These days, it's hard to know whether something is new, but a lot of web searching didn't turn up the result at the bottom here. Tiling an 8x8 with 4 cells removed, with the 12 pentominoes, is decades old. Usually the 4 are symmetric, but arbitrary placement has been discussed. If the 4 isolate 1-4 or 6 cells, tiling is obviously impossible. The one way to isolate 5 cells forces P in a corner, and has 410 solutions. There are 2 other impossible formations. One requires P in two corners. The other forces T or U in a position that isolates a cell. These are shown on the Wikipedia "pentomino" page. From the page edit trail, the second formation may have been discovered in January 2007 ("found another unsolvable"). I confirm these are the only non-isolating, impossible formations. Now for the maybe new thing. For the many other configurations of the 4 holes, what is the minimum number of solutions, and what configurations have that minimum? The minimum is 12 solutions, and only one pattern has 12 solutions: - - - - - - - - - - - - - - - - - - - - - - - - X - - - - - - - - X - - - - - - - - - - - - - - - - X - - - - - - - - X - - - - -- Mike Beeler