What the Brainteaser Is

As you are given two dice — one normal one, and one blank — you are to label the blank die so that you can roll both of the dice together and add them up to a whole number that is from one to twelve. All numbers should have an equal chance of being rolled.

You are allowed to use a number more than once or skip it entirely. For example — you can label the blank sides of the die with one, one, two, three, four, five while skipping number six altogether. However, you do have to label each one of the six faces.

## How to Solve the Brainteaser

There are 36 equally likely possible outcomes when rolling a pair of six-sided dice. This is figured out by multiplying six by six. For this particular brainteaser, you need to divide the 36 sums into 12 sums that are equally likely to occur since you want 12 numbers to have equal probability. This means that each sum has to appear three times or 36/12.

To have 12 occur three times, number six — the largest possible number — needs to appear three times. Using similar logic to get number one to appear three times, you have to use the smallest number that you have — zero — a total of three times.

To correctly solve the brainteaser, you need to label the blank die with the numbers 0,0,0,6,6,6.

There is also a way to solve this brainteaser using probability generating functions.