# A Dice Puzzle

Today I have a wonderfully counterintuitive puzzle to share!

You and a friend each throw a dice. Each of you can see how your own die landed, but not how your friend’s die landed. Each of you is to guess how the other’s die landed. If you both guess correctly, then you each get a reward. But if only one of you guesses correctly, neither of you get anything.

The two die rolls are independent and you are not allowed to communicate with your friend after the dice have been thrown, though you can coordinate beforehand. Given this, you would expect that you each have a 1 in 6 chance of guessing the other’s roll correctly, coming out to a total chance of 1 in 36 of getting the reward.

The question is: Is it possible to do any better?

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(Spoiler space)

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The answer is that remarkably, yes, you can do better! In fact, you can get your chance of getting the reward as high as 1 in 6. This should seem totally crazy. You and your friend each have zero information about how the other die roll turned out. So certainly each of you has a 1 in 6 chance of guessing correctly. The only way for the chance of both guessing correctly to drop below 1 in 36 would be if the two guesses being correct were somehow dependent on each other. But the two die rolls are independent of one another, and no communication of any kind is allowed once the dice have been rolled! So from where does the dependence come? Sure you can coordinate beforehand, but it’s hard to imagine how this could help out.

It turns out that the coordination beforehand does in fact make a huge difference. Here’s the strategy that both can adopt in order to get a 1 in 6 chance of getting the reward: Each guesses that the others’ die lands the same way that their own die landed. So if my die lands 3, I guess that my friend’s die landed 3 as well. This strategy succeeds whenever the dice actually do land the same way. And what’s the chance of this? 6 out of 36, or 1 out of 6!

1 1       2 1       3 1       4 1       5 1       6 1
1 2       2 2       3 2       4 2       5 2       6 2
1 3       2 3       3 3       4 3       5 3       6 3
1 4       2 4       3 4       4 4       5 4       6 4
1 5       2 5       3 5       4 5       5 5       6 5
1 6       2 6       3 6       4 6       5 6       6 6