At home that night, Ruth asked what the new teacher had taught him. Elijah explained that an infinite map was not the same thing as a very large map, and that rules could appear true for every example you drew while still failing somewhere you could not see. His grandmother listened while washing a saucepan, and when he finished she nodded solemnly. "Sounds like people," she said. "You can know somebody a long time and still find out you never really looked at them."
Elijah thought she had misunderstood the mathematics. Later, he would remember that she had understood something else perfectly. At ten, he still believed the most difficult part of his discovery would be the proof. He had not yet learned how hard it could be to persuade the people entrusted with listening to look at him long enough to hear it.
The next six months did not look like the life of a child destined for headlines. They looked like worksheets beside cereal bowls, library books returned two days late, and pencil marks on the back of supermarket receipts. Elijah still had spelling tests and social studies projects. He still argued with Nia over which television program to watch, sometimes forgot to take out the trash, and hated the way his winter boots squeaked on the classroom floor. Mathematics remained something he loved rather than an escape from being ten, although the two worlds began to overlap more often.
Okonkwo taught him to keep three notebooks. The first was for ideas that felt true, the second for examples that might prove those ideas false, and the third for statements he was prepared to defend. "A beautiful guess belongs in the first notebook," she told him. "A cruel counterexample belongs in the second. Only the things that survive both are allowed into the third." Elijah called them Maybe, Not Yet, and Check Again, because he thought formal titles would make them harder to use.
At first he wanted every arrangement to work. He drew expanding rows of squares, triangles, hexagons, and irregular regions, then assigned colors until the paper became crowded. Each failure made him angry for about a minute. Then he would turn the page over and try to understand which assumption had caused the failure. Okonkwo noticed that he did not avoid counterexamples; he hunted them like clues in a mystery story.
One Saturday she showed him a standard proof technique called compactness, explaining in accessible language why ordinary infinite planar graphs did not create the unsolved coloring mystery he had first imagined. If every finite collection of coloring constraints could be satisfied using four colors, the appropriate compactness principle could establish the existence of a coloring for the whole infinite graph. That was important because Elijah had been about to announce something experts had understood for decades. He listened without resentment, crossed out two weeks of work, and wrote at the top of a fresh page: EXISTENCE IS NOT THE SAME AS A REPEATING RECIPE.
"Good," Okonkwo said. "Now tell me what a recipe would have to do."