# A ridiculous puzzle: the end

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. In Round 6, 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 ….. ….. 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.

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 24. 24 can be 2×12 or 3×8 or 4×6.

1. If the numbers are 2 and 12 then Anna will never have said ‘I know’. (This is not visible in the table, which would have to be drawn bigger to show it).
2. If the numbers are 3 and 8 then Anna will never have said ‘I know’.
3. If the numbers are 4 and 6 then Anna will just have said ‘I know’.

If Anna has just said ‘I know’, then Mark can deduce that the numbers are 4 and 6. But if Anna has just said ‘I don’t know’, then Mark can deduce nothing at all, because the numbers could be 2 and 12 or 3 and 8.

Mark can deduce nothing new from Anna’s saying ‘I don’t know’, so the game is at an end.