Pentomino Solver
How to use it
Place as many pentominoes as you like, then press Solve the rest and the solver finishes the board around what you have put down. If your arrangement cannot be completed it will tell you so, which is often the more useful answer.
This is the part worth playing with. Dropping one piece in an awkward spot can make the whole board impossible, and finding out which placements kill it is a better way to understand pentominoes than watching a finished solution appear.
The twelve pentominoes
A pentomino is five squares joined edge to edge. There are exactly twelve, named after the letters they resemble, and together they cover sixty squares. That is why they fit a 6x10, a 5x12 and a 4x15 exactly, with nothing left over.
Why 3x20 is the famous one
There is a fourth rectangle, 3 by 20, and it is the one everybody talks about. The same twelve pieces fit it in only two distinct ways, against 2,339 for the 6x10.
You can feel that difference in software. The solver behind this page fills a 6x10 from empty in about three seconds and could not find a 3x20 packing in seventy, which is why that board is not offered here. Not because it is bigger, but because almost nothing fits.
The solution counts for the full set of rectangles are long-established published results: 2,339 for 6x10, 1,010 for 5x12, 368 for 4x15 and 2 for 3x20, each counted up to rotation and reflection of the whole board.
The background is in pentominoes explained, and if you would rather pack solid pieces than flat ones, the Soma cube shapes tool does the same job in three dimensions. The range itself is in mechanical puzzles.