Wythoff’s Queen

Losing positions and the golden ratio

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Wythoff’s game: move a queen left, down or diagonally toward the corner.
• Losing positions: $(\lfloor k\varphi \rfloor, \lfloor k\varphi^2 \rfloor)$ and their mirrors.
• A single extra kind of move can completely change a game’s structure.

Steps

1. A queen heading home
2. Next to the corner
3. Make the winning move
4. Sieving out the losing squares
5. The golden ratio appears
6. Predict the next
7. Not like Nim