Teaching objectives
A polyomino is a shape made by joining equal squares along full sides. The lab follows the series by size: dominoes (2 squares), trominoes (3), tetrominoes (4) and pentominoes (5), across 23 rooms. The student drags pieces onto a board (they can be rotated and flipped), paints cells to discover shapes, and answers multiple-choice questions or types numbers. There are no bets or simulated series: everything is solved by handling the pieces.
The rooms fall into four blocks: dominoes (rooms 1-7), trominoes (8-11), tetrominoes (12-16) and pentominoes (17-23). In the tromino, tetromino and pentomino blocks the first room is a discovery room (“paint a shape and check it”) and it never says how many shapes there are: only after all of them have been found does it ask whether the student thinks there are more. Challenges follow, covering shapes with an exact set of pieces. The last two rooms close the lab collaboratively: the student designs a challenge and solves a classmate’s.
What you'll learn
The lab takes the student from covering shapes by trial and error to explaining, with parity and colouring arguments, why some of them cannot be covered.
- Know what a polyomino is: pieces touch along a full side, not at a corner or along half a side (room 1).
- Rule shapes out by parity: a shape with an odd number of cells cannot be covered with dominoes (room 2).
- Use the chessboard colouring: a domino always covers one white and one black cell, so a board with different numbers of white and black cells cannot be covered (rooms 3 and 4), and the maximum number of dominoes that fit is set by the less numerous colour (room 7).
- Count all the ways of covering a rectangle and recognise the Fibonacci pattern: 3, 5, 8 and, as a prediction, 13 (room 5).
- Build a shape that meets a condition (connected, with at most 2 dominoes at a time) and look for the maximum, which is 9 cells (room 6).
- Discover systematically all the pieces of each size (2 trominoes, 5 tetrominoes, 12 pentominoes) and understand that rotations and reflections count as the same piece (rooms 8, 12 and 17).
- Classify shapes by which trominoes can cover them (straight, L, a mix or none) without trusting how they look (room 9).
- Split a shape into parts at a larger scale (the big L into four Ls) and use that idea to cover a 4 × 4 board with one cell missing (rooms 10 and 11).
- Apply the colouring to tetrominoes to see why five pieces with exactly the right area do not cover a 4 × 5 rectangle (room 13).
- Solve assembly challenges with given pieces and propose one of your own with a (nearly) unique solution (rooms 14-16 and 18-23).
Key mathematical ideas
- Polyomino: squares joined along full sides. Two shapes are the same piece if one can be obtained from the other by rotating or reflecting.
- Parity: each domino covers 2 cells, so a shape with an odd area cannot be covered. But an even area is not enough.
- Chessboard colouring: two neighbouring cells always have different colours, so each domino covers one of each colour. The 6 × 6 board without two opposite corners has 34 cells, 18 of one colour and 16 of the other, and cannot be covered with 17 dominoes; at most 16 fit. The star in room 7 (25 cells) has 9 cells of one colour and 16 of the other, and the maximum is 9 dominoes.
- Fibonacci: 2 × n rectangles can be covered with dominoes in 2, 3, 5, 8… ways (for 2 × 2, 2 × 3, 2 × 4, 2 × 5), and each number is the sum of the two before it. With 3, 5 and 8 in the table, the next one is 5 + 8 = 13.
- Trominoes: there are 2 (straight and L). A shape may be covered with only straight ones, only L, a mix or not at all, and this has to be checked piece by piece.
- Rep-tile: an L enlarged to twice the size on each side splits into 4 copies of the original. Applied again and again, it covers with Ls any 2n × 2n board with one cell missing (in room 11, the 4 × 4 one).
- Tetrominoes: there are 5 (I, O, T, S, L). The I, O, S and L pieces always cover 2 cells of each colour; the T covers 3 of one colour and 1 of the other. The 4 × 5 rectangle has 10 and 10, and that imbalance cannot be corrected: the five pieces do not cover it.
