Gathering data on the puzzle and deducing patterns to obtain solution for the puzzle
Through collection and analysis of data on the puzzle, we could devise a set of strategies that can be used to aid in solving the puzzle.
One type of data we collected was through examining the triangular pieces in greater detail.
1 - [0 0 0 0 1]
2 - [0 0 0 1 1], [0 0 0 0 2]
3 - [0 0 1 1 1], [0 0 0 1 2], [0 0 0 0 3]
4 - [0 1 1 1 1], [0 0 1 1 2], [0 0 0 1 3], [0 0 0 2 2]
5 - [1 1 1 1 1], [0 1 1 1 2], [0 0 1 2 2], [0 0 0 2 3], [0 0 1 1 3]
6 - [1 1 1 1 2], [0 1 1 1 3], [0 0 1 2 3], [0 1 1 2 2], [0 0 2 2 2], [0 0 0 3 3]
7 - [1 1 1 2 2], [0 1 1 2 3], [0 1 2 2 2], [0 0 1 3 3], [0 1 1 2 3], [0 0 2 2 3]
8 - [1 1 1 2 3], [1 1 2 2 2], [0 1 2 2 3], [0 2 2 2 2], [0 0 2 3 3], [0 1 1 3 3]
9 - [1 1 2 2 3], [1 2 2 2 2], [0 0 3 3 3], [0 1 2 3 3], [1 1 1 3 3], [0 2 2 2 3]
10 - [1 1 2 3 3], [1 2 2 2 3], [2 2 2 2 2], [0 1 3 3 3], [0 2 2 3 3], [1 2 2 2 3]
11 - [1 1 3 3 3], [1 2 2 3 3], [2 2 2 2 3], [0 2 3 3 3], [1 1 3 3 3]
12 - [1 2 3 3 3], [2 2 2 3 3], [0 3 3 3 3]
Shown above are all the possible ways the pins from 1 to 12 can be expressed as a sum of 5 integers with repititions allowed, the integers used being only 0, 1, 2 and 3. This set of data is used in deriving a few equations in the combinatorics part of our research.
Another type of data we collected was through using the solver to generate random puzzle configurations and solutions. An example is shown below.
Given,
Order of pins (A, B, C, D, E, F, G, H, I ,J ,K, L) => 4, 6, 12, 3, 1, 9, 10, 8, 5, 2, 11, 7,
then,
Solution ((x, y, z) on [p, q, r]) =>
(0, 2, 1) on [4, 12, 3]
(0, 0, 1) on [4, 3, 1]
(2, 0, 1) on [4, 1, 9]
(0, 2, 1) on [4, 9, 6]
(2, 1, 3) on [4, 9, 2]
(3, 1, 2) on [12, 8, 5]
(1, 0, 2) on [12, 5, 3]
(0, 0, 2) on [3, 5, 2]
(0, 0, 0) on [3, 2, 1]
(0, 0, 3) on [1, 2, 11]
(0, 1, 2) on [1, 11, 9]
(1, 3, 2) on [9, 11, 10]
(3, 3, 3) on [9, 10, 6]
(1, 2, 3) on [6, 10, 8]
(0, 3, 3) on [6, 8, 12]
(0, 2, 2) on [8, 7, 5]
(1, 1, 0) on [5, 7, 2]
(0, 1, 2) on [2, 7, 11]
(2, 2, 2) on [11, 7, 10]
(1, 1, 1) on [10, 7, 8],
where the letters A through L represent the different pin holes,
and where x is adjacent to pin p, y is adjacent to pin q, and z is adjacent to pin r, given a triangular tile (x, y, z) placed on a triangular face with 3 pins p, q, and r at the edges.
This set of data sheds some light on how the algorithm works. It is evident that the solver goes through a set path when working through the puzzle. It also helps to verify that all the ways we found the 12 pins can be expressed as a sum of 5 integers is indeed correct.
One type of data we collected was through examining the triangular pieces in greater detail.
1 - [0 0 0 0 1]
2 - [0 0 0 1 1], [0 0 0 0 2]
3 - [0 0 1 1 1], [0 0 0 1 2], [0 0 0 0 3]
4 - [0 1 1 1 1], [0 0 1 1 2], [0 0 0 1 3], [0 0 0 2 2]
5 - [1 1 1 1 1], [0 1 1 1 2], [0 0 1 2 2], [0 0 0 2 3], [0 0 1 1 3]
6 - [1 1 1 1 2], [0 1 1 1 3], [0 0 1 2 3], [0 1 1 2 2], [0 0 2 2 2], [0 0 0 3 3]
7 - [1 1 1 2 2], [0 1 1 2 3], [0 1 2 2 2], [0 0 1 3 3], [0 1 1 2 3], [0 0 2 2 3]
8 - [1 1 1 2 3], [1 1 2 2 2], [0 1 2 2 3], [0 2 2 2 2], [0 0 2 3 3], [0 1 1 3 3]
9 - [1 1 2 2 3], [1 2 2 2 2], [0 0 3 3 3], [0 1 2 3 3], [1 1 1 3 3], [0 2 2 2 3]
10 - [1 1 2 3 3], [1 2 2 2 3], [2 2 2 2 2], [0 1 3 3 3], [0 2 2 3 3], [1 2 2 2 3]
11 - [1 1 3 3 3], [1 2 2 3 3], [2 2 2 2 3], [0 2 3 3 3], [1 1 3 3 3]
12 - [1 2 3 3 3], [2 2 2 3 3], [0 3 3 3 3]
Shown above are all the possible ways the pins from 1 to 12 can be expressed as a sum of 5 integers with repititions allowed, the integers used being only 0, 1, 2 and 3. This set of data is used in deriving a few equations in the combinatorics part of our research.
Another type of data we collected was through using the solver to generate random puzzle configurations and solutions. An example is shown below.
Given,
Order of pins (A, B, C, D, E, F, G, H, I ,J ,K, L) => 4, 6, 12, 3, 1, 9, 10, 8, 5, 2, 11, 7,
then,
Solution ((x, y, z) on [p, q, r]) =>
(0, 2, 1) on [4, 12, 3]
(0, 0, 1) on [4, 3, 1]
(2, 0, 1) on [4, 1, 9]
(0, 2, 1) on [4, 9, 6]
(2, 1, 3) on [4, 9, 2]
(3, 1, 2) on [12, 8, 5]
(1, 0, 2) on [12, 5, 3]
(0, 0, 2) on [3, 5, 2]
(0, 0, 0) on [3, 2, 1]
(0, 0, 3) on [1, 2, 11]
(0, 1, 2) on [1, 11, 9]
(1, 3, 2) on [9, 11, 10]
(3, 3, 3) on [9, 10, 6]
(1, 2, 3) on [6, 10, 8]
(0, 3, 3) on [6, 8, 12]
(0, 2, 2) on [8, 7, 5]
(1, 1, 0) on [5, 7, 2]
(0, 1, 2) on [2, 7, 11]
(2, 2, 2) on [11, 7, 10]
(1, 1, 1) on [10, 7, 8],
where the letters A through L represent the different pin holes,
and where x is adjacent to pin p, y is adjacent to pin q, and z is adjacent to pin r, given a triangular tile (x, y, z) placed on a triangular face with 3 pins p, q, and r at the edges.
This set of data sheds some light on how the algorithm works. It is evident that the solver goes through a set path when working through the puzzle. It also helps to verify that all the ways we found the 12 pins can be expressed as a sum of 5 integers is indeed correct.