*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. In Round 4, 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 | …. | …. | …. | 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. |

Now it is Mark’s turn again.

Considering the numbers Mark doesn’t yet know the answer to, the only interesting case is if Mark sees 16. 16 can be 2×8 or 4×4.

- If the numbers are 2 and 8 then Anna will never have said ‘I know’.
- If the numbers are 4 and 4 then Anna will just have said ‘I know’.

So in both cases 1 and 2, Mark can now deduce the answer. Here is the new chart:

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

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

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

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

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

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

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

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

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

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

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

So we have the fourth interesting answer in the game.

**Mark:** Don’t know; **Anna:** I don’t know; **Mark:** I don’t know; **Anna:** I don’t know; **Mark:** I know – can only mean that the numbers are 2 and 8. Mark’s number is 16 and Anna’s number is 10.

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