Here is probably the hardest puzzle we’ve looked at so far in maths club …
Two perfect logicians, Sam and Polly, are told that integers x and y have been chosen such that 1 < x < y and x+y < 100. Sam is given the value x+y and Polly is given the value xy.
They then have the following conversation.
Polly: I cannot determine the two numbers.
Sam: I knew that.
Polly: Now I can determine them.
Sam: So can I.
Given that the above statements are true, what are the two numbers?
Starting points:
Before Polly and Sam say anything:
What is the range of numbers Sam could be given?
What is the range of numbers Polly could be given?
What special type of number can the product of x and y never be?
What about the square of these numbers?
After Polly’s first statement:
Give two examples of products that Polly can not be given.
After Sam’s first statement:
Give two examples of sum’s that Sam can not be given.
Next steps …
At this point you probably want to start using a computer to generate a list of numbers that they could be given.