Finding the number of cases for each configuration by using combinatorics
When researching, we found that combinatorics was unable to be utilised properly as there were too many limitations and constraints. However, based on what we could find, we could draw a conclusion. With the restriction being the fact that the possible numbers on a tile can only be 0, 1, 2 or 3, the total number of ways the puzzle can be fixed was:
1 * 2 * 3 * 4 * 5 * 6 * 6 * 6 * 6 * 6 * 5 * 3 = 13996800 (Refer to "Gathering data on the puzzle and deducing patterns to obtain solution for the puzzle" for derivation of this equation.)
The total number of pin configurations being:
11!/5 = 7983360
We get the factorial of 11 because we experimented with the puzzle with 1 pin designated as a nailed pivot point. We divide by 5 as due to rotations of the puzzle frame, there can be similarities.
Therefore, it can be concluded that there are more possible ways of placing the tiles than possible pin configurations.
1 * 2 * 3 * 4 * 5 * 6 * 6 * 6 * 6 * 6 * 5 * 3 = 13996800 (Refer to "Gathering data on the puzzle and deducing patterns to obtain solution for the puzzle" for derivation of this equation.)
The total number of pin configurations being:
11!/5 = 7983360
We get the factorial of 11 because we experimented with the puzzle with 1 pin designated as a nailed pivot point. We divide by 5 as due to rotations of the puzzle frame, there can be similarities.
Therefore, it can be concluded that there are more possible ways of placing the tiles than possible pin configurations.