The Colouring Trick That Proves a Floor Cannot Be Tiled
Learn to decide whether a grid can be covered by domino tiles using parity and colouring, find which shapes tile the plane, and see why rearranging tangram pieces never changes the area.
How can you prove a floor cannot be tiled without trying every arrangement?
By colouring it like a chessboard and counting. Each tile always covers one dark square and one light square, so if the dark and light counts differ, no arrangement can ever work — however long you try.
That is a proof, not a failed attempt. This page covers everything in the CBSE Class 7 Mathematics chapter's second half: deciding when a grid can be covered by domino tiles, which shapes tile the plane, and dissection puzzles such as the tangram.
That is a proof, not a failed attempt. This page covers everything in the CBSE Class 7 Mathematics chapter's second half: deciding when a grid can be covered by domino tiles, which shapes tile the plane, and dissection puzzles such as the tangram.
Can every grid be covered by 2 by 1 tiles?
No, and two different arguments rule cases out.
The first is parity. Each tile covers exactly 2 squares, so the total number of squares must be even. A grid has
an odd number, so it cannot be covered — one square must always be left over. In general an grid needs to be even, which means at least one of and must be even.
The second is colouring, and it catches cases parity misses. Colour the grid like a chessboard, dark and light alternating. Every tile covers one square of each colour, because neighbouring squares always differ in colour.
Now take an grid and remove two opposite corner squares. That leaves squares, an even number, so parity permits it. But opposite corners share the same colour, so the removal leaves 30 of one colour and 32 of the other. Covering 62 squares needs 31 tiles carrying 31 of each colour, and 30 is not 31 — so it is impossible.
That is the pattern to remember: an even count is necessary but not sufficient. Passing the parity test does not mean a tiling exists; the colour counts must balance too.
The first is parity. Each tile covers exactly 2 squares, so the total number of squares must be even. A grid has
an odd number, so it cannot be covered — one square must always be left over. In general an grid needs to be even, which means at least one of and must be even.
The second is colouring, and it catches cases parity misses. Colour the grid like a chessboard, dark and light alternating. Every tile covers one square of each colour, because neighbouring squares always differ in colour.
Now take an grid and remove two opposite corner squares. That leaves squares, an even number, so parity permits it. But opposite corners share the same colour, so the removal leaves 30 of one colour and 32 of the other. Covering 62 squares needs 31 tiles carrying 31 of each colour, and 30 is not 31 — so it is impossible.
That is the pattern to remember: an even count is necessary but not sufficient. Passing the parity test does not mean a tiling exists; the colour counts must balance too.
Which shapes tile the plane with no gaps or overlaps?
A shape tiles the plane if copies of it cover a surface completely, leaving no gaps and no overlaps. The test is what happens at a vertex: the angles meeting there must add to exactly , a full turn.
For regular polygons the arithmetic decides it:
- Equilateral triangle, angle : tiles meet at a point. It tiles.
- Square, angle : tiles meet. It tiles.
- Regular hexagon, angle : tiles meet. It tiles.
- Regular pentagon, angle : is not a whole number, so it cannot tile — three pentagons leave of gap and four would overlap.
So among regular polygons only those three work, which is why floor tiles, honeycombs and jaali screens keep returning to triangles, squares and hexagons.
Other shapes tile too, once they need not be regular. Any triangle tiles, and so does any quadrilateral, because four copies can be turned so their four angles meet at a point and sum to .
To build a pattern, take one tile and repeat it by sliding (translation), turning (rotation) and flipping (reflection). All three keep the shape and size identical, which is what lets the copies fit.
For regular polygons the arithmetic decides it:
- Equilateral triangle, angle : tiles meet at a point. It tiles.
- Square, angle : tiles meet. It tiles.
- Regular hexagon, angle : tiles meet. It tiles.
- Regular pentagon, angle : is not a whole number, so it cannot tile — three pentagons leave of gap and four would overlap.
So among regular polygons only those three work, which is why floor tiles, honeycombs and jaali screens keep returning to triangles, squares and hexagons.
Other shapes tile too, once they need not be regular. Any triangle tiles, and so does any quadrilateral, because four copies can be turned so their four angles meet at a point and sum to .
To build a pattern, take one tile and repeat it by sliding (translation), turning (rotation) and flipping (reflection). All three keep the shape and size identical, which is what lets the copies fit.
Why does rearranging tangram pieces never change the area?
Because cutting and rearranging moves area around without creating or destroying any.
A tangram is a square cut into seven pieces — five triangles, a square and a parallelogram. Those same seven pieces can be rearranged into a bird, a boat, a running figure and hundreds of other outlines.
