Abstract:

In this work we report a collaborative experience between industry and university. Specifically, it consisted in studying all the solutions of a puzzle type game called ICOXTREM.

In June 2022, the CEO of the company ICOXTREM S.L.U, contacted the Department of Mathematics of the University of Castilla-La Mancha to pose a curious and challenging problem. Some years ago, he had designed a puzzle type game, ICOXTREM, consisting of 20 pyramids of triangular base, joined along some of the edges, resembling a snake of pyramids. The aim of the game is to construct a convex regular icosahedron.

Before bringing the product to market, we needed to know for sure the number of different solutions of the game, namely, the different ways to construct an icosahedron. Over the years, he had found a certain number of different solutions for the game, all of them through trial and error. However, given the huge number of allowed relative positions for the 20 pieces of the puzzle, we could not be sure if he had found all the possible solutions, which meant an important setback. In fact, the game allows different solutions of different difficulty, and we needed to know all the solutions in order to devise a method to differentiate them and, additionally, to order them according to their difficulty.

Specifically, every two consecutive pyramids are joined along an edge of their base triangles allowing 4 different positions between them. Since there are 19 joint edges, there are 19^4 different configurations of the puzzle. As exhaustive search is computationally intractable given the number of possible configurations, in order to obtain all the solutions we devise an ad-hoc search method that exploits geometrical domain knowledge to prune the search space without losing the warranty of exploring all the feasible solutions.

The method combines a rule-based system with the backtracking search engine of Prolog, and takes the following key points into account:
- We define a coordinate system according to the geometry of the icosahedron, considering all its symmetries.
- The possible locations in the geometry of the icosahedron of a pyramid are determined from the location of the previous ones, so that it is verified that a pyramid is not located in an already occupied space.
- The geometry of the sequence of pyramids allows us to tackle the problem by considering segments with the same structure, which significantly reduces the number of possible configurations to be checked.

With all of this, we were able to find and describe all the solutions of ICOXTREM.

In conclusion, it leads to a fruitful collaboration which resulted in the resolution of the proposed problem after a mathematical and computational study of the game.

Keywords:

Puzzle game, collaboration industry-university, icosahedron.