## Math + Dice

Lee over at PrimePuzzle pointed me to this cool set of dice that a mathematician devised that decides in one roll who (for up to four players) goes first. The Go First dice are designed to produce no ties, and give everyone an equal chance at going first. What’s cool is that they work for 2, 3, and 4 players.

Questions: how would one construct such a set, and can we construct a set for larger numbers of players? Also, these dice look to be 12-sided. Could the same be done with only 4-sided dice?

This entry was posted on Monday, January 16th, 2012 at 10:27 am and is filed under math gear, question.
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Thanks so much for posting. The key question is “How do you construct the set?” I’ve managed to construct one set but I can’t seem to figure out how to construct all of them (I think there are 12 solutions).

45 41 39 38 (the largest 4 faces in die a)

42 46 37 40 (the largest 4 faces in die b)

34 36 47 44 (the largest 4 faces in die c)

35 33 43 48 (the largest 4 faces in die d)

You can get the next 4 faces by subtracting 16 from all the numbers and the next 4 faces by subtracting 32 from all the numbers.

Notice:

d>c>b>a in column 4

c>d>a>b in column 3

b>a>c>d in column 2

a>b>d>c in column 1

A simulation of this solution may be seen at

http://primepuzzle.com/tc/cards-doa.html

I am sooooooo looking forward to finding all the solutions. Especially the one that’s shown in the image above!

You are so cool! I love the show-rolls option. Listen, I’m curious about that 12 different solutions. I’m not sure what you are talking about. My intuition is that there are an infinite number of solutions. Just take any number and add it to all of the die faces above, wouldn’t that produce a “different” solution. Or do you have a more abstract notion of “different”?

The die may only have the numbers 1 thru 48 on their faces.

After studying the output, I noticed the odds weren’t quite even in the two person game. I then filled in the dice numbers in a slightly different way which still obeyed the inequalities listed above. I also added a new feature which lets you use another dice set which is generated from the first dice set by swapping 3 elements in die c with 3 elements in die d.

The highest values are now

41 47 38 36 (the largest 4 faces in die a)

45 43 34 40 (the largest 4 faces in die b)

37 35 46 44 (the largest 4 faces in die c)

33 39 42 48 (the largest 4 faces in die d)

I believe these dice are now fair. This was a difficult problem. And therefore a lot of fun!

I’ve removed the option which gives you another set of dice. I may bring it back as soon as I can figure out how to get it to produce another set of fair dice

The add one more die function would be awesome. I’m surprised your construction before didn’t produce “fair” results. When you say you filled the faces in differently, what do you mean? This is a great problem.

http://primepuzzle.com/tc/dice-block.html

The above link is my attempt to explain how all this works. If you study it, I hope and think you’ll understand it. It took a while for me to get to the point where I could explain it (like about 4 days!). As for my remark about filling the faces in differently, I might have filled in the numbers in such a way that the required inequalities didn’t hold. I’ve long since lost those original numbers so it’s probably best just to disregard the remark! Adding one more die scares me at this stage but maybe …