Back to Game Theory & Strategies
Game Theory Sandbox
Experiment with different game configurations, starting positions, and discover optimal strategies.
Game Configuration
Choose a game and adjust settings to explore different strategic scenarios
Symmetry Strategy: The second player can win by mirroring—whatever you take from one pile, they take the same from the other!
Nim Game
Remove any number of stones from one pile per turn. Take the last stone to win!
Pile 1
5 stones
Pile 2
5 stones
Your turn
Click any stone to take it and all stones above it from that pile
Key Concepts
Winning vs Losing Positions
A winning position is one where the current player has a move that guarantees victory. A losing position (cold square) is one where every move leads to a winning position for the opponent.
Working Backwards
To find optimal strategies, start from the goal and work backwards. Mark terminal positions, then mark positions that lead only to winning positions as "losing", and so on.
Symmetry Strategies
Many games can be won by the second player using symmetry: mirror the first player's moves across a line, point, or according to some transformation.