Problem 29 · 2021 Math Kangaroo
Stretch
Number Theory
sum-constraintcasework
In a group of 10 elves and trolls, each was given a token with a different number from 1 to 10 written on it. They were each asked what number was on their token, and all answered with a number from 1 to 10. The sum of the answers was 36. Each troll told a lie and each elf told the truth. What is the smallest number of trolls there could be in the group?
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Answer: B — 3
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Hint 1 of 2
If everyone told the truth the answers would total 1 + 2 + … + 10 = 55; the actual total is only 36.
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Hint 2 of 2
Each troll replaces its own token number with a smaller answer; how much total drop can just a few trolls create?
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Approach: cover the shortfall with the fewest liars
- Honest answers would total 1 + 2 + … + 10 = 55, but the answers added to 36, so the trolls' lies pulled the total down by 19.
- Each troll's biggest possible drop is from token 10 down to answer 1, a drop of 9; one troll can drop at most 9 and two trolls at most 9 + 8 = 17, both short of 19.
- Three trolls can manage it — for example tokens 10, 9, 8 answering 1, 1, 1 drops the total by 9 + 8 + 7 = 24, and other choices hit exactly 19 — so the smallest number of trolls is 3.
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