Well at least two of the people have new names this time

i) The product of x and y must be prime. Hence, y=xy and x=1. If xy is not prime, then there would be more possibilities for the values of x and y, as there are multiple prime factors, so Pam would not know the answer.

ii) xy is composite, as seen in the first part.

x+y=4 so either x=1, y=3 or x=2, y=2. Only the second pair gives a composite product. x=2, y=2 are thus the values of x and y.

iii) Either x=1, y=4 or x=2, y=2.

But if x=2, y=2, then x+y=4. By the previous part, Sam would know what x and y are. Ergo, x=1, y=4.

iv) xy=8=2^3. Therefore, x=1, y=8 or x=2, y=4

Sam knew Pam didn’t know the values. Hence, Sam knew the product xy is not prime. If x+y=6, then Sam cannot know that Pam didn’t know the values, since x=1, y=5 is a possibility, which gives xy=5, which is prime. Therefore, x=1, y=8.

v) x+y=5. Thus x=1, y=4 or x=2, y=3.

xy is composite since Pam does not know x and y. This does not add any new information, since in either case we have a composite number.

Thus, Sam does not know x and y

Pam then says he still doesn’t know. In order for that to be possible, the product of x and y must not be 4, for if it was Pam would deduce x\ne2, y=2, as in part iii and thus x=1, y=4.

Hence, the product is 6, so x=2, y=3.