Combinatorial Game Theory
who wins, and by how much
Concept

Prime factors — where it appears

The primes a whole number is built from, counted with multiplicity, written Ω when only the count is wanted. Maundy Cake is the game on this site whose values are read off them: the count decides the sign and the primes themselves decide the size.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Maundy Cake: the pieces must be equal. The same cake as Cutcake, cut by the same two players, with one extra rule: a cut must divide the cake into equal pieces, and every piece stays in play. The values are still whole numbers, but the arithmetic that decides them is not Cutcake's — it counts prime factors rather than binary digits.

Maundy Cake

Cutcake with one word added: a cut must divide the piece into equal parts. The values are still whole numbers, and the rule this site has been repeating about them is false — over all 1,296 cakes to 36×36 the largest-odd-divisor account has 946 counterexamples. What survives is a count of prime factors, and it says who wins without saying by how much.

positions · Cutcake Term by term. Every cut of one long side, with its value written as running products beside the terms of the largest-prime cut.

The short side is not in the lemma

The closed form for a two-sided Maundy Cake rested on one unproved statement: that no divisor beats the largest prime. Written out, that statement never mentions the short side — it is an inequality between a multiset of primes and a term count — and once it is stated that way it has a two-line proof, term by term. The ladder ends in a theorem rather than a grid.

positions · Cutcake

Named alongside it

The objects these essays reach for when they reach for this one.

CutcakeDisjunctive sumMaundy cakePartizanSimplicity ruleBinary bandClosed formCold gameCounterexampleEnumerationExhaustive searchInduction

All concepts