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 3 essays across 2 fields — 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 Heights from ω² down to nought. Thirteen Sylver Coinage positions with the gcd of the named numbers, its prime factors counted with repeats, the gaps of the semigroup scaled by the gcd, and the ordinal height those give: ω² for the empty position, ω·4 for {16}, ω for {2}, 3 for {3, 5} and 0 for {2, 3}.

No number bounds it

Sylver Coinage cannot be given a termination measure in the natural numbers: from {2} the game can last one move, two, or any number the second player chooses. The measure proposed for it — the gap count beside the number of distinct primes the named numbers share — rises on {4} + 6. Count the primes with their repeats and it falls on every one of 42,519 moves checked, and it is exact: the game tree is ω squared tall, and the height of a position is ω times the prime factors of its gcd plus the gaps that are left.

limits · Termination

Named alongside it

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

CutcakeDisjunctive sumExhaustive searchInductionMaundy cakePartizanProofSimplicity ruleBinary bandClosed formCold gameCounterexample

All concepts