A ridiculous puzzle: Round 6

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. In Round 5, Marl 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 ….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.

Now it is Anna’s turn again.

Considering the numbers Anna doesn’t yet know the answer to, the only interesting case is if Anna sees 10. 10 can be 2+8 or 3+7 or 4+6 or 5+5.

1. If the numbers are 2 and 8 then Mark will just have said ‘I know’.
2. If the numbers are 3 and 7 then Mark will have said ‘I know’ a long time ago.
3. If the numbers are 4 and 6 then Mark will never have said ‘I know’.
4. If the numbers are 5 and 5 then Mark will have said ‘I know’ a long time ago.

In both cases 1 and 3, Anna can now deduce the answer (in cases 2 and 4, she still can’t). Here is the new chart:

2 3 4 5 6 7 8 9 10 11
2 MA MA M.. MA ..MA MA ….MA ….. ….. MA
3 MA M. .AM MA ….. M. ….. M. ….. M.
4 M. .AM …AM ….. …..A ….. ….. ….. ….. …..
5 MA MA ….. M. ….. M. ….. ….. ….. M.
6 ..MA ….. …..A ….. ….. ….. ….. ….. ….. …..
7 MA M. ….. M. ….. M. ….. ….. ….. M.
8 ….MA ….. ….. ….. ….. ….. ….. ….. ….. …..
9 ….. M. ….. ….. ….. ….. ….. ….. ….. …..
10 ….. ….. ….. ….. ….. ….. ….. ….. ….. …..
11 MA M. ….. M. ….. M. ….. ….. ….. M.

So we have the fifth 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 don’t know; Anna: I know – can only mean that the numbers are 4 and 6. Mark’s number is 24 and Anna’s number is 10.

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