# A ridiculous puzzle: Round 1

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, it is Mark’s turn to speak. Here is a chart of what he will say, depending on the two numbers I thought of:

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

M indicates that Mark has been able to deduce the two numbers; a dot means that he hasn’t.

Mark can deduce the two numbers in the following circumstances:

1. Both the numbers are primes. For instance, if Mark is given 3×5=15, 15 can only be written as 3×5 or 5×3, so Mark will know that the numbers are 3 and 5. I have marked these cases in black in the table.
2. One number is a prime and the other number is its square. For instance, if Mark is given 2×4=8, 8 can only be written as 2×4 or 4×2, so Mark will know that the numbers are 2 and 4. The same thing does not apply to cubes. If Mark is given 2×8=16, that can be 2×8 or 4×4, and Mark can’t tell which is is. I have marked these cases in red in the table.

In every other case, Mark can’t tell what the numbers are. For instance, if he is given 3×4=12, he will see 12 but he will not be able to decide whether that comes from 3×4 or 2×6.

Next, it is Anna’s turn, and we shall see what she can deduce.