Elijah drew a little square in the center of the page and arrows on all four sides. "If I know the colors on this side, I need a way to go to the next one without breaking the rules," he said. "And if I keep going, I shouldn't have to come back to the beginning and discover the colors don't match." Okonkwo nodded slowly. She knew adults who used pages of notation to explain the same concern, but the boy's ordinary language revealed that he had identified a real distinction.
His fascination moved from individual colored regions to the information that crossed a boundary. Instead of drawing an endless floor, he represented each repeating patch by a small record of what its edges required. He started treating those records as states in a process, comparing what happened when a pattern moved right, left, up, or down. Some state changes could be reversed; others could not. He drew boxes and arrows until his grandmother complained that the dining table looked like a railroad yard.
The turning point came after Nia dropped a carton of colored beads beneath the sofa. Elijah helped retrieve them, noticing that some beads had disappeared into the same cracks and grooves each time. He thought about where a pattern could hide information, and about whether a long sequence of boundary choices might eventually return to a familiar state. That evening he invented a little chart with rows of colored marks and symbols for the allowable transitions. He called it the traveling edge chart, a name that made Okonkwo smile before she realized how much of the construction it captured.
There were mistakes, plenty of them. In April he confidently colored a network that turned out not to meet Hartwell's compatibility rules. In May he believed he had proved that a certain boundary state must eventually repeat, only to discover he had mistaken a finite list of possibilities for a guarantee that a compatible cycle existed. Okonkwo did not spare his feelings by pretending those errors were minor. She told him he had found the exact place where a serious proof might fail, and she made him keep the incorrect pages instead of tearing them out.
Elijah hated that part. He liked a clean notebook with neat lines and no embarrassing detours. Okonkwo explained that mistakes were not stains on mathematical work; they were a record of its development, and that an honest account could help another researcher understand why a tempting argument did not hold. The notebooks gradually filled with colored-pencil diagrams, formal statements written in Elijah's own improvised symbols, and tiny comments such as DON'T ASSUME and CHECK THE CORNER.
At school, the other children understood only that Elijah spent a lot of time drawing squares. Some teased him affectionately, asking whether he planned to build a house made of graph paper. Others stopped by to ask for help on fractions because he explained things without making them feel foolish. One afternoon a boy named Marcus became upset when his multiplication homework came back marked in red. Elijah sat beside him during recess and showed him how to break a hard problem into smaller ones, then stayed indoors until Marcus felt confident enough to try again.