Amazon Uses This Tricky Dice Brainteaser in Job Interviews

Candidates are often asked to answer obscure or tricky math and probability questions during job interviews at tech companies like Google and Amazon. This dice problem, as demonstrated on YouTube by Presh Talwalkar, is one such example.

You are given two dice: one is normal, the other is blank. Your task is to label the blank die so that when you roll both of them together, the sum shows each whole number from 1 to 12 with equal chance. You can use a number more than once, or skip it entirely. For instance, you could label the sides of your blank die 1, 2, 3, 4, 4, 5, and not include 6. But you do have to label all six faces.

When rolling a pair of six-sided dice, there are 36 (6 x 6) equally likely possible outcomes. In order to succeed in this particular game, you need to divide those 36 sums into 12 equally likely sums, because you want 12 to occur with equal probability. So each sum has to appear 3 (36/12) times.

In order to have three instances of 12, the biggest number, you need the number 6 to appear three times. By similar logic, in order to get three instances of 1, the smallest number, you will need to have a zero on your blank die. So in order to win the game, the blank die needs to be labeled 0,0,0,6,6,6.

Talwalkar breaks down the logic of this probability puzzle in his video, and also outlines a different strategy (probability generating functions) for approaching the problem.

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