Enumeration of Tsume-Shogi Diagrams by the Reverse Method

With the advances of modern computer technologies and elaborated algorithm design methodologies, it becomes important to enumerate all feasible solutions for given constraints. In this paper, we propose algorithms for enumerating Tsume-Shogi diagrams (i.e., Shogi mating problems) by the reverse method. Conventional algorithms always take the principle 'generate candidate diagrams and sieve them by Tsume-Shogi solvers,' which tends to require lengthy execution time. In our approach, the reverse method enables us to enumerate all diagrams without using Tsume-Shogi solvers. We can sieve the candidates easily and quickly, since they are generated in the ascending order according to the number of necessary moves for mating the defender's king. We have implemented the proposed algorithms, and enumerated all diagrams with several restrictions (e.g., those with only four knights). From this result, we prove many results for knight diagrams, e.g., i) the maximum number of moves is 11, ii) it is 13 for additional mate free, iii) it is 7 if at least one piece is in hand of the attacker, and iv) the maximum pieces in hand of the attacker is two.