They looked exactly as expected: a wall of white, peppered with black specks for smaller integers. “We expected the black dots to peter out,” Stange said. Rickards added, “I thought maybe it would even be possible to prove they peter out.” He speculated that by looking at charts that synthesized many packings together, the team would be able to prove results that weren’t possible when they looked at any one packing on its own.
While Stange was away, Haag wound up plotting every pair of remainders — about 120. No surprises there. Then she went big.
Haag had been plotting how 1,000 integers interact. (The graph is bigger than it sounds, since it involves 1 million possible pairs.) Then she cranked the dial up to 10,000 times 10,000. In one graph, regular rows and columns of black specks refused to dissolve. It looked nothing like what the local-global conjecture would predict.
The team met on a Monday after Stange returned. Haag presented her graphs, and they all focused on the one with the weird dots. “It was just a continual pattern,” Haag said. “And that was when Kate said, ‘What if the local-global conjecture isn’t true?’”