A ridiculous puzzle

I think of two numbers greater than 1 (they may or may not be the same number).

I add these two numbers together and give them to my friend Anna. I multiply the same two numbers together and give them to my friend Mark. Both my friends are perfectly truthful and perfect logicians.

  1. M says to A: ‘I don’t know what your number is’.
  2. A says to M: ‘I don’t know what your number is’.
  3. M says to A: ‘I don’t know what your number is’.
  4. A says to M: ‘I don’t know what your number is’.
  5. M says to A: ‘I don’t know what your number is’.
  6. A says to M: ‘I know what your number is’.
  7. M says to A: ‘I know what your number is’.

Question. What are Anna’s and Mark’s numbers? (Or, equivalently: what numbers did I originally think of?)

I came across a problem of this type in a magazine many years ago but never got round to investigating its ramifications. There really is a unique answer to this question. There is also a unique answer to the variant where there are one, two, three or four “don’t know”s instead of five. Unfortunately there are infinitely many answers to the variant where there are no “don’t know”s; but there is only one answer in which one of my numbers isn’t 2.

To see the answer for each case, follow the links in the previous paragraph.

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