Puzzles
Two piles of counters, one move that breaks all the rules of Nim — who wins?
Two players take turns removing counters from two piles: any number from just one pile, or — the twist — an equal number from both piles at once. Whoever takes the last counter wins. Is there a position you can always force your way into to guarantee victory?
Reveal the answer
Yes — the losing ('cold') positions follow a precise pattern built from the golden ratio: pile sizes of the form (⌊nφ⌋, ⌊nφ²⌋) for n = 0, 1, 2..., giving pairs like (1,2), (3,5), (4,7), (6,10). Dutch mathematician W. A. Wythoff published the analysis in 1907; the same game can be modeled as a chess queen sliding toward a corner.
— Willem Abraham Wythoff, A modification of the game of Nim — Nieuw Archief voor Wiskunde, 1907