count fourfold, the ones from categories five to seven count twice, and the last octomino counts only once. This results in 316 × 8 + (23+5+18) × 4 + (1+4+1) × 2 + 1 = 2,725 fixed octominoes. Packing and tiling Of the 369 free octominoes, 320 satisfy the Conway criterion and 23 more can form a patch satisfying the criterion. The other 26 octominoes (including the 6 with holes) are unable to tessellate the plane. Since 6 of the free octominoes have a hole, it is trivial to prove that the complete set of octominoes cannot be packed into a rectangle, and that not all octominoes can be tiled. References Polyforms