A pentomino is 5 squares joined edge-to-edge. There are exactly 12 distinct shapes. Together they cover 60 squares — and fit perfectly into a 6 × 10 rectangle, in 2,339 different ways.
A pentomino is 5 unit squares joined edge-to-edge. Counting flips and rotations as the same shape, how many distinct pentominoes are there?
Why exactly 12
List every way to stick 5 squares together, then throw out any that match another after a rotation or a mirror-flip. What survives is 12 — no more, no less. The 12 squares' worth of area (60 units) is why they fit rectangles like 6×10, 5×12, 4×15 and 3×20.
Warm up on the 6×10 box (lots of solutions), then try 3×20 — it has only 2 solutions and is genuinely hard. The narrow strip leaves almost no room to manoeuvre.
If 12 pentominoes cover 60 unit squares, which rectangles with whole-number sides could they exactly fill?
Why 12?
Count carefully: shapes that are the same after flipping or rotating count as one. The 12 are usually labelled by the letters they look like — F, I, L, N, P, T, U, V, W, X, Y, Z.
Try the 5×12, 4×15, and 3×20 rectangles too. The 3×20 has only 2 solutions — much harder.