Sprague–Grundy values of King on even-by-even generalized staircases

Determine the Sprague–Grundy value of the King game on a generalized staircase with distinct even dimensions r and c, for every k ≥ 1, showing that the value is 2 when k is odd and 3 when k is even.

Background

The paper establishes formulas for the King game on several families of partitions but leaves certain generalized staircases unresolved. Conjecture 5.2 addresses generalized staircases whose two dimensions are distinct even integers.

The conjecture proposes a particularly simple parity law: the Sprague–Grundy value is 2 for odd k and 3 for even k. Proving this would extend the paper’s partial analysis of King positions on generalized staircases.