How to Solve a Pentomino Puzzle: Area, Corners and Colours

A pentomino puzzle asks you to fill a board, usually a rectangle, with flat pieces made of five squares joined edge to edge, using each piece once. You solve one by checking area and colour first, marking the cramped spots, placing the most awkward pieces early and testing every leftover gap before moving on.

This guide covers the twelve shapes, the checks that rule boards in or out, a step-by-step method, a fully worked small box and a practice box with its answer.

What is a pentomino puzzle?

A pentomino is a shape made of five equal squares joined edge to edge into one connected piece. Mathematicians treat these shapes as one case of a wider idea: a 2003 paper on tiling rectangles describes a polyomino as a two-dimensional polycube, a finite union of unit squares whose corners sit on the points of a square lattice ^[1].

In a pentomino puzzle you get a set of these pieces and a board. The pieces may be turned and flipped over, they may not overlap, and together they must cover every cell of the board exactly. A good puzzle has one answer, or a small number of answers that are mirror images of each other.

How many pentominoes are there?

There are exactly twelve. If you join five squares in every possible way and treat turned or mirrored copies as the same shape, you end with twelve pieces. Our checking script confirms the count by growing the shapes one square at a time and removing repeats.

Each piece is named after a letter it resembles: F, I, L, N, P, T, U, V, W, X, Y and Z. The table shows each one as a small picture, how many different positions it can take on a grid, and how its squares fall on a grid coloured in alternating dark and light cells.

The positions column adds up to 63 for the whole set. Six pieces, F, L, N, P, Y and Z, look different from their mirror image. Five of them, all but the Z, have eight positions; the Z has only four, because a half turn brings it back to itself. Spotting those six early saves time, because they are the pieces you most often need to flip.

Which boards can pentominoes fill?

Classic boards use all twelve pieces, so they have 60 cells. Four rectangles at least three cells wide have that area: 3 by 20, 4 by 15, 5 by 12 and 6 by 10. Our script searched each one exhaustively, and the counts below treat a filling and its turned or mirrored copies as one.

The table shows something useful for solvers. The narrow 3 by 20 strip has only two fillings, so a solver working by hand has the fewest targets to aim for, while the squarer boxes offer far more ways to succeed.

Smaller puzzles use only a few pieces. A 3 by 5 box takes three pieces, and our script tried all 220 ways to choose three of the twelve. Only seven of those sets fill the box at all, and each of the seven does it in exactly one way, apart from turning or mirroring the whole box. That makes the 3 by 5 box a good place to learn.

Mathematicians have looked at the reverse question too ^[1]. The 2003 paper discusses the order of a polyomino, the smallest number of copies of one shape that can be put together into a rectangle ^[1]. It also surveys which rectangles can be filled with copies of a single small polyomino, covering shapes of fewer than seven squares ^[1].

Why does counting area come first?

The area check is the fastest test there is. Every piece covers five cells, so the board must have five times as many cells as you have pieces. A board of 23 cells cannot be filled by any set of pentominoes, and you can stop there.

Area is necessary, but it is never enough on its own.

A paper published on 25 September 2018 looked at the squares of sides 1 to 24, whose areas add up to exactly the area of a 70 by 70 square, and reported that a proof with the help of extensive computer work showed the best possible packing leaves out the 7 by 7 square ^[2]. Matching areas only tells you the puzzle is not ruled out yet.

How does the colour check work?

Colour the board in alternating dark and light cells, like a two-colour grid, and count the dark and light cells. Then look at the last column of the piece table. Eleven pieces cover three cells of one colour and two of the other, wherever you put them. The X is the odd one out: it always covers four cells of one colour and one of the other.

Each piece therefore tips the balance by one cell, except the X, which tips it by three. The tips of all your pieces must add up to the difference between the dark and light cells of the board.

Here is the check in action. A board with equal numbers of dark and light cells can never be filled by the X and just one other piece, because three plus or minus one is never zero.

On a 3 by 5 box with no X, which has 8 cells of one colour and 7 of the other, two pieces must lean towards the majority colour and one towards the minority colour. That tells you something about how each piece must sit before you place anything.

Which spots on the board are hardest to fill?

Corners and narrow strips are the tight spots. A corner cell has only two neighbours, so only certain parts of a piece can reach it. Of the twelve pieces, only the X can never cover the corner cell of a box. The corner would have to hold one of its four tips, and the centre square next to that tip would then need a square beyond the edge of the box.

