Math · Stage 5

Apply the pigeonhole principle to guarantee outcomes

A guarantee has to survive the unluckiest draw. Plan for the worst case, then force the result — a matching pair, three of a kind, or at least ⌈N ÷ holes⌉ in the fullest hole.

✦ Math📘 Stage 5 · S5⏱ ~12 min🧒 Ages 9–10★ Free to start

I can apply the pigeonhole principle to guarantee outcomes.

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Kavi's Impossible-to-Lose Bet 🎲

"I'll guarantee it," hoots Kavi, "not just hope for it!" A guarantee is a promise that holds even on your unluckiest day 😈. So Kavi always pictures the worst possible draw first — one in every hole — and then forces the result with just one more. Pair, triple, or a crowded hole: find the holes, fill them the meanest way, then add what it takes ✨.

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What your child will learn

1Lucky is not the same as guaranteedA drawer mixes socks of 3 colours. You might pull a matching pair on your 2nd grab — but that is luck, not a promise. To guarantee a pair, your answer must work even on the unluckiest draw. Tap each card.

2Guarantee THREE of a kindNow you need three socks the same, not just a pair. The meanest draw fills every colour twice first — still no triple. With 3 colours that is 2 × 3 = 6 socks. Sort them into colour-bins and watch the 7th sock land.

3The worst case + 1, for ANY targetWrite the guarantee as a number bond. To be sure of m of a kind across h holes, the worst case is (m − 1) in every hole, then add 1. Three of a kind from 4 colours: worst case (3 − 1) × 4 = 8, then one more.

4Share evenly → the fullest hole is forcedSometimes the count is fixed and you ask the other way: how full must the busiest hole be? Put 25 pigeons into 6 holes. Share them as evenly as you can — that keeps the busiest hole as small as possible.

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A peek inside

A bag mixes gloves of 4 colours. How many must you grab to be SURE of THREE that match?

target m = 3, holes h = 4 · worst case = two of every colour = (3 − 1) × 4 = 8, no triple yet · one more forces a third → 8 + 1

Try this at home: Socks come in 3 colours. How many must you grab to be SURE of a matching PAIR? Why?

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Part of Math · Stage 5 · S5 on KAVI Academy — a secular, whole-child learning platform for ages 5–12.