Math has unlocked a way to build a set of 'cheat' dice where you can let your opponent pick first and still guarantee you’ll beat them.
We usually think that if Die A beats Die B, and Die B beats Die C, then Die A must be the best of the bunch. This paper uses a complex math concept called 'Schutte's property' to prove that logic is a total lie in certain games of chance. The researchers created a formal recipe for sets of dice where every single die has a specific counter-die that crushes it. It’s not just a lucky guess—the math provides a controllable, unfair edge that ensures you always have the upper hand. This means you could build a game where the person who picks first is mathematically doomed to lose, regardless of how fair the dice look. It’s a literal blueprint for an 'unbeatable' game that proves our basic intuition about probability is fundamentally broken.
Schuttes property for sets of tournaments and an application to dice games
arXiv · 2604.08790
A tournament has Schuttes property $S_k$ if for every set of $k$ vertices, there is a vertex which dominates the set. In 1963, Erdos provided bounds for $f(k)$, the smallest order of an $S_k$ tournament. Schuttes property has various applications, including the design of unfair dice games. A set of dice introduced by James Grime motivates a generalization of Schuttes property to sets of tournaments: a set of tournaments on the same vertex set has property $S_k$ if for every set of $k$ vertices,