Problem 13 · 2015 Math Kangaroo
Medium
Algebra & Patterns
substitutionsum-constraint
Each inhabitant of a distant planet has at least two ears. Three inhabitants called Imi, Dimi and Trimi meet in a trendy crater. Imi says: “I can see 8 ears.” Dimi then replies: “I can see 7 ears.” Finally Trimi says: “Strange, I can only see 5 ears.” None of them can see their own ears. How many ears does Trimi have?
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Answer: C — 5
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Hint 1 of 2
Add up what all three claim they can see; every ear gets counted by the two who don't own it.
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Hint 2 of 2
Sum of the three counts = 2 × (total ears); subtract the count that excludes Trimi to get Trimi's ears.
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Approach: add the three counts and halve
- Each ear is seen by exactly the two others, so 8 + 7 + 5 = 2 × (total ears).
- Total ears = 20 / 2 = 10.
- Imi (sees 8) has 10 − 8 = 2; Dimi (sees 7) has 3; Trimi (sees 5) has 10 − 5 = 5.
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