Evan O’Dorney leaps from his seat and snatches the chalk from his teacher’s hand. “I think it goes like this,” the eighth-grader exclaims. He has come up with a formula for the number of ways to ...

In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight ...