A ridiculous puzzle: Round 2

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). Here is the chart showing what my numbers could be and what Mark’s answer would be in each case:

2 3 4 5 6 7 8 9 10 11
2 M M M M . M . . . M
3 M M . M . M . M . M
4 M . . . . . . . . .
5 M M . M . M . . . M
6 . . . . . . . . . .
7 M M . M . M . . . M
8 . . . . . . . . . .
9 . M . . . . . . . .
10 . . . . . . . . . .
11 M M . M . M . . . M

Now it is Anna’s turn. Anna sees her number, which is the sum of the numbers I thought of, and she has also heard Mark’s statement – ‘I know what the numbers are’ or ‘I don’t know what the numbers are’. Here are some possibilities:

  1. Anna sees 7, and hears Mark saying ‘I know’. 7 could be 2+5 or 3+4. From the chart, if the numbers were 2 and 5, Mark would have said ‘I know’ and if they were 3 and 4 Mark would have said ‘I don’t know’. So now Anna knows that the numbers must be 2 and 5.
  2. Anna sees 7, and hears Mark saying ‘I don’t know’. 7 could be 2+5 or 3+4. From the chart, if the numbers were 2 and 5, Mark would have said ‘I know’ and if they were 3 and 4 Mark would have said ‘I don’t know’. So now Anna knows that the numbers must be 3 and 4.
  3. Anna sees 8, and hears Mark saying ‘I know’. 8 could be 2+6 or or 3+5 or 4+4. From the chart, if the numbers were 3 and 5, Mark would have said ‘I know’ and if they were 2 and 6 or 4 and 4 then Mark would have said ‘I don’t know’. So now Anna knows that the numbers must be 3 and 5.
  4. Anna sees 8, and hears Mark saying ‘I don’t know’. 8 could be 2+6 or or 3+5 or 4+4. From the chart, if the numbers were 3 and 5, Mark would have said ‘I know’ and if they were 2 and 6 or 4 and 4 then Mark would have said ‘I don’t know’. So now Anna knows that the numbers must be 2 and 6 or 4 and 4, but she doesn’t know which. Anna doesn’t know the numbers, and never will.

Applying this reasoning throughout, here is the chart after round 2. The first character is M if Mark said ‘I know’ and a dot if he didn’t. The second character is A if Anna said ‘I know’ and a dot if she didn’t.

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.

So we have the first interesting answer in the game.

Mark: Don’t know; Anna: I know – can only mean that the numbers are 3 and 4. Mark’s number is 12 and Anna’s number is 7.

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

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