- Pentominoes: there are 12 (F, I, L, N, P, T, U, V, W, X, Y, Z). In room 18 the symmetry of a shape does not decide whether it can be split into two different pentominoes: three symmetric shapes cannot, and the last one, also symmetric, can.
Level: primary school (the lab is not tied to a curriculum year). It can be done without prior theory; the colouring rooms (3, 4, 7 and 13) ask for the most reasoning and tend to start a conversation. In the last block the teacher hands out the challenges designed by the class once there are at least two valid submissions; a student who is alone gets a fixed challenge instead of waiting.
Room-by-room contents
Room 1 · What is a polyomino?
A polyomino is made by joining equal squares along a full side. The room shows three examples: two squares joined along a side count; joined only at a corner, they do not; and offset by half a side, they do not either. Then it presents four pairs of squares and the student marks the ones that are a domino (the smallest polyomino). Once that is right, they practise the rule by covering a 3 × 2 rectangle with dominoes, dragging the pieces.
Student tasks
- Mark the pairs of squares that form a domino (those sharing a full side).
- Cover a 3 × 2 shape with dominoes.
Room 2 · Can it be covered with dominoes?
Four shapes: two with 5 cells, one 2 × 3 and one with 8 cells. The student marks the ones that can be covered exactly with dominoes, without knowing how many cells each one has; the count only appears after a miss. A domino always covers 2 cells, so a shape with an odd number of cells cannot be covered. With the two even-area ones, they can check by building them.
Student tasks
- Mark the shapes that can be covered with dominoes.
- Try building the covering of the chosen shapes.
Room 3 · The colouring trick
A 3 × 4 board painted like a chessboard, with semi-transparent dominoes so the colours underneath can be seen. The student places 6 dominoes until the whole board is covered and notices which colours each one hides. Then they answer which cells a domino covers. Each domino takes two neighbouring cells and neighbours on a chessboard have different colours, so, horizontally or vertically, it always covers one white and one black cell.
Student tasks
- Cover the 3 × 4 board with 6 dominoes.
- Choose how many white and black cells each domino covers (1 white and 1 black, always).
Room 4 · The mutilated board
A 6 × 6 board with two opposite corners cut off: 34 cells remain, an even number, and it looks as if 17 dominoes would cover it. First the student counts the cells of each colour. There are 18 of one colour and 16 of the other, because the two cut corners are the same colour. With that count they decide whether 17 dominoes cover it (no). Then, on the same board, they type the maximum number of dominoes that fit (16, the number of cells of the less numerous colour).
Student tasks
- Count the white and black cells on the board (18 and 16).
- Decide whether it can be covered with 17 dominoes (no).
- Type the maximum number of dominoes that fit (16).
Room 5 · How many ways?
The same rectangle can be covered with dominoes in several ways. The student looks for all the ways to cover a 3 × 2 board (3) and a 4 × 2 board (5), with a counter of those found and a warning if one is repeated. For 5 × 2 they do not have to look: all of them are shown at once (8). Then they see a table of widths and ways: each number is the sum of the two before it, the Fibonacci sequence. To finish, they predict how many ways there are on a 6 × 2 board.
Student tasks
- Find the 3 ways to cover a 3 × 2 board.
- Find the 5 ways to cover a 4 × 2 board.
- Look at the count of the 8 ways on the 5 × 2 board.
- Predict how many ways there are on a 6 × 2 board (13).
Room 6 · Build to resist
The student marks cells on a grid to build a shape that can be walked through entirely, going from cell to cell horizontally or vertically, and where, however much they try, no more than 2 dominoes ever fit at once. An example with 3 cells is shown in which only 1 domino fits. The room warns if a cell disconnects the shape, splits it into pieces or would already allow 3 dominoes, and keeps count of the marked cells. With 5 cells it is easy, because 3 dominoes need 6; the challenge is to reach 6 or more. The maximum is 9.
