Octal Game - Octal Games Specification

Octal Games Specification

An octal game is played with tokens divided into heaps. Two players take turns moving until no moves are possible. Every move consists of selecting just one of the heaps, and either

• removing all of the tokens in the heap, leaving no heap,
• removing some but not all of the tokens, leaving one smaller heap, or
• removing some of the tokens and dividing the remaining tokens into two nonempty heaps.

Heaps other than the selected heap remain unchanged. The last player to move wins in normal play. The game may also be played in misère play, in which the last player to move loses.

Games played with heaps in this fashion, in which the allowed moves for each heap are determined by the original heap's size, are called Taking and Breaking games in the literature. Octal games are a subset of the taking and breaking games in which the allowed moves are determined by the number of tokens removed from the heap.

The octal code for a game is specified as

0 . d₁ d₂ d₃ d₄ …,

where the octal digit d_n specifies whether the player is allowed to leave zero, one, or two heaps after removing n tokens from a heap. The digit d_n is the sum of

• 1 if leaving zero heaps is permitted, 0 otherwise;
• 2 if leaving one heap is permitted, 0 otherwise; and
• 4 if leaving two heaps is permitted, 0 otherwise.

Zero tokens are not counted as a heap. Thus the digit d_n is odd if a heap of n tokens may be removed entirely, and even otherwise. The specification of one-heap results in d_n applies to removing n tokens from a heap of more than n. The two-heap results in d_n apply to removing n tokens from a heap of at least n+2, and separating the remainder into two nonempty heaps.

Octal games may allow splitting a heap into two parts without removing any tokens, by use of the digit 4 to the left of the decimal point. This is similar to the move in Grundy's game, which is to split a heap into two unequal parts. Standard octal game notation, however, does not have the power to express the constraint of unequal parts.

Octal games with only a finite number of non-zero digits are called finite octal games.