To recapitulate: I think of two numbers greater than 1. I add them together and give them to Anna. I multiply them together and give them to Mark. Anna and Mark are perfect logicians and perfectly truthful. They have a conversation together about whether they know the numbers I thought of or not.

In Round 1, it is Mark’s turn to speak. Here is a chart of what he will say, depending on the two numbers I thought of:

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
---|---|---|---|---|---|---|---|---|---|---|

2 | M | M | M | M | . | M | . | . | . | M |

3 | M | M | . | M | . | M | . | M | . | M |

4 | M | . | . | . | . | . | . | . | . | . |

5 | M | M | . | M | . | M | . | . | . | M |

6 | . | . | . | . | . | . | . | . | . | . |

7 | M | M | . | M | . | M | . | . | . | M |

8 | . | . | . | . | . | . | . | . | . | . |

9 | . | M | . | . | . | . | . | . | . | . |

10 | . | . | . | . | . | . | . | . | . | . |

11 | M | M | . | M | . | M | . | . | . | M |

M indicates that Mark has been able to deduce the two numbers; a dot means that he hasn’t.

Mark can deduce the two numbers in the following circumstances:

- Both the numbers are primes. For instance, if Mark is given 3×5=15, 15 can
*only*be written as 3×5 or 5×3, so Mark will know that the numbers are 3 and 5. I have marked these cases in black in the table. - One number is a prime and the other number is its square. For instance, if Mark is given 2×4=8, 8 can
*only*be written as 2×4 or 4×2, so Mark will know that the numbers are 2 and 4. The same thing does not apply to cubes. If Mark is given 2×8=16, that can be 2×8 or 4×4, and Mark can’t tell which is is. I have marked these cases in red in the table.

In every other case, Mark can’t tell what the numbers are. For instance, if he is given 3×4=12, he will see 12 but he will not be able to decide whether that comes from 3×4 or 2×6.

Next, it is Anna’s turn, and we shall see what she can deduce.