Pentomino Puzzle Solver

For the moment a child is stuck on a pentomino board: it shows solutions of the 6×10, 5×12, 4×15 and 3×20 rectangles, the 8×8 board with a hole in the middle, or a board you draw, one piece at a time. Copy the child's board and it tells you whether it can still be finished and which piece to try next. It also prints pieces to cut out.

How to use

1. Choose the child's board under "Board". The "Solve" tab shows the first solution at once; use "Start" and "Next" to see it piece by piece.
2. Press "Another solution" for a different answer on the same board, or "Count solutions" to see how many there are.
3. If the child is stuck, open "Hint" and drag the same pieces onto the board in the same places. Use "Turn" and "Flip" to match their direction.
4. Press "Check and get a hint". It says whether the board can be finished and, if so, shows one next piece with a dashed outline; "Place the suggested piece" puts it there.
5. No pieces at home? In "Printable pieces" choose the paper and "Square size", then press "Save as PDF" or "Print" and cut along the thick lines.

How it works

• The solver is an exact-cover search: it fits the 63 turned and flipped shapes of the 12 pieces into the first empty square, one after another, until the board is full. It runs in your browser.
• "Another solution" and the counts treat solutions that match after turning or flipping the whole board as one. The 6×10 board has 2,339; counting them can take tens of seconds depending on the device.
• "Draw your own" takes boards up to 12×12 and the pieces you choose under "Pieces to use". The board must have 5 squares for every chosen piece.
• Your drawn board, the pieces on the board and your settings are kept only in this browser.

Pentomino hint: where to restart when a child is stuck

In pentomino puzzles a single early piece often makes the rest impossible, however the others are moved. The "Hint" tab keeps the pieces that are on the board and searches to the end for any way to fill the remaining squares with the remaining pieces. If there is none, it suggests taking pieces out, starting with the last one placed.

A frequent dead end is a small pocket of one to four empty squares trapped against an edge or corner. An empty area whose size is not a multiple of 5 can never be filled, so counting the trapped squares together is a useful habit.

Pentomino solutions for each board

Counting solutions that match after turning or flipping the board as one, the 6×10 board has 2,339, 5×12 has 1,010, 4×15 has 368, 3×20 has 2, and 8×8 with a 2×2 hole in the middle has 65. "Count solutions" counts them here so you can compare.

With only 2 solutions, 3×20 is the hardest for a child to find alone, while 6×10 has many and is a good first board. On the small 3×5 board with three pieces, 7 different sets of three pieces work.

Questions and answers

Pentomino 6x10 solution

Choose "6×10" under "Board" and the "Solve" tab shows solution 1. "Next" goes piece by piece and "Another solution" shows a different one.

How many solutions does the 6x10 pentomino have

2,339 when turned or flipped copies of the board count as one, and 9,356 counted separately. "Count solutions" checks it.

Can I continue from where my child stopped

Yes. Copy the child's pieces in the "Hint" tab and press "Check and get a hint". It shows one next piece if the board can still be finished, and says so if it cannot.

How do I turn a piece on a phone

Tap a piece on the board to choose it, then tap it again to turn it. The "Turn" and "Flip" buttons work on the chosen piece too.

Can I make pentomino pieces from paper

In "Printable pieces" you can save or print the 12 pieces and an empty 6×10 board on A4 or Letter paper, with 2 cm or 1.5 cm squares.