1. Covering the 8x5 board
There are many puzzles involving tetrominoes, e.g.: Tetrominoes Puzzles, by T. Gottfried. Here I wanted to analyze an easy puzzle which consists of twice the fundamental tetrominoes to cover a 8x5 board without any extra rules:
If you take all tiles as different it has 99392 solutions. If you assume the equal pairs are identical plus you remove all the symmetric solutions then there are only 783 solutions, in the following gif:
13 of the 783 solutions are symmetric with respect to the center point inversion. There are no solutions with up-down or right-left symmetry (can you prove this?). Note that 99392 = 770*2^7 + 13*2^6. The 13 symmetric solutions follow:
This was computed using the famous algorithm X from Knuth. I used the python version from Ali Assaf. After finding unique cover with duplicated tetrominoes it is important to remove the equal solutions and then the symmetric ones for fun. My code: cover.py, plots are done with gnuplot: gplot.symm, solutions.dat.
Rogelio Tomas home web page
2. Covering the 10x4 boardTaking all tiles as different it has 57472 solutions. If you assume the equal pairs are identical there are 449 solutions, in the following gif:
However, contrary to the 8x5 case, there are no solutions featuring any possible symmetry. Note that 57472=449*2^7.