Insane Math Questions. Those With Weak Hearts Beware.

Ok, I did get a few extras in there.

But there are 3 distinct ways to place the first big L - sitting up, and clockwise/counterclockwise.



If you take black as "top" and grey as "side" these are definitely distinct ways of laying them out.

P(3,2) only applies to the clock/counterclockwise ones - the sitting up version is not rotationally symmetric so it has 9 options since either of the 2 Ls can take all 3 subsets and give a distinct arrangement.


Describing this is easier if I just number the boxes:

1 2 3
4 5 6
7 8 9
10 11 12
Assume it is reconstructed by "sliding" the top down to cover the bottom so 1 is above 7, 2 is above 8, etc.

124 356 and 145 236 are distinct ways of filling the top layer. To avoid being Ls the bottom is then 7 10 11, 8 9 12, and 7 8 10, 9 11 12.

The other 2 "flats" are
1 7 8, 2 3 9, 4 5 10, 6 11 12.
1 2 7, 3 8 9, 4 10 11, 5 6 12.
These are also not rotationally identical so there are definitely 4 possibilities there.


So my current answer is: 6*2 + 9 + 4 = 25.