Puzzles
Add 1, then subtract 1, forever. What does the sum equal?
Consider the never-ending series 1 − 1 + 1 − 1 + 1 − 1 ... Group the terms one way, (1−1) + (1−1) + ..., and it looks like it sums to 0. Group them another way, 1 − (1−1) − (1−1) − ..., and it looks like it sums to 1. Which grouping is correct?
Reveal the answer
Neither, technically: the series has no ordinary sum, because its partial sums oscillate forever between 0 and 1 rather than settling down. Italian mathematician Guido Grandi first published the paradox in 1703. It took over a century of debate, and the later invention of more careful summation methods, before mathematicians agreed the naive question itself was flawed.
— Guido Grandi, Quadratura circuli et hyperbolae per infinitas hyperbolas geometrice exhibita — 1703