1. Introduction

The main result of this paper states that all the short combinatorial games are game values of particular positions of KONANE (Theorem 14 in Section 4). To prove this result constructively, two instrumental lemmas giving the needed building “pieces” are used (Lemmas 12 and 13 in Section 3). Some fundamental results of combinatorial game theory, like reduction concepts and the largest game value of the day n, are also used. Readers can find these results in Section 3. The next paragraphs of this introduction and Section 2 have some basics: the rules of KONANE, the state of the art and motivation for this research, and the definitions of habitat and universality of a ruleset (Definitions 3 and 7). Readers who know the rules of KONANE and are fluent in combinatorial game theory may wish to proceed to Lemma 12, Lemma 13 and Theorem 14 (Sections 3 and 4).

One of the principal goals of combinatorial game theory (CGT) is the study of combinatorial rulesets with the following properties ([1; 2; 5; 15] are fundamental references, [7] is a complete survey):

• • There are two players who take turns moving alternately.

• • No chance devices such as dice, spinners, or card deals are involved, and each player is aware of all the details at all times.

• • Even ignoring the alternating condition, the play must end in a finite number of moves, and the winner is often determined on the basis of who made the last move. Under normal play, the last player wins, while in misere play, the last player loses.

We distinguish between multiple meanings of the word game by using the words ruleset and game. The word ruleset has a concrete meaning related to some particular set of rules (KONANE, AMAZONS, NIM are examples of rulesets).