Student tasks
- Mark a connected shape in which 3 dominoes do not fit.
- Reach at least 6 cells.
- Try for the maximum, 9 cells.
Room 7 · The star challenge
A star-shaped figure of 25 cells that cannot be covered entirely with dominoes. The student places as many dominoes as they can without overlapping them and presses Check. If it is not the maximum, the room says so and they keep trying. Once they find it (9), the room explains why it is the maximum: it paints the star like a chessboard and the student counts the black and white cells (9 and 16). Since a domino covers one of each colour, there cannot be more dominoes than black cells.
Student tasks
- Place as many dominoes as possible on the star (9).
- Count the black and white cells of the star (9 and 16).
Room 8 · Trominoes
A tromino has 3 cells. On a 3 × 3 grid, the student paints a shape and presses “Check shape”. A new shape is saved as a thumbnail and repeating one already found is flagged. The room does not say how many there are: after the two (the straight one and the L) have been found, it asks whether the student thinks there are more. If they try another, it is explained that rotating or reflecting a shape does not turn it into another one.
Student tasks
- Paint and check shapes of 3 connected cells.
- Find the two tromino shapes (straight and L).
- Answer whether they think there are more and, if they try, see why rotations and reflections count as the same piece.
Room 9 · Classify the shape
Five shapes, one after another. For each, the student decides how it can be covered with trominoes: only straight ones, only L, a mix or not at all. It is not always obvious: a 6-cell shape has three in a row but can only be covered with two Ls; another 6-cell shape cannot be covered at all; a 9-cell one needs a straight piece and an L; another 9-cell one looks mixable but can only be covered with Ls; and a T-shaped one is covered with two straight pieces. Each wrong option comes with the reason why it fails.
Student tasks
- Classify shape 1 (only with L).
- Classify shape 2 (it cannot be covered).
- Classify shape 3 (mixing straight and L).
- Classify shape 4 (only with L).
- Classify shape 5, T-shaped (only with straight pieces).
Room 10 · The big L
A big L that is twice the size, on each side, of an L-tromino. The student splits it into 4 equal L-trominoes by dragging the pieces. At the end, the room says that an L enlarged to twice the size can always be split into 4 copies of the original and that this is why it is called a rep-tile.
Student tasks
- Place the 4 L-trominoes to cover the big L.
Room 11 · The board with a hole
A 4 × 4 board with one cell missing. The student covers it entirely with 5 L-trominoes, with no cell left over or missing. It is the same idea as in the previous room applied twice: the board is split into smaller quadrants. At the end it is noted that the trick works for any 2^n × 2^n board with one cell missing, wherever it is.
Student tasks
- Cover the 4 × 4 board with one cell removed using 5 L-trominoes.
Room 12 · Tetrominoes
A tetromino has 4 cells. As in room 8, the student paints shapes on a 4 × 4 grid, checks them and they are saved as thumbnails. How many there are is not said. Repeated shapes, whether rotated or reflected, are flagged. When all 5 are found their names are shown: I, O, T, S and L; if the student thinks there are more, it is explained that rotations and reflections do not count as new.
Student tasks
- Paint and check shapes of 4 connected cells.
- Find the 5 tetrominoes (I, O, T, S and L).
- Answer whether they think there are more.
Room 13 · The 4×5 rectangle
The 5 tetrominoes add up to 20 cells, exactly those of a 4 × 5 rectangle. The student tries to fit them in without anyone telling them it is not possible. After 15 actions (placing or removing a piece) the board is shown painted like a chessboard, with the count of white and black cells (10 and 10). Then they are asked which of the five pieces behaves differently with the colouring: the T covers 3 cells of one colour and 1 of the other, and the rest cover 2 and 2. That imbalance cannot be fixed with the others, which is why there is no solution.
Student tasks
- Try to fit the 5 pieces into the 4 × 5 rectangle.
- Look at the count of white and black cells of the rectangle (10 and 10) and choose which tetromino splits the colour differently (the T).
