# Sunday Times Teaser 3098 – Wonky Dice

*by Michael Fletcher*

#### Published Sunday February 06 2022 (link)

I have two octahedral (eight-faced) dice. There are positive whole numbers (not necessarily all different) on each of their faces. I throw the two dice and add the two numbers that are showing on the top faces. The probability of a total of 2 is 1/64, the probability of a total of 3 is 2/64 and so on. The probabilities of the totals are the same as for a pair of standard octahedral dice (each numbered 1 to 8). Interestingly, however, the highest number on the first die is 11.

What are the eight numbers (in ascending order) on the second die?

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Fast to write, slow to run.

There is an elegant theory for analysing dice combinations using generating functions. This program uses this approach to give a faster solution using my Python polynomial library. This runs in 15 milliseconds (profile timing).

There are four fair octagonal dice pairs:

using recursion seems fast

Quite fast and even a lot faster (10ms) with: