Problem 30 · 2014 Math Kangaroo
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Logic & Word Problems
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In the forests of a magical island kingdom there are three kinds of animals: lions, wolves and goats. Wolves can eat goats, and lions can eat both wolves and goats. Since it is a magical island kingdom, a wolf that eats a goat changes into a lion, a lion that eats a goat changes into a wolf, and a lion that eats a wolf changes into a goat. To begin with there were 17 goats, 55 wolves and 6 lions on the island. After some time no more eating is possible. What is the maximum number of animals that can still be on the island?
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Answer: D — 23
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Hint 1 of 2
Every meal removes exactly one animal, so the total only goes down.
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Hint 2 of 2
Eating stops only when a single species remains — which species can grow largest?
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Approach: track the invariant; eating ends with one species
- Each eating event removes one animal, so the herd can only shrink, and it stops only when just one kind of animal is left (any two different kinds can still eat).
- Following the allowed transformations from 17 goats, 55 wolves, 6 lions, the largest single-species end state reachable is all lions.
- That leaves a maximum of 23 animals.
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