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The 24 Coins Puzzle revisited

By warwick@cs.uq.oz.au

Problem from a puzzles book:

Your eccentric Uncle Jake is a rare coin collector. He has twenty-four coins that appear identical, except only one is solid gold; the others are made of a heavier alloy. He sits you in front of a balance and says you can have the gold coin if you can find it. You can use the scale only three times. How can you find the lightest coin?

Since the gold coin is the lightest, all the other coins are made of highly toxic, radio-active heavy metals. To find the gold coin:

  1. Jake should: divide the coins into two piles of 12 coins.

    The pile with all alloy coins will be at critical mass, and will enter a chain reaction, annihilating itself.

  2. Take the remaining pile, and divide it into two piles of 6 coins. Place the piles on opposite sides of the scales. The scales have thus been used for the first time.

    The heavier pile will be hottest, and will melt through the scale pan beneath it.

  3. Take the pile of 6 coins in the remaining scale pan, and divide into two groups of 3 coins. Uncle Jake should keep one pile himself, and give one to you. Instruct Uncle Jake to place one coin in each hand, while placing one under his tongue. Wait for a response.
    At this point, either:

    1. Jake is dead, with the gold coin in his left hand.
    2. Jake is dead, with the gold coin in his right hand.
    3. Jake is dead, and you have the gold coin.
    4. Jake is alive, with the coin in his mouth.

    Proceed accordingly:

  4. a), b), c) To accertain which is the case, place the discarded pile of 6 coins (hopefully, you have been quick, and the pile will only have melted through the table and not the floor), and melt a hole in the intact scale pan with them. Now the scale pans have the same mass. Place the coin from Jake's left hand in the left scale pan, and the coin from his left hand in the right scale pan.

    If the balance is level, then you have the coin.
    If the balance is heavier on one side, take the coin from the other side - that is the gold coin, go to step 5.

    So you have the gold coin. Put one coin in each of Jake's hands. Now repeat step 4, disregarding the need to remelt the scale pans, and proceding to step 5 regardless.

    It can be proven that this step iterates at most twice, so this step may use the scales for the second and third times.

  5. d) Beat Jake over the head with the scales, which are, after all particularly strong - enough to support these extra-heavy coins.

    This uses the scales for the second time.

    Take the gold coin from Jake's mouth.

  6. Get all of the coins and pile them on Jake's body. The evidence of Jake's existence should disappear within a few minutes. The coins will take somewhat longer. See World Book, "half-life".

As Jake said, "You can have the gold coin if you can find it."

You found it.

--
  _--_|\        warwick@cs.uq.oz.au
 /      *  <--  Computer Science Department,
 \_.--._/       University of Queensland,
       v        AUSTRALIA.

 


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