The Number of Inequality Signs in the Design of Futoshiki Puzzle

In this paper, we study how many inequality signs we should include in the design of Futoshiki puzzle. A problem instance of Futoshiki puzzle is given as an n × n grid of cells such that some cells are empty, other cells are filled with integers in [n] = {1, 2, . . . , n}, and some pairs of two adjacent cells have inequality signs. A solver is then asked to fill all the empty cells with integers in [n] so that the n2 integers in the grid form an n × n Latin square and satisfy all the inequalities. In the design of a Futoshiki instance, we assert that the number of inequality signs should be an intermediate one. To draw this assertion, we compare Futoshiki instances that have different numbers of inequality signs from each other. The criterion is the degree to which the condition on inequality is used to solve the instance. If this degree were small, then the instance would be no better than one of a simple Latin square completion puzzle like Sudoku, with unnecessary inequality signs. Since we are considering Futoshiki puzzle, it is natural to take an interest in instances with large degrees. As a result of the experiments, the Futoshiki instances which have an intermediate number of inequality signs tend to achieve the largest evaluation values, rather than the ones which have few or many inequality signs.