Wythoff's game

An equivalent description of the game is that a single chess queen is placed somewhere on a large grid of squares, and each player can move the queen towards the lower left corner of the grid: south, west, or southwest, any number of steps.

Martin Gardner in his March 1977 "Mathematical Games column" in Scientific American claims that the game was played in China under the name 捡石子 jiǎn shízǐ ("picking stones").

[1] The Dutch mathematician W. A. Wythoff published a mathematical analysis of the game in 1907.

Wythoff discovered that the cold positions follow a regular pattern determined by the golden ratio.

Specifically, if k is any natural number and where φ is the golden ratio and we are using the floor function, then (nk, mk) is the kth cold position.

Wythoff's game is played with two piles of counters
A visualization of Wythoff's game of Nim. The bottom leftmost square is position (1,1) and the red squares are cold positions. Note that the winning square is not included in the picture.