Subject: Re: Old-Ball puzzle
Date: Sat, 29 Oct 96 21:54:00
From: Alex Bogomolny

There is one solution that depends on your ability to mark and thus distinguish the balls without, of course, affecting their weights.

Let the balls be numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c. Split them into three groups 1234, 5678, 9abc. Now compare 1234 and 5678. We'll consider two possible outcomes.

  1. They are even. We have to locate a deffective ball among 9abc. Compare 9 and a. Two outcomes are possible.
    1. They are even. Compare b with any of the good balls. If it's good then c is deffective. Otherwise, b is.
    2. a is heavier. Compare it to any good ball. Conclude as before.
  2. 1234 is heavier, 5678 is lighter. The rest are known to be good! Compare 1259 and 36ab. Three outcomes are possible
    1. 1259 and 36ab are even. Then either 4 is deffective and heavy or either of 7 or 8 is deffective and light. Compare 7 and 8. Two outcomes are possible:
      1. They are even. Then 4 is deffective.
      2. 7 (or 8) is lighter. Then 7 (or 8) is deffective.
    2. 1259 is heavier. In this case either 1 or 2 is heavy or 5 is light. Compare 1 and 2. If they are even, 5 is deffective. Otherwise, deffective is the heavier of the two.
    3. 1259 is lighter. Either 5 is light or 6 is heavy. Compare either to a good ball to find out.

Best Regards

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