Narrow corridors are the other trouble spot. In a box that is three cells tall, any piece that is three cells tall in every direction, namely F, T, V, W, X and Z, must stretch from the top edge to the bottom edge. That splits the box into a left part and a right part, and each part must then be filled on its own.

On a pentomino board, ask one question about every edge and corner: which pieces can possibly touch this edge or fill this corner?

What is the step-by-step method?

1. Check the area. The number of cells must equal five times the number of pieces.

1. Check the colours. Count dark and light cells and make sure the tips of your pieces can add up to the difference.

1. Mark the tight spots. Note the corners, the narrow strips and any cell with few open neighbours.

1. Place the awkward pieces first. The X, F and W are hard to fit anywhere, and long pieces such as the I and the L need room. Try them early, while the board is still open.

1. Test every gap. After each placement, count the cells of every enclosed empty region. If any region has a number of cells that is not a multiple of five, that placement is wrong.

1. Fill the gaps that only one piece fits. A region of exactly five cells must match one of your remaining pieces, which you can check by eye.

1. Back up when you get stuck. Remove the most recent piece, try its next position, and keep a note of what you have ruled out.

Step 7 is what puzzle software does at speed, and it is called backtracking. Computers are no shortcut in general, though.

The 2003 paper recalls an earlier result: for a given set of shapes, the question of whether any rectangle at all can be tiled by them is algorithmically undecidable ^[1]. Its main theorem shows that the rectangles a fixed set of shapes can tile are built from a finite list of basic rectangles, even though the proof gives no way to construct that list ^[1]. For small boards like the ones below, careful checking by hand is quite enough.

How do you solve a 3 by 5 pentomino puzzle step by step?

Here is the worked example. Fill a box 3 cells tall and 5 cells wide with the F, P and U pieces, each used once. Try it yourself before reading on.

Step 1: area. Three pieces cover 15 cells, and the box has 3 times 5, which is 15 cells. The puzzle passes.

Step 2: colour. The box has 8 cells of one colour and 7 of the other. None of the three pieces is the X, so two of them must cover three cells of the majority colour. Nothing is ruled out, so we move on.

Step 3: the awkward piece. The F is the hardest of the three to fit. In every position it is 3 cells wide and 3 cells tall, so it must run from the top row to the bottom row. With its eight positions and three possible column starts, there are 24 ways to put it in the box.

Step 4: test the gaps. Any F that touches the left or right end of the box leaves a pocket of one or two cells, which no piece can fill. That rules out 16 placements at once. The F must sit in the middle three columns.

Step 5: test the gaps again. Of the eight middle placements, four leave gaps of 4 and 6 cells. Neither number is a multiple of five, so those fail too. The four survivors are copies of one another under turning or mirroring the whole box, so we keep just one. It leaves two gaps of exactly five cells.

Step 6: fill the gaps. The left gap covers all of column 1 plus the top and bottom cells of column 2. That is the U shape, opening to the right. The right gap covers the top two cells of columns 4 and 5 plus the bottom cell of column 5, which is the P shape.

Step 7: check. Each letter appears five times and forms the right shape, and every cell is covered once. Our script tried every arrangement of these three pieces in the box and confirmed this is the only answer, apart from turning or mirroring the whole box.

Notice how little trial and error the solve needed. Two quick gap checks cut 24 options down to one, and the last two pieces simply fell into place.

Can you solve this practice pentomino puzzle?

Now try a slightly bigger box. Fill a box 4 cells tall and 5 cells wide with the N, T, V and Y pieces, each used once. Pieces may be turned and flipped.

Hint: four pieces cover 20 cells, so the area works. Start with the V. In the answer it fills one corner of the box, with one arm along a long side and the other along a short side.

Answer: here is the only filling, apart from turning or mirroring the whole box. Our script tried every arrangement, and no other one works.

Look at how the pieces lock together. The V wraps the top right corner, the Y lies along the bottom edge with its bump pointing up, the T stands in the lower left, and the N zigzags across the top left to fill what is left.

What should you try next?

Once the 3 by 5 and 4 by 5 boxes feel easy, move up to the 60-cell rectangles. The 6 by 10 box has the most answers, so it is the kindest place to start with all twelve pieces. The 3 by 20 strip, with only two answers, is a real test.

The skills you have practised here, fitting shapes into a space, turning them in your head and checking a gap before you commit, are exactly what a good spatial riddle asks for. Try this one next and see which piece fits the hole.