Damien was a little surprised Celia didn't try to follow him in, but after a moment of the door staying closed, he decided to let it go and enjoy his space while he had it. He sighed and started undressing for bed. He pulled on his usual pajama wear and a long robe before uncovering his boards and settling down to work. He had a lot to do and so little time to do it in!
He stood before his biggest board and stared at the white lines marching across the dark surface. They lay there, flat and unmoving, mocking him. The answer was here, somewhere, hidden among the letters, numbers, and symbols, but he couldn't find it! He had to find it. He was supposed to find it. Why else had he been given the gift he had? If he could not figure this out, then it was wasted... he was a failure... or perhaps this was their way of torturing him. The problem that would never be solved because it had no answer.
With a tiny shriek, Damien grabbed his head in both hands and pounded his forehead into the board twice. "Why?" he keened. "What is it! I have to find it! I need to know! Why won't you tell me!" He let his head hit the board again then stood there panting, his forehead pressed against the cool, gritty surface.
He took a deep breath then coughed as he inhaled some chalk dust. He straightened and ran a hand over his forehead, smearing the dust more. Never mind. He would find this answer. No question was truly unsolvable. Nightmarishly difficult, perhaps, but not unsolvable. All he had to do was find just one clue, and it would be like the key in the lock. He could do this. The chalk had fallen. He picked it up, wiped down his board, and started again. Letters, symbols, and numbers dashed into being under his frantic hand only to be wiped away and rewritten in a different order. He would solve this. He had to! For the sake of humanity.
"Okay, let's start again from the beginning. Let P be a problem. Assume that given a solution, it is easy to check P is correct. I need to show that P then is easily solved. Assume P is not easily solved. Then we must show the solutions are not easy to verify. Equivalently, we have to show that there is a problem that is hard to solve, but easy to check. Which means we have to show every possible means of solving it. But that is completely absurd, at least in first-order of logic. So that line of proof seems for the moment to be impossible..." He scribbled down the line of proof in the corner of the board and slashed an asterisk next to it. "Later. Come back to this later."
He stood back from the board and crossed one arm over his torso, the other hand picking at his chin stubble fretfully, his brow furrowed. Then he leaned forward. "But is it impossible? Let us try attacking it in a different way. Assume that such a problem exists. Then every algorithm that solves P..." he trailed off for a second, thinking, "is of nonpolynomial time. Let A be the algorithm that varifies solutions of P that has polynomial time, and let A1, A2, A3, and so forth be all the algorithms that solve P."
He stopped, chalk dangling from his fingers, and pondered the white mass on the blackboard. "This is two more assumptions that I need to prove that I can make. First, is it true that a problem that has an algorithm that can check solutions must have an algorithm that can solve it? Secondly, Would the set of those algorithms be countable?" He stared for a moment longer.
In a sudden fit of frustration, he threw the piece of chalk at the board. "More questions! More assumptions! I cannot get anywhere without unearthing more! Every time I think I am getting close, it seems every answer dissolves into three more questions." He ran his hands through his hair in frustration and paced in the room within a room he had created by setting up the blackboards in a square. "What am I missing?"