Puzzles we wrote
Colouring arguments: proving something is impossible
Colour a board like a chessboard, in alternating dark and light squares. Many moves, tiles and paths must always switch colour or always cover one of each. Counting colours can then prove that a task can never be done, without trying every possibility.
Colour first
Shade the grid in a checkerboard pattern. Every square now has exactly one colour, and neighbours always differ.
Find what never changes
A domino always covers one dark and one light square. A step to a neighbouring square always changes colour.
Count and compare
If the task needs equal numbers of each colour and the board has unequal numbers, the task is impossible.
Original puzzles solved with colouring| Puzzle | The colouring idea | Answer |
|---|
| An 8 by 8 board has two opposite corners removed. Can 31 dominoes cover the remaining 62 squares? | Opposite corners share a colour, leaving 30 of one colour and 32 of the other; each domino covers one of each | No |
| Can dominoes exactly cover a 5 by 5 board? | 25 squares is odd, and every domino covers 2 | No |
| A 4 by 4 board loses the two corners at the ends of its top row. Does colouring rule out a domino covering? | Those corners are different colours, leaving 7 of each, so colouring cannot rule it out | No; and a covering does exist |
| A knight jumps around a chessboard. Can it return to its start in an odd number of moves? | Every knight move changes colour, so it is back on its starting colour only after an even number of moves | No |
| On a 3 by 3 grid, a path visits every square once, stepping to neighbours. Can it start on an edge square that is not a corner? | Shade corners and centre dark: 5 dark, 4 light. A 9-square path alternates colours, so it must start and end on dark | No; it must start on a corner or the centre |
The first puzzle is a classic for a reason. Trying to place 31 dominoes by hand takes ages and never settles the question. Colouring settles it in one line: the board has 30 squares of one colour and 32 of the other, but 31 dominoes would cover exactly 31 of each. No amount of clever placing can fix that.
The grid path in the last row rewards slow reading. Shade the four corners and the centre dark; the four edge squares in between are light. Any path that steps only to neighbours must alternate dark, light, dark, light. A path through all nine squares therefore has five squares in one colour and four in the other, so it must begin and end on the colour that appears five times. Dark appears five times, so a full path has to start on a corner or the centre. Starting on a light edge square would need five light squares, and there are only four. One count rules out every attempt at once, which is exactly what makes colouring arguments so satisfying.
Parity, odd and even, is the simplest relative of these ideas. The 5 by 5 board fails before any colouring at all, because 25 squares cannot be split into pairs. Encourage students to try the cheapest argument first: count, then colour, then search for a construction only if neither settles the question.
The third puzzle teaches the limit of the method, and it is just as important. When the colours balance, colouring proves nothing either way. A student then has to find an actual covering, here a domino across the middle two top squares and three rows of dominoes below, to show the task is possible. Knowing when an argument works, and when it does not, is the heart of mathematical reasoning.
