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.
April 15, 2026
Original Paper
Schuttes property for sets of tournaments and an application to dice games
arXiv · 2604.08790
The Takeaway
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.
From the abstract
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,