"},{"startTime":3446.19,"endTime":3448.62,"body":"we're looking at the"},{"startTime":3446.19,"endTime":3448.62,"body":"mathematics of game theory,"},{"startTime":3448.62,"endTime":3451.8,"body":"which is how maths can help"},{"startTime":3448.62,"endTime":3451.8,"body":"us to find the best strategies"},{"startTime":3451.8,"endTime":3454.41,"body":"when we're buying, selling, bargaining,"},{"startTime":3454.41,"endTime":3456.33,"body":"bartering and auctioning."},{"startTime":3456.33,"endTime":3459.284,"body":"So thank you very much and"},{"startTime":3456.33,"endTime":3459.284,"body":"I hope to see you then."},{"startTime":3459.284,"endTime":3463.266,"body":"(audience applauding)"},{"startTime":3463.266,"endTime":3467.28,"body":"- When you were talking earlier"},{"startTime":3463.266,"endTime":3467.28,"body":"on about the people cheating"},{"startTime":3467.28,"endTime":3470.67,"body":"with various fake coinage,"},{"startTime":3470.67,"endTime":3472.59,"body":"there was a question that came in about"},{"startTime":3472.59,"endTime":3477.06,"body":"how about an alloy of gold,"},{"startTime":3472.59,"endTime":3477.06,"body":"silver, and a denser metal"},{"startTime":3477.06,"endTime":3479.43,"body":"in such a proportion"},{"startTime":3477.06,"endTime":3479.43,"body":"that the overall density"},{"startTime":3479.43,"endTime":3481.56,"body":"is the same as that of gold."},{"startTime":3481.56,"endTime":3482.799,"body":"- Yes."},{"startTime":3482.799,"endTime":3484.29,"body":"So, and that's, yeah,"},{"startTime":3482.799,"endTime":3484.29,"body":"that's a great question."},{"startTime":3484.29,"endTime":3488.82,"body":"So you could imagine potentially"},{"startTime":3484.29,"endTime":3488.82,"body":"being able to put in,"},{"startTime":3488.82,"endTime":3491.58,"body":"manage to mix things, maybe"},{"startTime":3488.82,"endTime":3491.58,"body":"do a bit of allegation,"},{"startTime":3491.58,"endTime":3493.92,"body":"but for bad and nefarious purposes,"},{"startTime":3493.92,"endTime":3496.53,"body":"would allow you to get something"},{"startTime":3493.92,"endTime":3496.53,"body":"that matches the density"},{"startTime":3496.53,"endTime":3498.18,"body":"and therefore the mass."},{"startTime":3498.18,"endTime":3500.16,"body":"So depending on what metals you use,"},{"startTime":3500.16,"endTime":3503.19,"body":"potentially you might be"},{"startTime":3500.16,"endTime":3503.19,"body":"able to detect in the color,"},{"startTime":3503.19,"endTime":3506.58,"body":"the luster, what it actually"},{"startTime":3503.19,"endTime":3506.58,"body":"looks like, how soft it is."},{"startTime":3506.58,"endTime":3509.91,"body":"So lead for example, is"},{"startTime":3506.58,"endTime":3509.91,"body":"potentially quite soft."},{"startTime":3509.91,"endTime":3512.31,"body":"If you're really worried"},{"startTime":3509.91,"endTime":3512.31,"body":"about a batch of coins,"},{"startTime":3512.31,"endTime":3515.7,"body":"then you can do an assay,"},{"startTime":3512.31,"endTime":3515.7,"body":"you can test them chemically."},{"startTime":3515.7,"endTime":3518.01,"body":"If the testing of the"},{"startTime":3515.7,"endTime":3518.01,"body":"densities and the volumes"},{"startTime":3518.01,"endTime":3518.94,"body":"doesn't do it for you."},{"startTime":3518.94,"endTime":3521.61,"body":"So there are ways still"},{"startTime":3518.94,"endTime":3521.61,"body":"to detect those things."},{"startTime":3521.61,"endTime":3523.58,"body":"It might just be if someone"},{"startTime":3521.61,"endTime":3523.58,"body":"was super clever like that,"},{"startTime":3523.58,"endTime":3525.36,"body":"it might be a bit harder, yeah."},{"startTime":3525.36,"endTime":3526.83,"body":"- Thank you so much, Professor Hart."},{"startTime":3526.83,"endTime":3529.08,"body":"I'm very sorry to say"},{"startTime":3526.83,"endTime":3529.08,"body":"we are out of time now"},{"startTime":3529.08,"endTime":3532.26,"body":"for any further questions,"},{"startTime":3529.08,"endTime":3532.26,"body":"but thank you and please,"},{"startTime":3532.26,"endTime":3533.73,"body":"I hope we will see you all again"},{"startTime":3533.73,"endTime":3537.36,"body":"on Tuesday the 22nd of"},{"startTime":3533.73,"endTime":3537.36,"body":"November, for Professor Hart's"},{"startTime":3537.36,"endTime":3538.92,"body":"Maths of Game Theory."},{"startTime":3538.92,"endTime":3540.715,"body":"Thank you so much."},{"startTime":3540.715,"endTime":3543.059,"body":"(audience applauding)"}]}