Problem
Two-coloring decides domino tiling
A domino is two squares joined edge to edge. Four odd-shaped boards are painted like a checkerboard. Dominoes must cover the board with no gaps or overhang. Pick the board that can be covered.
Reasoning · Grade 5-2 Parity
Your answer
Watch out Equal colours feels conclusive, but you still have to build a covering.
How to solve
Strategy Draw a Diagram — Trying to lay dominoes on all 4 boards by trial and error could take forever, because failing once only means that one attempt failed, not that every attempt must. So I use the colouring that is already drawn on the boards: it turns 'can I cover this?' into a question about counting, which is quick and certain. I count each colour on each board row by row, throw out every board where the counts do not match, and then — because matching counts are not by themselves a promise — I actually build a covering for the survivor, laying the dominoes one by one the way you would with real tiles.
1STEP 1
See what one domino always covers
One domino always covers one black and one white.
1 domino = 1 black + 1 white
This is the whole idea in one sentence, and it needs no calculation at all — just the observation that the colours alternate, so touching squares never match.
3.OA.D.9Draw A DiagramWhatever way a domino is laid, it always covers one dark square and one light square.
Why?
On a checkerboard the colours alternate, so two squares sharing an edge never have the same colour.
🧱Alternating two-coloringColor a board like a checkerboard and touching squares never match, so any piece covering two neighbours always takes one of each color.
Why?
A domino covers exactly two neighbouring squares, so it takes exactly one of each colour every single time.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
2STEP 2
Turn that into a test the board must pass
So a board needs equal numbers of each colour.
n dominoes cover n black + n white = 2n squares
Counting by colour instead of by position converts a hard picture puzzle into simple whole-number bookkeeping, and a count that comes out wrong rules out every possible attempt at once.
3.OA.D.9Organize Information In More Ways3STEP 3
Count board ① by rows
Board 1 has 8 and 6, so it cannot be covered.
3 + 4 + 4 + 3 = 14, 2 + 2 + 2 + 2 = 8 black, 14 - 8 = 6 white, 8 ≠ 6
Counting a shape row by row and adding is a Grade 2 array skill, and doing it the same way on every board keeps a square from being missed or double-counted.
2.OA.C.4Make A Systematic List4STEP 4
Count board ② by rows
Board 2 has an odd number of squares, so no.
5 × 5 = 25 (odd), 13 white ≠ 12 black
Deciding whether 25 objects can be paired off with none left over is the plain odd-and-even question from Grade 2 — an odd total is by itself enough to rule the board out, before colours are even considered.
2.OA.C.3Make A Systematic List5STEP 5
Count board ④ by rows
Board 4 has 10 and 12, so no.
4 + 5 + 5 + 5 + 3 = 22, 1 + 3 + 2 + 3 + 1 = 10 black, 22 - 10 = 12 white, 10 ≠ 12
Board ④ shows why the colour test is worth doing: an even number of squares is not enough, and only the colour count catches this board.
2.OA.C.4Make A Systematic List6STEP 6
Count board ③ by rows
Only board 3 has 13 and 13.
2 + 6 + 6 + 6 + 6 = 26, 12 + 1 = 13 black, 12 + 1 = 13 white
A row of 6 alternating squares splits evenly 3 and 3 without any counting effort, so the only careful work left is deciding the colours of the 2 bumps on top.
2.OA.C.4Make A Systematic List7STEP 7
Passing the test is not yet a cover — build one
Build a cover with 13 dominoes.
2 + 1 + 2 + 2 + 3 + 3 = 13 dominoes
Working top-down and always dealing with the leftmost uncovered square keeps the placing honest — and 13 real dominoes on paper settle the question in a way no amount of counting can.
2.OA.C.4Create A Physical Representation8STEP 8
Check the built cover
All 26 squares get covered, so board 3 it is.
13 × 2 = 26
Multiplying the number of dominoes by 2 and matching it against the square count is a one-line proof that nothing was left out or covered twice.
3.OA.A.1Create A Physical RepresentationAnswer
3
13 + 13 = 26
The two halves of the argument are of different kinds and both are needed. For boards ①, ② and ④ the conclusion is a proof of impossibility: since every domino covers 1 black and 1 white, unequal colour counts (8 against 6 on board ①, 10 against 12 on board ④) or an odd total (25 on board ②) cannot be fixed by any arrangement at all. For board ③ the equal counts 13 and 13 only mean nothing is obviously wrong, which is why an actual layout of 13 dominoes was exhibited; 13 times 2 = 26 matches the square count exactly, so the cover is complete. As a size check, each board's total agrees with the picture: 16 - 2 = 14, 25, 24 + 2 = 26 and 25 - 3 = 22, and these all match the row-by-row counts.
Takeaway
Colour the board like a checkerboard: a domino always eats 1 of each colour, so if the 2 colours are not tied the board is hopeless — and if they are tied, go ahead and lay the tiles to be sure.
- See what one domino always covers
- Turn that into a test the board must pass
- Count board ① by rows
- Count board ② by rows
- Count board ④ by rows
- Count board ③ by rows
- Passing the test is not yet a cover — build one
- Check the built cover