The Complexity of Some Combinatorial Puzzles

Two puzzles that repeatedly appeared in the Munich newspaper tz are Knossos and The Hour-Glass (“Das Stundenglas”) [Bul10a, Bul10b, Bul10c, Bul11], see also [Pre]. Knossos asks the reader to insert walls into a grid of cells representing the famous palace of Knossos. Every room is described by a number that indicates the length of its walls in terms of segments enclosing the room (we assume that the walls form a connected curve and thus rooms are not nested). Note that there is only one shape for rooms with walls of lengths 4 or 6, where the room with length 6 contains two cells and can appear in two orientations. For lengths 8 and more there are rooms of different shapes containing different numbers of cells. As an example consider the initial grid in Figure 1. A possible solution is depicted in Figure 2. The Hour-Glass consists of two triangular shapes containing numbers and a target sum. The goal is to find a path from the first row to the last row such that the numbers traversed add up to the target sum. Each step moves diagonally down the structure. An example is shown in Figure 3, where a solution for the target sum 53 is indicated by bold-face numbers. We generalize these puzzles in several ways: