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Puzzled: Weighed in the Balance

a balanced balance scale

A balanced balance scale for Puzzle 1.

  1. You have 13 coins with the property that any 12 of them can be split into two piles of six each that balance perfectly on the scale (see the figure here). Now prove all the coins have the same weight. (Advice: Try this for integer weights first, then rational, then—for the brave—arbitrary positive real weights.)
  2. Eight coins have at most two different weights; now show that with three weighings, you can determine whether all the coins have the same weight.
  3. Following the same rules as in the second puzzle, now solve it with 10 coins.

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Peter Winkler ( is William Morrill Professor of Mathematics and Computer Science at Dartmouth College, Hanover, NH.

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Readers are encouraged to submit prospective puzzles for future columns to

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UF1Figure. A balanced balance scale for Puzzle 1.

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