Sunday, July 21, 2013

The Blank and Regular Die Puzzle

Q:You have two dice. One of them is a regular one numbered from 1 to 6. The other one is blank on all sides. How do you number the blank one with two distinct numbers such that when you roll both dice, the sum of the number that shows up is almost evenly distributed?

A Book on Statistical Inference

A: If you assume that the blank dice was a regular one, then it is easy to see why the distribution of the sum would look like a triangular wave. There are more combinations of numbers between 1 and 6 that sum to 6 than there are to 2 and 12. For example, {1+5, 2+4, 3+3} all add up to 6 whereas only {1+1} add up to 2 and {6,6} add up to 12 respectively. Interestingly, if you insert the numbers 1 and 2 on three faces each you will get a fairly even distribution of the numbers. A comprehensive case by case enumeration is shown below.

First Roll Total @ Second Roll = 1 Total @ Second Roll = 2
1 2 3
2 3 4
3 4 5
4 5 6
5 6 7
6 7 8

Notice, the distribution of numbers 3 through 7 is exactly two whereas 2 and 8 would occur just once.

A Course in Probability Theory, Third EditionCovered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online


  1. The problem asks for an evenly distributed result, not almost evenly distributed.

    0 and 6

  2. Hmmm. You are right. Will make that edit. Thanks for the feedback