MHB Generating Solutions for a Challenging Puzzle

[JamesBwoii]

Hi, I'm doing some programming work with regards to this puzzle.

Impossible Puzzle - Wikipedia, the free encyclopedia

I have to generating the solutions to the the resulting solutions after the four following sentences are said.

1. P says "I cannot find these numbers."
2. S says "I was sure that you could not find them. I cannot find them either."
3. P says "Then, I found these numbers."
4. S says "If you could find them, then I also found them."

I am creating a list of quads, [X,Y,S,P] where X and Y are the two integers where

• 0 < X < Y
• S = X + Y
• S < 100
• P = X * Y

So firstly I generated every single possibility which is 2352 possible quads.

Then after sentence 1 is said I remove a quad if P is a prime number or if P has two unique factors. In other words if P can only be made from 2 possible numbers it is removed. This leaves 1747 quads. I know this is correct.

From there on I am a little confused. I know that unique sums need to be removed but I have been told that that is not enough and other quads need removing, but I can't think what else would mean that it would need to be removed.

Thank you.

 

[JamesBwoii]

Pretty sure I've figured it out now.

• S is odd(Goldbach Conjecture).
• S is not of the form Q+2 where Q is prime.
• S is less than 55.