Impartial and partisan games

Abstract

Conway has recently developed a theory particularly well suited to the analysis of two-person games that are completely determined. Using this theory we consolidate some results due to Conway and Guy about the partisan game Col, as well as proving some new results for take and break games. In Chapter 4, the results obtained by Guy and Smith, and Kenyon for octal games are generalized to arbitrary take and break games. Chapter 5 discusses subtraction games. We show that all subtraction games are periodic, and prove that in certain circumstances it is possible to determine the period length exactly. We also state the rules, due to Conway and Guy respectively, for writing down the period of the games S(a,b), S(a,b,2b-a). Using Ferguson's Pairing Property, we give the analysis, again due to Conway and Guy, of S(a,b ,a+b). Chapter 6 deals with arithmetico-periodicity. Conway's proof that no octal game is arithmetico-periodic• is given. We prove new arithmetico-periodicity theorems for sedecimal and infinite recurring octal and tetral games. Chapter 7 contains Tables that list the G-sequence of certain types of games. With the exception of Table 7.7, the basis for these was provided by Guy. Table 7.1 was expanded by the author to include all subtraction games in which the subtrahends do not exceed 8. The games .55, .165, .356 and .644 were also solved by the author.