Subject:        AFLB talk
Date:           Sat, 22 Oct 88 18:20:21 PDT
From:           Anil R. Gangolli <>

        Periodicity Results for Some Simple
               Octal Games

              Thane Plambeck

  Taking and breaking games are a class of impartial games played by
removing beans from a pile and leaving the pile in zero or more
parts.  Following Conway, the rules for a taking and breaking game can
specified as a numeric code which describes the number of beans one
can take and the number of piles into which one can break a pile.
The games which have Conway codes consisting only of the octal digits
0...7 are known as octal games.

The theorem of Sprague and Grundy tells us that every instance of an
impartial game is equivalent to an instance of the game Nim.  The
instances of a taking and breaking game thus specify a sequence of Nim
values.  It is conjectured [R. Guy] that all finitely-specified octal
games have ultimately periodic Nim-value sequences, but the conjecture
is not known to hold even for all octal games with 3 or fewer code
digits.  Slightly fewer than sixty of these now remain unsettled.
We present techniques for giving computational proofs of the periodicity
of some of these sequences and consequent periodicity results for three such

No background in the theory of impartial games will be assumed.

This is joint work with Anil Gangolli.