Heap game

Heap games are a subclass of impartial games that involve the disjunctive sum of various single-heap games. Single-heap positions, or Γ-heaps are games represented naturally by the ordinal amount of a heap of tokens, where players play according to a specific ruleset on that single heap.

Every option of a heap game must be representable as $H_{n}^{'}=H_{a_{1}}+H_{a_{2}}+H_{a_{3}}+...$, where each $H^{a_{n}}$ is another heap game.

The nim-value of a game is represented as the nim-addition of each heap in a heap game.

Examples

• Octal games
• Nim
• Kayles