Ring six spheres around three that all touch each other — will the ring ever close up?
Start with three spheres, all mutually touching each other, sitting inside (or alongside) one more enclosing sphere. Now pack a chain of smaller spheres into the gap, each one touching the same three spheres plus its two chain neighbors. Keep adding spheres around the loop. Does the chain always close up into a neat ring — and if so, after exactly how many spheres, regardless of where the first one was placed?
Reveal the answer
The chain always closes after exactly six spheres, forming what's now called Soddy's hexlet — and it closes for any starting position at all, a rare bit of mathematical serendipity called a porism. Japanese temple mathematicians carved the result onto a wooden sangaku tablet in Kanagawa Prefecture in 1822; the British chemist and Nobel laureate Frederick Soddy rediscovered it independently in a 1937 paper, unaware of the earlier Japanese proof, and it now carries his name.