These 27 tickets guarantee a win on the Lottery

[Music] new maths you need 27 tickets to guarantee a win on the UK National Lottery it checks out let's go [Music] in the UK National Lottery there are 59 numbers of which you have to select six and of course get them right to win the main prize and the paper absolutely does show come on that there are hang on oh it's the dog here we go there are 27 tickets such that if you buy those you are guaranteed to win except you're not guaranteed to win the main prize you're guaranteed to win hey you're not getting a coffee come on come on you're guaranteed to win a prize and the smallest prize you can win is matching two numbers and the fine people over at the a periodical random numbers and worked out that 75 of the time that's all you're going to win two numbers in fact in a further 23.5 the time the best you'll have is three numbers matched and you're still going to lose money and only 1.5 of the time included a strategy where you actually make money hey future Matt here yes Stephen I got to work on this one day thing so I misspoke slightly it's 98.5 chance you will match uh three or fewer numbers across all the tickets but that might happen multiple times so actually you can match three numbers across enough tickets to turn a profit so you don't just make a profit one point five percent of the time Peter roll it a periodical very kindly run the numbers again for me it's actually five percent of the time it's still not good but is that the point but that's not the point the point is the maths is super interesting and it's a nice challenge see if you can work out for 27 sets of six numbers how do you guarantee that for any new set of six numbers there's definitely at least one set that has at least two matches uh I mean it's not straightforward it does use math that we've had in the channel before if you want to give it a go and you've got until I buy 27 lottery tickets and go back to the office to work it out all right let's go this way come on foreign [Music] lottery tickets by copying numbers meticulously off a scrap of paper when people are staring I felt like saying it's okay I have a system but I didn't know if that would help but anyway I'm now back in the office and we have uh there's all my games all set ready to go now I've got one last confession to make in this video and that is that the math behind why this works for 27 games is not the most impressive part of this paper in fact the first three pages covered the introduction and the math behind why that works the remaining oh goodness 13 14 pages of very complicated mathematics is proving that you cannot do this with 26. yes the real achievement is just showing that 26 doesn't work otherwise you got 27 people are like well hang on can you can you do it even better and that makes to prove you cannot so 27 is interesting not because it's possible because that's the smallest number for which it is possible and uh time is up if you're trying to work out how it's done it is using a bit of mathematics obviously on the channel before they're choosing a finite projective plane it's the Pharaoh plane from before from the double video you know the double it's the same idea a double is how do you get it such that any two cards always have a symbol in common and this is how do you have a bunch of you know tickets such that whatever the six numbers you get they two of them will have like a ticket in common so it's a very similar problem but slightly different in how you set it up so the failure plane if you have not met it before you've got seven boxes linked by seven lines one of which is a circle and if you go through all seven lines you'll see there's one there one two three four five six seven and if you pick any two boxes on there they are definitely joined by one line and what we can do is fill some numbers in and it means if you choose any two numbers and they go from one to fourteen on here they will be joined by one line and because each line has three boxes and each box has two numbers it means each of these lines now has six numbers each of those lines is one game one set of six on a ticket and so if we have one two three four five six seven tickets we will have covered every single option here and that's got us from one through to 14 we just need a few more of those so we do two more final planes and each one's got the next 14 numbers we're now up to 42 numbers and if you're thinking well you just have a fourth fanoplane each of these represents seven tickets so we're up to 21 tickets you could do it with another one but then you're up to 28 tickets in fact you wouldn't quite be all the way to 59 and they were trying to prove you could do it with 27 because that's the smallest number for which it was possible and so all you do is you take advantage of the facts you don't have to have only two numbers in each box so they then did a really reduced finite plan it's only got three points they're all joined each of them have three numbers so now you've got three lines in each of those three lines has six numbers so what are we up to now that's another uh three tickets and then you have one more but this time you've got a vanishing point and the vanishing point is just included in every single one of the three tickets and these will have two so they're all six tickets each and now we're up to 59 altogether and that's it there are 27 tickets and when the draw happens if you've bought all 27 lines as individual tickets like I have done you just wait for two numbers from the draw to both be in the same one of these shapes and the moment there are two numbers in the same shape we know there's definitely a ticket corresponding to a line that will join them together we have one on two numbers except what if the draw doesn't give us two numbers in the same shape well they have to draw six numbers and there's only five shapes so worst case scenario there's one in each of these shapes and the sixth one has to be the second number in one of them it's the pigeonhole principle so many lovely delightful bits of mathematics means that you can guarantee these 27 games will win at least two numbers okay another confession so the numbers I've done are different to the ones in the original paper so I numbered the thano planes first starting at one then I did the two unusual cases whereas they numbered the unusual cases first from one and then did the final planes and so it means my numbering is slightly different to this numbering which actually is good because if there is one bit of mathematics behind winning the actual main prize on the lottery which is incredibly unlikely is that you don't want to pick the same numbers as someone else otherwise you have to share the price and this mathematics was only done this week tonight is the first lottery draw since this mathematics was done and I can't be the only mathematical Dingus who thinks it's hilarious to go out and buy the tickets even though it's in here he was thinking well I said was a