Let T be a unit square admitting the standard square tiling of the plane T . A pentomino is a connected patch made up of 5 squares, so that the patch is itself a tile (that is, the union of the 5 squares is a topological disk).
(a) There are 12 distinct pentominoes that, when viewed as a tile, admit a planar tiling. Showing all your work, find each pentomino. How do you know your list is complete, that is, can you conclude from your work that there are exactly 12 distinct pentominoes?
(b) Show that one of your pentominoes admits two distinct tilings. Justify your answer.
(c) By analogy with the definition given above, give a definition for a hexomino. Considering the enumeration you gave in part (a), explain why there are at least twice as many distinct hexominoes as distinct pentominoes.