Generalized genus sequences for misère octal games

Two approaches have been used to solve impartial games with misère play; genus theory, and Sibert–Conway decomposition, which has been used to solve the octal game 0.77 (known as Kayles). The main aim of this paper is to publish (for the first time) results extending genus theory beyond the applications to which it has previously been applied. In addition, we extend an earlier result to misère play by adapting it to use the extended genus theory. The resulting theorems require extensive calculations to verify that their preconditions hold for any particular games. These calculations have been carried out by computer for all two-digit octal games. For many of these games, this has resulted in complete solutions. Complete solutions are presented for four games listed earlier as unsolved.