Room 14 · Tetromino challenge (I)
An 8-cell board that is covered exactly with the pieces shown, an I and an L. Every assembly challenge in this block was built by placing real pieces, so it has a solution. A counter shows the pieces placed and, at the end, a message and a button to continue appear.
Student tasks
- Cover the 8-cell shape with an I and an L.
Room 15 · Tetromino challenge (II)
A somewhat bigger shape, of 12 cells, covered with an O, a T and an S. It is done with the same engine: the pieces are dragged, rotated and flipped until they fit.
Student tasks
- Cover the 12-cell shape with an O, a T and an S.
Room 16 · Tetromino challenge (III)
The hardest in the block: an elongated 16-cell shape covered with four pieces, L, T, S and I.
Student tasks
- Cover the 16-cell shape with an L, a T, an S and an I.
Room 17 · Pentominoes
A pentomino has 5 cells. Same mechanism as rooms 8 and 12, with a 5 × 5 grid and thumbnails of the shapes found, without saying how many there are. When all 12 are found their names are shown: F, I, L, N, P, T, U, V, W, X, Y and Z, each after the letter it resembles. If the student thinks there are more, it is explained that rotating or reflecting does not create a new shape.
Student tasks
- Paint and check shapes of 5 connected cells.
- Find the 12 pentominoes.
- Answer whether they think there are more.
Room 18 · Pentomino pairs
Seven 10-cell shapes, one after another. For each, the student decides whether it can be built from two different pentominoes: they paint 5 cells to propose one half (the other is the unpainted cells) or press “It’s impossible”. The room says whether the split forms two valid, distinct pentominoes or two equal ones, and at the end of each shape it shows the solution. Shapes 2, 4 and 6, which are symmetric, cannot be split into two different ones. Shape 7, also symmetric, can: symmetry alone does not decide.
Student tasks
- Split the shapes that can be split (1, 3, 5 and 7) into two different pentominoes by painting 5 cells.
- Press “It’s impossible” on the shapes that cannot be split (2, 4 and 6).
- Say, for shape 7, whether they think it will also be impossible because it is symmetric (it can be split).
Room 19 · Pentomino challenge (I)
A 10-cell shape covered with two pentominoes, a P and an L. The pieces are dragged, rotated and flipped on the board.
Student tasks
- Cover the 10-cell shape with a P and an L.
Room 20 · Pentomino challenge (II)
A somewhat bigger shape, of 15 cells, covered with three pentominoes: T, U and N.
Student tasks
- Cover the 15-cell shape with a T, a U and an N.
Room 21 · Silhouette challenge
A silhouette with more pieces in play: five pentominoes, V, W, X, Y and Z. It serves as a warm-up before the student invents their own.
Student tasks
- Cover the silhouette with a V, a W, an X, a Y and a Z.
Room 22 · Design your challenge
The student chooses 5 different pentominoes out of the 12 and fits them together, side by side with no gaps, in a free 11 × 11 area to create their own silhouette. On submitting, it is validated: the silhouette must be connected, cannot be a perfect rectangle (too easy) and cannot be solvable in more than 3 different ways with those pieces. If it does not meet these conditions, it says why and the student can change it; if it is accepted, the teacher can hand it out to classmates.
Student tasks
- Choose 5 different pentominoes and join them into a silhouette with no gaps.
- Submit the challenge and, if it is rejected, adjust it according to the reason.
Room 23 · A classmate’s challenge
The student solves the silhouette designed by a classmate, with the same 5 pieces they used, and sees the name of who created it. Until it arrives, a waiting message appears. The teacher hands out the challenges once there are at least two valid submissions, so each student gets one that is not their own. If the student is alone in the lab, they get a fixed challenge instead of waiting.
Student tasks
- Wait for the teacher to hand out the class challenges.
- Cover the classmate’s silhouette with their 5 pentominoes.
Rooms to project
The most striking ones to show and discuss in class.