Sicherman Dice Sum Factorization
Unusual six-sided dice can reproduce every sum probability of ordinary dice while having very different faces.
Free theory articleUnusual six-sided dice can reproduce every sum probability of ordinary dice while having very different faces.
Concept & interpretation
Can two six-sided dice have completely different faces than ordinary standard dice, yet produce exactly the same probabilities for the sum of their rolls? Surprisingly, yes. Sicherman dice are a unique pair of non-standard dice that have this property, demonstrating how generating polynomials can be factored to discover identical probability distributions.
experiment
Explore this convolution interactively: https://riano.app/module/sicherman-dice-sum-factorization
- Select "Standard" in the "Dice pair" control. Observe that a sum of 7 has a probability of ~0.1667 (6 out of 36 outcomes).
- Change "Dice pair" to "Sicherman". Note that the histogram remains perfectly aligned with the standard distribution, and the probability of 7 is still ~0.1667 (6 out of 36), even though the individual means are 2.5 and 4.5.
- Select "Changed last face" in the "Dice pair" control. This changes Die B's faces to (1, 3, 4, 5, 6, 7). The "Same Entire Sum Law" indicates "No", and a sum of 12 now has 0 probability, proving that a specific factorization is required.