How many billiard balls can touch one central ball at the same time?
Arrange identical spheres so that as many as possible each touch one central sphere, without any of them overlapping. In one dimension the answer is obviously 2; in two dimensions, arranging circles around a central circle, the answer is 6. In three dimensions, the real-world case of stacking oranges or cannonballs, what's the maximum number of spheres that can simultaneously touch one sphere in the middle?
Reveal the answer
Twelve. Isaac Newton and David Gregory argued about this exact number in a 1694 letter exchange, with Newton guessing 12 and Gregory holding out for 13, but neither could actually prove it. The value of 12 wasn't rigorously settled until 1953, nearly 260 years later. The problem generalises to higher dimensions, where the answer is known for only a handful of cases, including a famously difficult proof for 8 and 24 dimensions tied to exceptional geometric structures.