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Puzzled: Weighed in the Balance

Many of us have pondered puzzles involving a set of n coins and a balance scale, the idea being typically to find the counterfeit coin and determine whether it is lighter or heavier than the others using the fewest possible weighings. Here we take a slightly different tack, but the equipment is familiar: a set of coins and a balance scale that can tell us which of two sets of coins is heavier or that they are of equal weight.
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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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UF1 Figure. A balanced balance scale for Puzzle 1.

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