Two Circles in a Square

Two Circles in a Square

Looking at lists of unsolved problems in maths always provides a strange combination of things that seem very difficult (as well they should) to misleadingly trivial. It is those in the latter category that are the most interesting. I want to introduce you to a packing problem which feels like it should be solvable with a bit of brute force from a computer.

You have two circles of unit diameter. By cutting one of them into two pieces along a chord, what is the smallest square that can contain the three pieces?

Without a cut the answer is shown below. If you want to give this problem a go it feels like splitting one of the circles into two equal pieces may generate some very small squares with each semi circle hiding in opposite corners. 

 Size 1+ √2/2

Size 1+√2/2

I know that unsolved problems are a little unsatisfying to read an article about because I can't give you a neat conclusion, but I feel like posting only the problems that actually work out gives an unrealistic portrayal about what the process of figuring things out is actually like. I look at dozens of problems each week and a large proportion of them are ones I don't make much progress on at all. So here's a reminder that sometimes there is joy to be found in having a cursory play with something you know you will never solve.

Dartboard Density

Dartboard Density

Volume of a Torus

Volume of a Torus