Welter's Game Nim where no two heaps may be equal
Coins sit on a strip of squares numbered 0, 1, 2, … . On your turn slide one coin left to any empty square (jumping is fine). Whoever cannot move loses. Read the strip as a Maya diagram and the coins become a Young diagram: a coin moving from a to b removes a hook of length a − b. The value of the game is the nim-sum of 2ord₂ h + 1 − 1 over all hook lengths h (Sato), and the page uses it to mark every legal move ✓ winning or ✗ losing.
Young diagram λ — each cell shows its hook length
notes
Why it is Nim with a twist. One coin on square a is a Nim heap of size a. Several coins are Nim heaps that may never become equal, and that single rule breaks the plain nim-sum: Welter (1954) found the value is x¹ ⊕ … ⊕ xᵐ ⊕ ⊕i<j N(xⁱ − xʲ) with N(d) = d ⊕ (d − 1). Sato rediscovered it as the Maya game and rewrote it over hook lengths, the same numbers that give the dimension of an irreducible representation of the symmetric group.
Full positions. A position is full when its value equals the number of cells |λ|. By Irie (2018, using Macdonald) that happens exactly when the representation χλ has odd dimension, so the number of full positions of size n is OEIS A059867 — this page recounts it both ways.
p-saturations. Irie lets up to k − 1 coins move at once under a p-adic rule and
proves the value becomes the base-p hook count. The least k that suffices is unknown for
3 ≤ p ≤ m; the repository's tools/compute.mts sat computes it exactly on finite windows.
Source: github.com/sen-ltd/welters-game · MIT · by SEN LLC