Every one of those shapes has exactly the area of the original square. If the square had side 8 cm, its area is
and each rearrangement covers cm too, whatever its outline looks like.
This is the key idea behind dissection: area is conserved under cutting and rearranging, so a shape's area can be found by cutting it into pieces you already know how to measure.
The perimeter, however, is not conserved — and this is the point students miss. A long thin tangram figure has a much greater perimeter than the compact square, because rearrangement exposes edges that were previously joined. Same area, different boundary.
That distinction reappears throughout later geometry: cutting a shape up preserves how much surface there is, not how much edge.
A tangram is a square cut into seven pieces — five triangles, a square and a parallelogram. Those same seven pieces can be rearranged into a bird, a boat, a running figure and hundreds of other outlines.
Every one of those shapes has exactly the area of the original square. If the square had side 8 cm, its area is
and each rearrangement covers cm too, whatever its outline looks like.
This is the key idea behind dissection: area is conserved under cutting and rearranging, so a shape's area can be found by cutting it into pieces you already know how to measure.
The perimeter, however, is not conserved — and this is the point students miss. A long thin tangram figure has a much greater perimeter than the compact square, because rearrangement exposes edges that were previously joined. Same area, different boundary.
That distinction reappears throughout later geometry: cutting a shape up preserves how much surface there is, not how much edge.
Exam tip
Exam tip: proving impossibility instead of listing failed attempts
When a question asks whether a covering is possible, the marks are in the argument, not the verdict.
Run the two tests in order. First parity: count the squares and check the total is even. If it is odd, say so and stop — that is a complete proof.
If the count is even, colour the grid and count the two colours. State the imbalance explicitly: each tile covers one dark and one light square, so 31 tiles cover 31 of each, but the board has 30 dark and 32 light — hence impossible.
For a tiling question about shapes, quote the vertex angle and divide into . *A regular pentagon has angles of , and is not a multiple of , so pentagons cannot tile.*
And never answer an impossibility question with "I tried several arrangements and none worked". Trying shows effort; only the counting argument settles every arrangement at once.
Run the two tests in order. First parity: count the squares and check the total is even. If it is odd, say so and stop — that is a complete proof.
If the count is even, colour the grid and count the two colours. State the imbalance explicitly: each tile covers one dark and one light square, so 31 tiles cover 31 of each, but the board has 30 dark and 32 light — hence impossible.
For a tiling question about shapes, quote the vertex angle and divide into . *A regular pentagon has angles of , and is not a multiple of , so pentagons cannot tile.*
And never answer an impossibility question with "I tried several arrangements and none worked". Trying shows effort; only the counting argument settles every arrangement at once.
Did you know
Why do hexagons appear so often in natural patterns?
Because divides exactly three times, so three hexagons close up around every corner with nothing left over.
That makes the hexagon the regular shape with the most sides that still tiles — pentagons fail at , and every regular polygon with more than six sides has angles too large for even three to fit.
Honeycomb cells, the packing of bubbles in foam and many jaali screen designs all settle into the same arrangement for the same arithmetic reason.
That makes the hexagon the regular shape with the most sides that still tiles — pentagons fail at , and every regular polygon with more than six sides has angles too large for even three to fit.
Honeycomb cells, the packing of bubbles in foam and many jaali screen designs all settle into the same arrangement for the same arithmetic reason.
Key takeaways
Tiling, parity and dissection: quick revision
- A tiling needs an even number of squares, so a grid with 25 squares is impossible on parity alone.
- Chessboard colouring is the stronger test: each tile covers one square of each colour, so the two colour counts must be equal.
- An board with opposite corners removed has 62 squares but 30 and 32 of the two colours, so it cannot be tiled — an even count is necessary, not sufficient.
- A shape tiles when the angles meeting at a vertex total : triangles, squares and regular hexagons do; regular pentagons at do not.
- Any triangle and any quadrilateral tiles, and patterns are built by sliding, turning and flipping one tile.
- Dissection preserves area but not perimeter, so every tangram figure has the area of the original square.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Chessboard colouring is the stronger test: each tile covers one square of each colour, so the two colour counts must be equal.
- An board with opposite corners removed has 62 squares but 30 and 32 of the two colours, so it cannot be tiled — an even count is necessary, not sufficient.
- A shape tiles when the angles meeting at a vertex total : triangles, squares and regular hexagons do; regular pentagons at do not.
- Any triangle and any quadrilateral tiles, and patterns are built by sliding, turning and flipping one tile.
- Dissection preserves area but not perimeter, so every tangram figure has the area of the original square.
You will remember all of this far better after answering five questions on it than after reading it twice.