Pair the rungs from each end. The first and the last hold 101 pigeons between them, so do the second and the 99th, and so on in to the 50th and 51st. That’s 50 pairs of 101.
A ladder has 100 rungs. One pigeon sits on the first rung, two on the second, three on the third and so on, up to 100 on the top rung. How many pigeons are on the ladder?
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Pair the bottom rung with the top one.
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5,050
Source: Alcuin of York (attributed), Propositiones ad Acuendos Juvenes, no. 42, c. 800