Answer:
Let's break down the information we have:
1. Sam has three 2-digit numbers: let's call them A, B, and C in the order.
2. Two of them are perfect squares of prime numbers, let's assume these are A and C.
3. The middle number B is the sum of those two prime numbers.
Let's start with the prime numbers. We're looking for two prime numbers whose squares are two-digit numbers and whose sum is also a two-digit number.
The 2-digit perfect squares of prime numbers are: 4 (2^2), 9 (3^2), 25 (5^2), and 49 (7^2).
Given that the middle number is the sum of the two prime numbers, we can immediately rule out 7^2 (49) since adding 7 to any of the other available primes would result in a 3-digit number.
So let's see the remaining possible combinations:
2^2 (4) and 3^2 (9) --> Sum of the primes is 5, which is a single digit number.
2^2 (4) and 5^2 (25) --> Sum of the primes is 7, which is a single digit number.
3^2 (9) and 5^2 (25) --> Sum of the primes is 8, which is a single digit number.
There's no way to get a 2-digit middle number from these combinations, which seems to be a contradiction.
It is likely that the problem contains a mistake or misunderstanding. The conditions as stated do not appear to allow for a solution. Can you check the problem again?