[Math Lair] Metapuzzles

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Metapuzzle is a term coined by Raymond Smullyan to describe puzzles that are about puzzles. In a metapuzzle, you are given a puzzle to solve but not given sufficient data to solve it. You are also told that someone else was able to solve the puzzle under certain circumstances and/or knowing certain information (which is often not precisely specified). Knowing that someone else was able to solve the puzzle allows you to solve the puzzle.

Here are some examples of metapuzzles:

  1. There was a crime investigation that was focusing on identical twins. It was known that at least one of them always lied, but it was not known which. It was known that one of the twins, named John, had committed the crime. The purpose of the investigation was to find out which one was John.

    "Are you John?" the judge asked the first twin.

    "Yes, I am," was the reply.

    "Are you John?" the judge asked the second twin.

    The second twin then answered, and the judge then knew which one was John.

    Was John the first twin or the second?

  2. A police officer brought five persons of interest, Abby, Bernard, Claire, Damian, and Edward, to the police station for questioning in the murder of Fabian Foot. It was known that only one person committed the murder, but it was not known at that time whether it was one of these five or whether someone else did it. Each was given a lie-detector test, at which time they made the following statements: After the tests, the police officer asked the lie detector technician how many of the five told the truth. The lie detector technician told him the number. With that information, the officer was able to place the murderer under arrest. Who committed the crime?
  3. (more difficult) In an espionage trial, there were three defendants, A, B, and C. One of them was a truth-teller who always told the truth, one of them was a liar who always lied, and the third, the spy (the guilty party) sometimes lied and sometimes told the truth. The judge at the trial had the task of finding the spy.

    First, A stated either that C was the liar or that C was the spy, but we are not told which. B then stated which one of the three A was (truth-teller, liar, or spy), but we are not told what he said. C then stated which of the three B was, but we are not told what he said either. The judge then determined who the guilty party was and convicted him.

    The case was described to a logician as above. He looked at the problem, but found that there was not enough information for him to determine who the spy was. The logician was then told what A said, and he determined the spy's identity. Who was the spy?

For solutions, see the answers page.

If you enjoyed the second puzzle, see also Lie Detection Puzzles.