How many solutions for each hole position?
Each number counts truly different answers: a solution and its rotations or mirror images that keep the hole in the same place are counted once. The centre hole is the most constrained, so it has by far the fewest solutions.
| Hole (column, row of its top-left square) | Same as | Unique solutions |
|---|---|---|
| Centre hole (4,4) | 1 position | 65 |
| Hole at 3,3 | 4 positions | 582 |
| Hole at 2,4 | 4 positions | 721 |
| Hole at 3,4 | 4 positions | 768 |
| Hole at 2,2 | 4 positions | 987 |
| Hole at 1,4 | 4 positions | 1,288 |
| Hole at 2,3 | 8 positions | 1,662 |
| Hole at 1,3 | 8 positions | 1,839 |
| Hole at 1,2 | 8 positions | 3,207 |
| Corner hole (1,1) | 4 positions | 5,027 |
| All 10 positions | 49 positions | 16,146 |
| Four corners removed | — | 2,170 |
Solution map: where the hole is easiest
Each square below is one place for the top-left corner of the 2×2 hole. Darker wood means more solutions. The map is symmetric, because turning or flipping the board moves the hole without changing the puzzle.
The hardest spot is the exact centre (65 solutions); next comes the hole at 3,3 with 582. A hole in the corner leaves the most freedom: 5,027 solutions.
The first centre-hole solution
Letters name the 12 pieces (F, I, L, N, P, T, U, V, W, X, Y, Z); the empty cut-out is the hole. Not sure which piece is which? See the pentomino shapes guide.
Tips for solving the 8×8 board
- Put the X next to the hole. In 46 of the 65 centre-hole solutions the X touches the hole — it fits poorly anywhere else.
- Run the I along an edge. It touches the border in 65 of the 65 centre-hole solutions.
- Fill the four corners early with the V, L, and other pieces that have a right angle.
- Work in rings: the outer ring of 28 squares, then the inner ring around the hole. A single trapped square means undo.
- Stuck? Load your half-finished board into the pentomino solver to see whether it can still be completed.
The four-corners board
Cut one square from each corner instead of a 2×2 hole and the board still has 60 squares. It is much easier: 2,170 unique solutions. Here is the first one; open the “Four corners” tab above to browse them all.
Try it yourself
- Print a blank 8×8 board and cut-out pieces from printable pentominoes.
- Compare with the rectangles: all 2,339 pentomino 6×10 solutions.
- Learn a method in how to solve pentomino puzzles, or warm up with 4-square pieces in the tetromino puzzle.