Gathering data and obtaining a solution
There were noticeable patterns on the tiles of the puzzle and these patterns were used to obtain the pieces to use for our other methodologies on “Usage of the Tetrahedron” and “Usage of the Octahedron”. Through observation of these patterns, a manual solution to the puzzle could be obtained.
Decoding algorithm of solver
There was an existing online solver that can generate a solution when the configuration of the pins was keyed into the programme. The programme was examined and the approach the solver took to obtaining the solution was found. This approach could potentially help us obtain a manual solution to the puzzle that did not require a programme. The existing online algorithm, created by SethosII, (Refer to appendix 1), utilises the backtracking algorithm, enabling it to come up with a complete solution for the Icosoku. This provides the orientation and pieces to be placed on a specific segment in the puzzle. This solver is based on backtracking with a check to cut off subtrees. Before the backtracking starts, a list with the areas and the sum of the edge numbers is created and sorted ascending. The areas will be filled with pieces in the order of this list. This was done because the pieces are sorted from few to many dots at their edges and therefore the pieces with fewer dots have a higher chance to be set at the right place in less attempts. The check is doable. This method checks whether the edge numbers can be reached with the remaining areas filled with pieces with three or less dots and compares the sum of pieces with the number at full edges.
Usage of the Tetrahedron
The tetrahedron is a platonic solid that has the same shape for a face compared to the icosahedron. It has 4 vertices and 4 triangular faces. Hence, due to their similarities in properties, it was used as a simplified version of the Icosoku for simpler calculations. Having the shape of the tetrahedron, the types of tiles used had to be decided. The property of tiles that was noticed earlier was used to select the pieces to use for the tetrahedron. All solutions to all possible configurations of the tetrahedron puzzle were obtained through brute force. This is because the puzzle was simple enough such that we can run tests on the puzzle with the brute force method, unlike in the actual icosahedron puzzle, which was far too complex. This information could help solve our research question “Why is there at least one solution for each pin configuration?” and “Which pin configurations have one solution only and which have multiple solutions?”.
Finding information with Combinatorics
We attempted to obtain the number of solutions possible for each configuration through combinatorics. The combination of 0s, 1s, 2s and 3s that can surround a certain pin was found. We then utilised combinatorics to obtain the total number of configurations possible for the puzzle. We compared their values and derived a conclusion.
Devising strategies
From the data that we collect, we devised a set of strategies to promote better understanding of the puzzle and the process to undertake when solving it. The strategies tell how to utilise tiles safely without creating major problems, and how to work around editing the puzzle in the editing phase.
Extension: Usage of the Octahedron
The octahedron is a platonic solid that has the same shape for a face compared to the icosahedron. It has 6 vertices and 8 triangular faces. Due to the similarities between the octahedron and icosahedron, the octahedron was used as a simplified version of the icosahedron, much like what the tetrahedron was used for in our previous methodology. The pieces for the octahedron puzzle were selected through the same process that was used in selecting the pieces for the tetrahedron. The solutions to all configurations of the octahedron puzzle were attempted to be derived through brute force, like with the tetrahedron puzzle. This information could help solve our research question “Why is there at least one solution for each pin configuration?” and “Which pin configurations have one solution only and which have multiple solutions?”.
Extension: Value system
In order to improve our solution for the Icosoku to be more reliable and efficient, a value system for tiles was formed. This value system allowed the player to know which tiles should be placed into the puzzle first and which tiles should be placed last in a strict and logical manner. This allows players to attain a higher rate of success in solving the puzzle within the first try, reducing the need for the editing phase. By observing the number and pattern of dots on each tile, a tile’s value could be derived. This value is decided based on the adaptability and flexibility of the tile in different situations of the puzzle. Flexibility is defined as the number of situations a tile can be placed into the puzzle in a logical manner. For example, tile 333 is not flexible as it should only be placed in the triangular space with the pins surrounding it being the largest numbers. Tiles that are more flexible are assigned a higher value as they are more valuable and can be used in more situations. Hence, an order for players to place their tiles could constructed as it is known that lower value tiles should be placed first.