Cutcake — the series · Combinatorial Game Theory

Cutcake: every value an integer. The value of an m by n cake, for every small m and n. Left cuts down, Right cuts across, and neither player ever gains by moving — so nothing is ever at stake, every value is a whole number, and the number says exactly how many spare moves one player has.

Cutcake, where every value is a whole number

A partizan game in which no position is ever worth a fraction, a star or a fight. Every value is an integer, the integer is a count of spare moves, and the pattern it follows is decided by binary digits.

part 1 · positions

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.

part 2 · positions

Seven, ten, thirteen and sixteen. The four values the rung below named, each read off the prime factorisation: one plus the largest prime plus the product of the two largest.

The size of a cake

Ω gives the sign of a Maundy Cake and says nothing about the size, and the rung below left four values — 7, 10, 13 and 16 — unaccounted for. For a one-row cake they are a formula: write the prime factors largest first and add up their running products. The rule behind it is greedy — cut by the largest prime — and it is exact on every one-row cake to two hundred and wrong on a fifth of the two-sided ones.

part 3 · positions

Cut small unless you are behind

The rung below found the greedy rule — cut at the largest prime — wrong on 104 of 552 Maundy Cakes and asked for a description of them. On all 104 the best cut is at the smallest prime, the exact opposite. A middle divisor is never needed on any cake in a sixty by sixty grid, and which of the two extremes wins is decided by Ω alone: cut small when Ω(m) + 1 ≥ Ω(n), large otherwise, and that is exact on all 3,540.

part 4 · positions

Running products, and where to stop. Six Maundy Cakes with the prime factors of the longer side, the running products those primes make, and the value the sum of them gives.

The short side only says how many

The rung below settled which cut to make in a Maundy Cake and left the value open. With the cut settled the recursion is a walk, the walk unrolls, and what it unrolls into is the running products of the long side's prime factors, largest first. The short side never enters the products at all — it decides how many of them there are and nothing else, so sixty-two different short sides give one value.

part 5 · positions

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.

part 6 · positions

The two proofs, beside each other. Maundy Cake's rule was proved by restating its lemma so the short side vanished into a multiset of primes and a term count. Cutcake's rule takes the same five steps, with the multiset replaced by a binary length — one integer instead of a multiset — and the closing argument correspondingly shorter. The one line where they differ is which cut a reader would guess.

The obvious cut is the wrong one

Maundy Cake's rule was proved by restating its lemma so the short side vanished. Cutcake's collapses the same way — into a binary length instead of a multiset of primes — but the cut the argument needs is not the one the ladder predicted. Halving is wrong on a third of all cakes, and the smallest counterexample is six squares by two.

part 7 · positions