*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. In Round 3, Mark 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 | ..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. |

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

Considering the numbers Anna doesn’t yet know the answer to, the only interesting case is if Anna sees 8. 8 can be 2+6 or 3+5 or 4+4.

- If the numbers are 2 and 6 then Mark will just have said ‘I know’.
- If the numbers are 3 and 5 then Mark will have said ‘I know’ a long time ago (and Anna will have said ‘I know’ as well).
- If the numbers are 4 and 4 then Mark will not have said ‘I know’ at all.

So in both cases 1 and 3, Anna will now have deduced the answer (and in case 2 she has known it for some time). Here is the new chart:

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

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

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

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

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

6 | ..MA | …. | …. | …. | …. | …. | …. | …. | …. | …. |

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

8 | …. | …. | …. | …. | …. | …. | …. | …. | …. | …. |

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

10 | …. | …. | …. | …. | …. | …. | …. | …. | …. | …. |

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

So we have the third interesting answer in the game.

**Mark:** Don’t know; **Anna:** I don’t know; **Mark:** I don’t know; **Anna:** I know – can only mean that the numbers are 4 and 4. Mark’s number is 16 and Anna’s number is 8.

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