Five Girls, Ten Weighings in Pairs — Find Each One's Weight
Five schoolgirls step on a scale two at a time, in every possible pairing, giving ten different combined weights. None of them is ever weighed alone. From just those ten pair-totals, can you work out exactly how much each girl weighs on her own?
Reveal the answer
Yes: since every girl appears in exactly four of the ten pairings, adding up all ten totals and dividing by four gives the sum of all five girls' weights; from there, comparing the heaviest and lightest pairings lets you peel off each individual weight one at a time. In the original puzzle the ten pair-weighings are 114, 116, 118, 120, 121, 122, 123, 124, 125 and 129 pounds, giving individual weights of 56, 58, 60, 64 and 65 pounds. From Sam Loyd's "Outwitting the Weighing Machine," collected in his Cyclopedia of Puzzles (1914).
— Sam Loyd, Sam Loyd's Cyclopedia of Puzzles — Posthumous compilation edited by Sam Loyd Jr., 1914; puzzle "Outwitting the Weighing Machine"
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