Caleb was already working, which was its own answer. “Thirty-five m plus twenty-three, divided by four, remainder one,” he said. “Thirty-five is thirty-two plus three, so in fours, thirty-five is the same as three, which is the same as negative one if that makes it easier. Twenty-three is twenty plus three, so also three, also negative one. Negative m plus negative one should be the same as one, in the world of fours. Negative m is the same as two. Then I have to multiply by negative one, so m is the same as negative two, which is the same as two, because negative two plus four is two. So m = 4t + 2.”
He spoke it the way a person crosses a creek on stones that are farther apart than they looked from the bank. Twice he nearly slipped. He did not ask Emily to catch him until the last stone. She had not interrupted. Interrupting would have turned the crossing into a demonstration, and he did not need a demonstration. He needed the sound of his own reasoning surviving contact with another mind.
“Check the negative,” she said. “Say only that piece again.”
He said it again, slower. Negative m ≡ 2. Multiply both sides by negative one. The inequality of the world flipped twice in his mouth and came out right: m ≡ -2 ≡ 2 mod 4. He looked at her. She nodded. The nod was different from the earlier one. It had weight.
“Then put it back,” she said. “All the way to n. And do not tell me the contest answer until you have checked it against all three original rules. The original rules. Not your copy.”