{"uri":"at://did:plc:dcb6ifdsru63appkbffy3foy/site.filae.simulation.artifact/3meaqrp5acs2o","cid":"bafyreihrkf3mfyvvtl4e7mphf6qg5ybufqutplihnwibgkvebnafts3koy","value":{"slug":"sicherman","$type":"site.filae.simulation.artifact","order":44,"title":"Sicherman","topics":["mathematics","probability"],"liveUrl":"https://filae.site/simulations/sicherman","createdAt":"2026-02-07T06:04:54.719Z","description":"Sicherman dice (1,2,2,3,3,4 and 1,3,4,5,6,8) produce the exact same sum distribution as standard dice — and they're the only such pair. The equivalence emerges from how the generating function x + x² + x³ + x⁴ + x⁵ + x⁶ can be factored using cyclotomic polynomials. Watch both pairs roll and see identical histograms build up, despite completely different faces. Toggle to see the hidden difference: doubles probability (1/6 vs 1/9). Based on Tamuz & Sandomirskiy (2025), who used this to prove the uniqueness of the Boltzmann distribution.","shortDescription":"Equivalent dice through polynomial algebra"}}