Using the tetrahedron and octahedron as miniature IcoSokus
The idea to use the tetrahedron came about when we were exploring platonic solids in detail. There are 5 platonic solids, with the IcoSoku modelled after the icosahedron. The tetrahedron was the smallest platonic solid and the simplest. Since they were geometrically similar, we figured that some conclusions that could be drawn from experimentation with the tetrahedron could be extrapolated to the actual toy.
Through experimentation with the tetrahedron, we came up with a new way of classifying the triangular tiles: Rotated and unrotated tiles. Rotated tiles are not unique, and have "mirror image" tiles. They also have duplicate tiles. Examples include [0 2 1], whose mirror image is [0 1 2]. Unrotated tiles do not have "mirror images" of themselves, and are therefore unique. Examples include [0 0 2] and [0 1 1]. Note that they have a drawable line of symmetry through them.
We found that there were 2 distinct solutions for each pin configuration of the tetrahedral puzzle, thereby possibly proving that in the actual icosahedral puzzle, the fact that there is more than 1 solution for all configurations of the puzzle is indeed true.
The octahedron is another platonic solid, simpler than the icosahedron. We contrived 8 types of tiles for use in the octahedral puzzle: 4 rotated tiles, and 4 unrotated tiles. The process of deciding on which tiles to use is based on the type of tiles the actual IcoSoku uses. The IcoSoku has 10 rotated and unrotated tiles each, and therefore, to keep things as similar as possible, we decided that the octahedral puzzle should have 4 rotated and unrotated tiles each as well. In fact, this applied to the choosing of tiles for use when we experimented with the tetrahedral puzzle as well.
Once again, we found that for random pin configurations of the octahedron, there are solutions that can be generated from the tiles we hand-picked. However, it is not feasible to use the brute-force method to find the number of ways to solve an octahedral puzzle as it is much more complex than the tetrahedral one. However, observing the tiles we used for the octahedron gave us the idea of implementing a value system for the tiles. Please see the page entitled "Creating a value system" under Results for details.
Through experimentation with the tetrahedron, we came up with a new way of classifying the triangular tiles: Rotated and unrotated tiles. Rotated tiles are not unique, and have "mirror image" tiles. They also have duplicate tiles. Examples include [0 2 1], whose mirror image is [0 1 2]. Unrotated tiles do not have "mirror images" of themselves, and are therefore unique. Examples include [0 0 2] and [0 1 1]. Note that they have a drawable line of symmetry through them.
We found that there were 2 distinct solutions for each pin configuration of the tetrahedral puzzle, thereby possibly proving that in the actual icosahedral puzzle, the fact that there is more than 1 solution for all configurations of the puzzle is indeed true.
The octahedron is another platonic solid, simpler than the icosahedron. We contrived 8 types of tiles for use in the octahedral puzzle: 4 rotated tiles, and 4 unrotated tiles. The process of deciding on which tiles to use is based on the type of tiles the actual IcoSoku uses. The IcoSoku has 10 rotated and unrotated tiles each, and therefore, to keep things as similar as possible, we decided that the octahedral puzzle should have 4 rotated and unrotated tiles each as well. In fact, this applied to the choosing of tiles for use when we experimented with the tetrahedral puzzle as well.
Once again, we found that for random pin configurations of the octahedron, there are solutions that can be generated from the tiles we hand-picked. However, it is not feasible to use the brute-force method to find the number of ways to solve an octahedral puzzle as it is much more complex than the tetrahedral one. However, observing the tiles we used for the octahedron gave us the idea of implementing a value system for the tiles. Please see the page entitled "Creating a value system" under Results for details.