A finite partizan combinatorial game is defined in Berlekamp et al. [1] as follows: there are two players, Left and Right, who move alternately; there are rules that specify moves from a position to its options, and Left and Right do not necessarily have the same options; a player unable to move loses, and the play ends after a finite number of moves; and the players have complete information, and there are no chance moves.
The game { a | b } = x denotes that the option for Left is a position of value a, the option for Right is a position of value b, and x is the number of free moves for Left. A game is born on or before Day n if all its Left and Right options are born on or before Day n - 1. Thus the game { | } = 0 is the only game born on or before Day 0. The games born on or before Day 1 are 0 , { 0 , | 0 } = *, { 0 | } = 1 and { | 0 } = - 1. Twenty-two distinct games are born on or before Day 2, and J 2, the additive Abelian subgroup of games generated by them, is isomorphic to Z 3 ⊗ Z &slash; 4Z ⊗ ( Z &slash; 2Z ) 3. I 2 = Z 2 ⊗ Z &slash; 4Z ⊗ Z &slash; 2Z, where I 2 is the group of infinitesimals within J 2, and J 2 &slash; I 2 = Z ⊗ ( Z &slash; 2Z ) 2. The basic purpose of this paper is to scharacterize the subgroup of games generated by the games born on or before Day 3. It has been computed through computer search that 1474 distinct games are born on or before Day 3, which are listed up to infinitesimals. The main result of the paper is that J 3 &slash; I 3 is isomorphic to Z T ⊗ Z &slash; 4Z ⊗ ( Z &slash; 2Z ) 8, and the author computes a corresponding basis. The paper’s best feature is the proofs, which are precise and clear. Its worst feature is that most of the terms used are not explained, and hence the paper caters to a small group of readers with good backgrounds in games and mathematics. A few more references to related work would be useful. The paper presents a significant result.
