Sherlock Holmes and Dr Watson, H and W, are told that two positive integers x and y have been chosen such that 1 < x < y and x + y < 100. H is given the value x + y and W is given the value xy. They then have the following conversation:

W: I cannot determine the two numbers.
H: I knew that.
W: Now I can determine them.
H: So can I.

Given that Holmes' and Watson's logic is impeccable and that the above statements are all true, what are the two numbers?

Originally posted by ThudanBlunder Sherlock Holmes and Dr Watson, H and W, are told that two positive integers x and y have been chosen such that 1 < x < y and x + y < 100. H is given the value x + y and W is given the value xy. They then have the following conversation:

W: I cannot determine the two numbers.
H: I knew that.
W: Now I can determine them.
H: So on's logic is impeccable and that the above statements are all true, what are the two numbers?

Great problem, got a chuckle from me! No idea how to actually solve it!
Ah, simultaneous equations.

Originally posted by sonhouse Great problem, got a chuckle from me! No idea how to actually solve it!
Ah, simultaneous equations.

The problem here isn't the simultaneous equations but the logical conditions that Holmes and Watson demonstrate. Holmes knows that Watson cannot solve the equations just based on the information he is given and once Watson knows that that is the case then he can solve the system. And once Holmes knows that Watson can now solve it so can he.