I am trying to figure out a permutation question. It is related to an art project I am doing with recombinable panels.
I have a grid of $64$ panels in an $8\times8$ arrangement. The panels can be placed in any square in the grid, so $64$ possible places for $64$ possible panels. So $64\times64=4096$ possible arrangements so far.
However the panels each have 2 states “up” or “down”. Does this mean I multiply the $4096$ by $2$, or by another $64$? My logic being I multiply by $2$ as I am effectively making $128$ source panels for the grid of $64$, since an individual panel can not be in the grid in each state simultaneously, as opposed to multiplying by another $64$ which would be valid if a panel could be in the grid in both states simultaneously. i.e. Panel A can be in Grid Square 1 but nowhere else in the grid, and Panel A can be "up" or "down" but not both at the same time. This is true for all other 63 panels at the same time to compose one permutation of the 64 panel grid.
However, once I have that number ($\#$ of possible permutations for the $64$ panel grid), I think I multiply it by $9$, as the entire grid of $64$ panels can be recombined with itself in a meta-grid of $9$ meta-panels ($3\times3$).
So ultimately I am also trying to figure out how many permutations of the $3 \times 3$ meta-grid the 64 panel set could generate, considering their two states.
Any help sorting that out?