one and a half percent chance five percent to make a profit that's not the point point is the maths but if I took the ones from the paper and they did they write them all up so there's actually a whole list here I could have just taken that list of numbers from the paper into the news agent and ordered those anyway so for a start I've got my renumbering now I could have just you know done a substitution on that instead I wanted to renumber the planes because I think that's more interesting here you've got nine tickets and eight tickets that's 17 spots available and there are 17 prime numbers below 60. two there's exactly 17 prime numbers to choose from and there's 17 spots to be filled here so this bottom panel here is all prime numbers so nice which means that the tickets that correspond to those those games only have primes and then realized there are seven square numbers put them in here annoyingly there are 10 triangle numbers but if you exclude three because we've already used that as a prime and you realize that both 1 and 36 are square and triangle it means there are precisely 14 non-prime square or triangle not exclusive or numbers so I can put them all on the same ones this is their primes square triangle numbers and these two are the rest um I I was I was about to start coding up some python to use the online encyclopedia of integers sequences API to work out two interesting sets they could be partitioned into and then I realized I've only got one day to make this video and maybe I'm getting distracted so I just put them all in here I did put all the ones that end in zero eight or two sorry zero eight or four together because I thought that was hilarious and the rest are there so they're my numbers there what I hope will win me some money tonight and given these tickets were paid for by my patreon supporters it feels only fair to say if I happen to win any money I promise to spend it all on making more videos and so I can't rescind that I'm going to Tweet the numbers before the draw so if you follow me on Twitter you'll see oh on X uh it's a weird time for social media I will also thread them on threads I will uh what do you do on Tick Tock I'll take them on Tick Tock I will reel them on Instagram I will toot them on Mastodon oh this is so ridiculous all those accounts are real by the way some of them quite new so if you are on any of those platforms please do go and follow me so I can get my numbers up but I will put these out into the world before the draw so you can make sure I definitely spend the money that I win on making more ridiculous videos with a higher budget than this one now we just have to wait until I think it's like 7 45 tonight and we'll see you here five one okay we are on I got my tickets at the ready uh there we go the draw is about to happen there's been one in pricing and they reckon there's millions of pounds it could be one in prizes so we're very excited by that cost of ability well I'll do those I'll take it back around to me I'm going to write the numbers down as they come out tonight four million four million four million pounds that feels deeply unlikely and the first number out is first number is 20 I'm glad I'm glad this is on the second Channel because we've reached the point where it's just 56 random numbers live and if you're thinking that the audio quality 37 he's better now than it was before eight it's because my microphone I recently updated the firmware and it reset the settings which included the automatically record setting so I was turning it on and off and recording all day but 33 and uh that wasn't enough for me at all so 25 is a supplementary number right which goes on which goes on the very end um as a supplementary number but I don't know I mean this is the pure maths so I think you know what out of principle I'm not gonna use 25 no wait no I'm an absolute purist let's color in uh the legitimate six numbers and see what I've won first up 20. I've done 20 and it only appears three times it's only though that's it that's it for 20s the other two no 20s that's the dog at the door hang on wanna help me check the numbers that one the dog realizes she's part of the come on Happy over here not over there over here round okay all there you make yourself comfortable that's better good girl good girl let's send here we go there we are who's that come to see me not make any money so where was I was doing 56 which again is not looking super promised oh there's one at least one yeah it makes sense if there is a number anywhere they're all clumped together because they're only in one place on one of the graphs so they have to be clumped all right 37 8. 8 21s 33. okay that's it and I why okay well you know what uh it's disappointing from a winning money point of view but it's great from a Math's point of view I won exactly once I have a single there's like there's none on there no two numbers that one's oh there's one at the top there that's once so they get twice then nope that was the ones none on there none on there single win not using the supplementary number so there it is as promised one win and guess what you get for two numbers another ticket so um I get one lucky dip in the lottery so that's a substantial loss for all involved so thank you so much for watching this ridiculous video again sorry about the bad audio quality before I only had a day to put it together no time for reshoots and I've had a lot of fun recently like I like the main Channel my highly produced videos I love making them so much and I have such an incredibly talented team I mean camera person Alex is way better at filming and editing than I am and producing a coal is way better at producing and project managing um and physics than I am and so like I love working with the team I love making fantastic videos and I've been so busy recently like we've filmed a bunch of videos I've just finished the first draft of my next book which will be out if the plan is April 2024 and if you're watching this far into this kind of ridiculous video low quality second Channel you're the kind of Hardcore fan who uh if you want there's a link below if you want to pre-register no obligation to be uh to be announced ahead of time when pre-orders are up for the special editions of my next book I will link to that below you feel like the kind of people that might what are you doing any other dog's trying to get my attention so uh I mean look at look at this do you think she's just disappointed are you disappointed I actually can't see where the camera's pointing so I'm guessing is that the dog there I feel like the camera's pointing at the dog now look at you do you want these do you want these now you're rightfully disgusted by them so uh anyway from myself and Skylab thank you so much for watching um this ridiculous video and hooray maths and don't Gamble

Channel: Matt_Parker_2

Views: 278,650

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Id: zYkmIxS4ksA

Length: 15min 31sec (931 seconds)

Published: Sun Jul 30 2023