1. Introduction

Officers is an impartial game played with beans arranged into heaps. On each move, a player selects a heap with at least two beans and removes exactly one bean from the heap. If at least two beans remain in the heap, then the player may also optionally split the remaining heap into two. Officers is an example of an octal game [Berlekamp, Conway and Guy 2001]: a take-and-break game where each legal move can be described as removing k beans from a heap, then possibly splitting any remaining beans, leaving behind exactly 0, 1 or 2 nonempty heaps. The name “octal game” comes from the fact that such games can conveniently be described as a string of octal digits (traditionally preceded by a decimal point with trailing zeros omitted), where the k-th digit after the decimal point encodes the legal moves involving the removal of k beans from a heap. The binary representation of a digit has three bits, with bit m indicating whether or not it is legal to leave exactly m nonempty heaps. Thus, if the k-th digit is zero then it is never legal to remove exactly k beans. Under this encoding, Officers is the game .6: only the first digit after the decimal point is nonzero (only one bean can be removed), and this digit has bits 1 and 2 set (exactly 1 or 2 nonempty heaps can remain).