Problem 21 · 2012 Math Kangaroo
Hard
Logic & Word Problems
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Initially the side length of a talking magic square is 8 cm. Every time it speaks the truth its sides each decrease by 2 cm. If it lies its perimeter doubles. It says four sentences, two of which are true and two are false, in which order is unknown. What is the biggest possible perimeter it can have after those four sentences?
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Answer: D — 112
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Hint 1 of 3
A true sentence shrinks each side by 2; a false one doubles the whole perimeter (so it doubles each side too).
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Hint 2 of 3
To make the perimeter biggest, decide the best order for the two lies and two truths.
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Hint 3 of 3
A subtraction of 2 hurts less when the side is small, so think about when to place the truths.
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Approach: order the operations to maximise
- Start with side 8 (perimeter 32). A truth lowers the side by 2; a lie doubles the side (and so doubles the perimeter).
- Doubling first makes each later −2 a smaller fraction lost, so do both lies first: 8 → 16 → 32, then both truths: 32 → 30 → 28.
- The biggest side reachable is 28, giving perimeter 4 × 28 = 112, answer D.
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