Wythoff's Game (Get Home)

Wythoff's Game is played on a chessboard. Two players take it in turns to move a piece. That piece can move any number of square to the left, and number of squares down, or any number of squares on a down-left diagonal. The winner is the player who moves the piece to the bottom-left square. What are the losing squares?

See my first video with Katie Steckles here https://youtu.be/pzlpi7lJi4k

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If we call the bottom-left square (0,0) then the losing squares are (1,2), (3,5), (4,7) and their reflections that swap the coordinates.

The losing squares can be generated one at a time using the following two conditions: First, for the nth losing square, the difference between its coordinates is n. And second, each positive integer appears once and only once as either the x or y coordinate of a losing square.

These two conditions have the effect of putting all losing squares on different rows, columns and diagonals.

In 1907, Willem Wythoff proved that the nth losing square has coordinates (n*phi, n*phi^2) where phi is the golden ratio (1.618), and the two coordinates are rounded down to the previous integer. He showed that the golden ratio is the only number that will work in this way, giving the desired two properties of losing squares.

Play an interactive version of the nim version of Wythoff's Game (called Last Biscuit here) on nrich: https://nrich.maths.org/1186

Wythoff's Game on Wikipedia:
https://en.wikipedia.org/wiki/Wythoff%27s_game

Wythoff's Game on Mathworld: http://mathworld.wolfram.com/WythoffsGame.html

Wythoff's Proof: https://archive.org/stream/nieuwarchiefvoo03genogoog#page/n219/mode/2up

An excellent series of blog posts by Zachary Abel, read in reverse order (Thanks to Daniel Kelsall for this link) http://blog.zacharyabel.com/tag/wythoffs-game/

Willem Wythoff: https://en.wikipedia.org/wiki/Willem_Abraham_Wythoff

Rufus Isaacs: https://en.wikipedia.org/wiki/Rufus_Isaacs_(game_theorist)

Видео Wythoff's Game (Get Home) канала singingbanana