If I tell you the radii of the spheres ๐ and ๐ in this picture, can you figure out the radii ๐โ,...,๐โ of the six spheres that touch them and snugly fit inside the big sphere? Can you at least do it if you know ๐โ?
Irisawa Shintarล Hiroatsu did it in 1822! He was a merchant who sold tea, textiles and ingredients for traditional Chinese medicine - and he had a hobby of solving math puzzles.
In 1932 his technique was rediscovered by a Nobel-prize-winning chemist, so it's often called Soddyโs Hexlet Theorem. But Hiroatsu did it earlier as part of a Japanese mathematical tradition called ๐ค๐๐ ๐๐ - and as part of this tradition, he donated a plaque containing this result to a shrine!
He wasn't the only one who did this sort of thing. This kind of plaque is called a ๐ ๐๐๐๐๐๐ข. These plaques were used to commemorate newly discovered solutions to hard math problems during the Edo Period from 1603 to 1868. There's a lot of interesting math in these ๐ ๐๐๐๐๐๐ข, and you can see some of them here:
You can also learn more about the solution to the puzzle I gave! The most surprising thing is that the reciprocals of the opposite pairs of spheres in the "hexlet" of 6 spheres add up to the same number:
