30 | August | 2015 | RobertLovesPi.net

This is a puzzle I made up not long ago. After trying to solve it for a bit (no success yet, but I haven’t given up), I decided to share the fun.

A small circle of radius r is centered on a large circle of radius R. It is a given that 0 < r < R. In terms of r and R, what fraction of the smaller circle’s circumference lies outside the larger circle?

I am 90% certain there is an extremely simple way to do this, using only things I already know. It’s frustrating that the answer isn’t simply leaping out of the computer screen, at me. For simple math problems, that’s what usually happens . . . so either this is merely deceptively simple, or I am missing something.

The figure above, rotating in hyperspace, is an orthogonal projection of a four-dimensional polychoron known as the truncated tesseract. It is analogous to the truncated cube, one of the Archimedean solids. The image below is of the same figure, but is shown as a perspective projection.

Both images were created using Stella 4d, software you can buy (with a free trial download available, first) at http://www.software3d.com/Stella.php. It’s great software, and a friend of mine wrote it — but no, he doesn’t pay me to give his program free advertising, as some have wondered.