We know why Fermat discovered his theorem: yes, that theorem, the one about power residues modulo a prime that underlies all of number theory today. We know because his letters written in 1640 reveal what problem from antiquity he was working on, why he was working on it, and what he wanted to know. But this doesn't tell us how he discovered his theorem, which is an unexpected amazement and wouldn't just be guessed. Participants will be invited to think like Fermat, look at the very problem and data he was studying, and conjecture the patterns he saw that led to his theorem. Absolutely no background is required! But this doesn't mean it won't be challenging.