Can You Win the Dice Tournament?

January 29, 2016 | Elizabeth Knowles

A friend challenges you to a tournament. You can pick any die you want. Both of you roll and the person who rolls highest wins a point. Do you pick your die first or last?

Better think before you pick!

According to rumor, Warren Buffett once challenged Bill Gates to a similar game. Each die was equally weighted so it was just as likely for each side to be rolled.

The question isn’t which die you should pick to play with, but rather whether to pick first or last. You may be thinking that choosing first is always to your advantage — you have more choices after all — but I’ll give the answer away straight away and tell you that that isn’t always the case.

If you think about a game of rock-paper-scissors for example, it is obvious that you don’t want to be the person to choose first if you aren’t playing simultaneously. Each option has another option in the cycle that can beat it: If you choose rock, your opponent will choose paper. If you choose paper, your opponent will choose scissor, and if you choose scissors, your opponent will choose paper. There is no one strategy that will win.

These dice form a similar sort of cycle. Let’s consider them one pair at a time.

The green die (3, 3, 3, 3, 3, 3) is likely to lose when played against the red die (0, 0, 4, 4, 4, 4). Why? Because green always rolls 3s and red has four faces that beat 3s.

The red die (0, 0, 4, 4, 4, 4) is likely to lose against the yellow die. (1, 1, 1, 5, 5, 5). Why? Because in one third of the cases, red will roll 0 and lose against yellow. In the other two thirds of the cases, they are evenly matched — each is just as likely to win. So, red will only win in one third of all of the cases.

The yellow die (1, 1, 1, 5, 5, 5) is likely to lose against the blue die (2, 2, 2, 2, 6, 6) Why? In half of the cases, yellow rolls a 1 and loses against blue. In the other half of the cases, yellow still only wins against four out of the six numbers blue can roll. Thus, yellow only wins ⅔ * ½ = ⅓ of the time.

The blue die is likely to lose against the green die. Why? Green always rolls a 3, so we know that that beats four out of six of blue’s possible rolls.

We end up with a cycle — just like in rock paper scissors.


Red die beats green, green beats blue, blue beats yellow

So, if you pick first, your opponent can always pick a die that is apt to do better than yours. Of course, you are only planning for ten rounds so there is no way to say for sure who would win. But, if you had all the time in the world and wanted to play an infinite number of rounds, you could figure it out before you started.

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