A room contains more than one Martian. Each Martian has two hands, with at least one finger on each hand, and all Martians have the same number of fingers. Altogether there are between 200 and 300 Martian fingers in the room; if you knew the exact number, you could deduce the exact number of Martians. How many Martians are there, and how many fingers does each one have?
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The key is that the number of fingers can tell us uniquely the number of Martians. That’s a tall order. It eliminates, say, 246 fingers, because that’s too ambiguous: There might be 82 Martians with 3 fingers each, or 3 Martians with 82 fingers each, and so on. The only possibility that avoids this uncertainty is that the quantity of Martians and the quantity of fingers per Martian are expressed by the same number, and that this number is not composite. That means we’re looking for the square of a prime number, and the only such number in the range 200-300 is 289, or 172. So there are 17 Martians, each of which has 17 fingers. This is the only number between 200 and 300 that permits such a definite assertion.
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