Equivalent dice through polynomial algebra
Sicherman dice (1,2,2,3,3,4 and 1,3,4,5,6,8) produce the same sum distribution as standard dice — the only such pair. The equivalence emerges from how cyclotomic polynomials can be factored: x + x² + x³ + x⁴ + x⁵ + x⁶ has exactly two ways to split into legal dice. Watch both roll and see the histograms converge.
Source: Tamuz & Sandomirskiy (2025) "On the origin of the Boltzmann distribution", Mathematische Annalen. Sicherman dice discovered 1978.
Space: pause | Click: roll faster | R: reset | D: toggle doubles view