Math · Stage 6

Use parity, colouring and invariant arguments

Some puzzles are impossible — and you can prove it without trying every case. Watch a hidden quantity that every move must keep the same: its parity (even/odd), its colour balance, or an invariant.

✦ Math📘 Stage 6 · S6⏱ ~14 min🧒 Ages 10–12★ Free to start

I can use parity, colouring and invariant arguments.

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Kavi and the Impossible Tiling 🧩

Kavi tried to cover a chessboard with its two opposite corners snipped off using 2×1 dominoes — and kept failing. So Kavi coloured it like a checkerboard and counted: 32 white, 30 black. “Every domino covers one white and one black — so two white squares will always be left lonely. It's impossible, and I didn't try a single tiling!” That shortcut — a parity check, a colouring, an invariant that never changes — is today's superpower.

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What your child will learn

1Even or odd? Sort each sumBefore any clever argument comes the atom: is a number even (a count-by-2 number) or odd? Work out each sum and drop it in the right box →

2A parity argument: counting the oddsTap each card. Adding only odd numbers — the answer's parity depends on how many odds there are, not on which ones.

3Colour the board, then countColour a chessboard like a checkerboard and remove the two opposite corners (both are black). Every 2×1 domino must cover one black AND one white square — so the colours have to match.

4The switch that only has two statesTap along. A light flips OFF → ON → OFF → ON … it only ever cycles between two states.

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A peek inside

Is 7 + 7 + 7 (three odd numbers) even or odd? Don't add — use a parity argument.

odd + odd = even, then even + odd = odd; an odd count of odds gives an odd total

Try this at home: Parity of a sum: is <b>2 + 4 + 6 + 9</b> even or odd? → __

Ready to play and learn?

Start the free interactive lesson, or print the handout for screen-free play.

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Part of Math · Stage 6 · S6 on KAVI Academy — a secular, whole-child learning platform for ages 5–12.