Problem 14 · 2024 Math Kangaroo
Medium
Counting & Probability
careful-countingcasework
Lucas has these five puzzle pieces (shown on the right): a smiling head, a banana-tail, and three middle pieces. They snap together only where a bump fits into a notch. He wants to make a caterpillar with a head, a tail, and 1, 2 or 3 pieces in between. How many different caterpillars can Lucas build?
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Answer: B — 4
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Hint 1 of 3
The head only joins on one side and the tail only joins on one side; the pieces fit only where a bump (tab) meets a notch (socket).
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Hint 2 of 3
A caterpillar is head, then 1, 2, or 3 middle pieces, then tail — build the chains by matching bumps to notches.
Still stuck? Show hint 3 →
Hint 3 of 3
Go case by case: count the legal chains with exactly 1 middle piece, then 2, then 3, and add them up.
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Approach: match the connectors (bump-to-notch) and count the legal chains of 1, 2, or 3 middle pieces
- A piece can join another only where a bump fits into a notch, and the head and tail each connect on just one side.
- Try 1 middle piece between head and tail: only the pieces whose bumps and notches line up both ways work.
- Now try 2 middle pieces, then 3 middle pieces, keeping every join a bump-into-notch fit.
- Adding up all the chains that connect properly gives 4 (B) different caterpillars.
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