The Impossible Problem Solution

Though it seems that not enough information is provided to solve the problem, it is strangely the case that not only can Patty and Sam solve this problem but we can do so as well.

The impossible problem relies on two basic tools. 1) The nature of prime numbers and 2) deductive logic.

When Patty says that she can not determine the two numbers, we may deduce that Patty's number (the product) is not a pair of prime numbers. For if they were prime numbers, she would immediately know the numbers and their sum. For instance, if Patty's product was 35, she would know that the numbers were 5 and 7 and that the sum was 13.

When Sam says that she knew that Patty could not solve the problem, she indicated that she had thought about the problem and realized that Patty's product can not be a product of two prime numbers. We may deduce from this that Sam's sum can not be written as a sum of two distinct primes ( the case that the two primes could be equal is not allowed in the problem).

We can deduce this fact, but this is also apparent to Patty. Patty then looks at the sums that can be made from the factors in her product. Patty realizes that there is only one product that satisfies the condition that the sum can not be made with two distinct primes.

Can you find the sum? Check all sum values less than twenty to see if any such numbers satisfy the condition that a pair which sum to that number can not be made with two distinct primes. As an instance consider 13. Below we have broken 13 into the allowable sums

2+11

3+10

4+9

5+8

6+7.

Observe that 2 + 11 would give product 22, which Patty would easily deduce to be the product of 2 and 11. Thus a sum of 13 does not satisfy the condition needed for a solution.

There is one exception. Namely when the sum is 6 and the product is 8, but Patty would immediately know that the numbers must be 2 and 4.