*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, Mark spoke. He said whether or not he knew what numbers I had thought of (or, equivalently, what Anna’s total was). In Round 2, Anna spoke. Here is the chart showing what my numbers could be and what Mark’s and Anna’s answer would be in each case. M means that Mark says ‘I know’, A means that Anna says ‘I know’. A dot indicates a ‘don’t know’.

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

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

3 | MA | M. | .A | MA | .. | M | .. | M | .. | M. |

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

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

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

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

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

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

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

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

Now it is Mark’s turn again. Mark knows more than he did last time, because he knows whether or not Anna had managed to deduce the numbers, taking into account what she saw in front of her and what Mark said before.

Considering the numbers Mark doesn’t yet know the answer to, the only interesting case is if Mark sees 12. 12 can be 2×6 or 3×4. If the numbers are 3 and 4 then Anna has just said ‘I know’ in Round 2. If the numbers are 2 and 6 then Anna has just said ‘I don’t know’ in Round 2. In both cases, Mark can now deduce what the numbers are, and say ‘I know’.

So here is the new chart:

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

2 | MA | MA | M.. | MA | ..M | MA | … | … | … | MA |

3 | MA | M. | .AM | MA | … | M | … | M. | … | M. |

4 | M. | .AM | … | … | … | … | … | … | … | … |

5 | MA | MA | … | M. | … | M. | … | … | … | M. |

6 | ..M | … | … | … | … | … | … | … | … | … |

7 | MA | M | … | M. | .. | M. | .. | .. | .. | M. |

8 | … | … | … | … | … | … | … | … | … | … |

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

10 | … | … | … | … | … | … | … | … | … | … |

11 | MA | M. | … | M. | … | M. | … | … | … | M. |

So we have the second interesting answer in the game.

**Mark:** Don’t know; **Anna:** I don’t know; **Mark:** I know – can only mean that the numbers are 2 and 6. Mark’s number is 12 and Anna’s number is 8.

Next, it is Anna’s turn. Now that she has heard what Mark has said, we shall see what she can deduce.