Glasser's master theorem
Imagine you have a puzzle with 4 pieces, and you want to know how many different ways you can put them together to make a complete picture. Glasser's Master Theorem can help you figure out the answer.
The theorem says that if you have n puzzle pieces, and each piece can be rotated in four different directions, then the total number of possible solutions is:
4^n - 2n - 1
Now, let's break that down:
- The 4^n part means that for each puzzle piece, there are 4 different possible orientations (up, down, left, and right). So if you have 2 pieces, that's 4*4=16 possibilities, and if you have 3, it's 4*4*4=64.
- The 2n part accounts for the fact that the first piece can be oriented in any direction, but each subsequent piece will be limited by the orientation of the previous piece. So there are only 3 orientations possible for the second piece, then 2 for the third piece, and so on. This part of the formula is saying that you need to subtract out all of the redundant combinations that result from this limitation.
- Finally, the -1 is there because there's always one solution where all of the pieces are lined up in a row, and that's not considered a true puzzle solution.
In summary, Glasser's Master Theorem gives you a way to calculate the total number of possible solutions to a modular puzzle with n pieces and 4 possible orientations per piece, while taking into account the limitations caused by the orientation of previously placed pieces.
The theorem says that if you have n puzzle pieces, and each piece can be rotated in four different directions, then the total number of possible solutions is:
4^n - 2n - 1
Now, let's break that down:
- The 4^n part means that for each puzzle piece, there are 4 different possible orientations (up, down, left, and right). So if you have 2 pieces, that's 4*4=16 possibilities, and if you have 3, it's 4*4*4=64.
- The 2n part accounts for the fact that the first piece can be oriented in any direction, but each subsequent piece will be limited by the orientation of the previous piece. So there are only 3 orientations possible for the second piece, then 2 for the third piece, and so on. This part of the formula is saying that you need to subtract out all of the redundant combinations that result from this limitation.
- Finally, the -1 is there because there's always one solution where all of the pieces are lined up in a row, and that's not considered a true puzzle solution.
In summary, Glasser's Master Theorem gives you a way to calculate the total number of possible solutions to a modular puzzle with n pieces and 4 possible orientations per piece, while taking into account the limitations caused by the orientation of previously placed pieces.