Math · Stage 6

Apply the pigeonhole principle and extremal reasoning to prove a result

If there are more pigeons than holes, some hole MUST hold two — you don't even have to look. That one idea, plus 'think about the worst case', lets you PROVE things that are impossible to check one by one.

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

I can apply the pigeonhole principle and extremal reasoning to prove a result.

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Kavi's Dark Sock Drawer 🧦

Kavi reached into the sock drawer in the dark 🌙. “There are only 3 colours in here — red, blue, green. If I grab 4 socks, two of them must be the same colour… and I don't even need to switch on the light to be sure!” 💡 That is the pigeonhole principle: when you have more things than boxes, some box has to hold at least two. No checking — it is guaranteed.

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What your child will learn

14 socks, only 3 coloursKavi has socks in 3 colours and grabs 4. Tap each card — one tries to dodge a repeat, the other tells the truth.

2Even the fairest sharing leaves one fullPut 10 balls into 3 boxes as evenly as possible. You try hard to keep them level — but 10 does not split into 3 equal piles.

3Extremal reasoning: picture the worst caseTo PROVE a 'must happen' claim you don't test every case — you test the hardest one. Tap each card.

4Forced, or just possible?Pigeonhole only fires when items > boxes. If they are equal or fewer, a repeat is possible but not forced. Drop each claim in the right box →

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A peek inside

A class has 13 children. Every birthday falls in one of the 12 months. Show that at least two children share a birth month.

months are the boxes (12), children are the pigeons (13); 13 > 12, so the children cannot each take a different month

Try this at home: Pigeons: 7 pigeons fly into 6 holes → at least how many share a hole?

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.