The Three Indistinguishable Dice Puzzle


This week I have a puzzle for you that I haven’t solved myself. But, I still know there is definitely a solution. I just don’t know if it’s a nice solution or not. When my maths buddy Lucas showed me this dice… Basically it’s one clear cube which has three tiny cubes Inside it and the idea is instead of rolling three dice which could scatter in all different directions, you just roll one cube and there you are, three dice, eleven. When Lucas showed me this three dice in one, he said he was wondering “let’s say you were at home and you want to play a game that requires a single dice” [ha] well actually he would have said “a single die”, because he’s more correct than I am whereas personally I never say die, and so he said “how could you use the three-in-one dice to simulate a single six-sided dice”. So that’s the first half of the puzzle How could you use this three dice in one to simulate a single D6 dice and I don’t want any cheating by which I mean thinking outside of this tiny box, right, this is not a lateral thinking problem It’s a maths problem. I want a mathematical solution of course in the real world if you had this and you wanted to have a normal six sided dice you just get a marker and write the numbers one to six on the outside, and then roll it normally. No, I want a maths solution. To state it a bit more precisely, you have three indistinguishable dice. You roll them simultaneously, can you have an unambiguous way that everyone playing the game can agree on to look at the three results and immediately know, “oh!, that’s equivalent to a” you know four or six or something, right? How can you do that? I do have a solution to this first half of the puzzle and I’m going to give it to you in just a moment. If you want to have a go yourself pause the video now and, uh, have a go because Lucas and I stood out there for a while rolling the dice and he also had a solution to this simple case where you’ve got to take three down to one and what you do is after you’ve rolled it you simply add the values together. Now, of course that might give you a number bigger than 6 and if you ever get a number bigger than 6 You just take whatever the remainder is once you’ve divided 6 into the number. We say it’s the answer Mod 6 or Modulus 6. Ok, so you roll the dice, and you have a look at the result and you go look at that I’ve just rolled a 13 which mod 6 is one piece of cake. After a while you can get pretty quick because used to ignore the sixes. I’ve just rolled a three, ignore the fact there’s a dot missing on the two. That’s the problem with a sealed container You can’t get in there to fix any of the dice. On to the slightly harder second half of the puzzle that I don’t have a solution to and It’s exactly what you’d expect. How would you use this to completely fairly simulate… two dice being rolled. So you roll three indistinguishable dice how can you collapse that down, fairly, to two dice the same that you would get if you [just] rolled two originally and again, no physical solutions. I mean, I guess in reality if you had to roll three you know to you could just say oh I’ll take whichever two are closest together or you could mark a dot or a line in whichever ones are closest to that. There are ways around the problem. I want to know mathematically, how do you do it in theory? How do I know there is definitely a solution even though I haven’t found it. Well, I know there’s definitely an ugly, crude, solution because when you roll three dice you’re going to get one of 216 possible results, and you could in theory make a giant list of one of those 216 results, and then you can use that as a kind of lookup table. So you roll the dice, you go “oh!, I just got a 4 1 1”, you look up 4 1 1 in the massive list and go “Oh, that’s a 3” or something. Now, there are some slight problems here. You need to put 216 possible results into 36 different categories and some of them are indistinguishable, so this one 1 1 4, that I just rolled, well I mean, I don’t know which dice is which and so I need to put all the results I can’t distinguish between in the same category. So if I put them all in buckets where all the indistinguishable ones are in the same bucket and then I assign each bucket to one of the possible 36 normal results from two dice. And then you end up with this massive table that you and your friends can just look up while you’re playing the game. But you’ve got to admit, that’s a bit disappointing. Just listing all the possible results of matching them up. We need a better way, and that’s what we did for the one dice solution. When you add them together and mod six there is a shortcut to effectively memorize the way you’ve put all the values in buckets. You could then use that to write out the complete list of how every possible result matches to a new result but because you can add and divide by six fairly quickly you don’t need to. And so the true puzzle here is, is there a way to take the three results from the indistinguishable dice and map them quickly and easily onto the possible results from just rolling two dice, so every result has the same probability of coming up. So there you are have a go I have shown this puzzle to a few other people I did a talk at the university of the west of England in Bristol last week and a few of the guys there got very excited about this they’ve been emailing me their findings so far, And it looks like there are better ways than just memorizing a lookup table So have a go. If you think you’ve got a solution just put it in the comments underneath don’t worry about spoiling if everyone else we can all work together I wanna find the best way to solve a problem. Which isn’t really a problem. And by the way, I haven’t forgotten and to announce who won the, guess how many times I had to flip a coin to get ten in a row, From this video here when I passed ten to the five subscribers. I will get that done soon. I guarantee it it’s just I’ve been ridiculously busy recently partly, very excitingly, because I’m writing a, uh, Radio program for BBC radio four, vestal the Spoken Nerd which I’m a part of. We’ve got a four part series On Radio 4 which will be coming out in july. Which is very exciting but it means I’m now brain deep in writing math jokes for the radio so, but that’ll be out soon. It’ll be great I’ll keep making videos just I’m sorry everyone who sent me an email or a message And I haven’t responded in a timely fashion. I will get right onto it, hopefully soon I’ll have more time to do more videos, but I will announce the winners to the competition soon

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