# “Complex” Solutions to a “Complex” Problem – Amazing Answers from the #MTBoS

On Tuesday, I finally turned some thought into a problem. In pretty typical fashion, I put out this problem to the Maths Twitter Blog-o-Sphere (MTBoS) for advice about how to better present it, as I felt the wording wasn’t quite right. Here’s the revised problem (thanks to @DavidKButlerUoA for reframing the question):

I then started trying to find some solutions with the hope to develop a simulation or version of a Complex Number Analogue Clock. To do this, I need to start coming up with some solutions, which I knew there would be many. See one of my solutions below, which shows a possible answer for every 2-minute interval (1 degree change in the hour hand).

After running some (pretty inefficient) code on Excel, I stumbled upon quite an interesting pattern. Below is the angles of all right angled triangles that can be made with integer legs from 1 – 100 that are within 0.1 degree of a whole number. For example, a triangle with side lengths 15 and 8 has an angle of 28.07 degrees. It was when I removed all other angles (e.g. 62.5 degrees) when something fascinating was revealed:

After sharing this online, two people in particular took this to the next level…

What I particularly love about this is the shared enthusiasm that communities can have towards quite specific things. Getting excited about maths is not anything out of the ordinary for the MTBoS, in fact I would be confident in saying that it’s actually quite unusual for people in the MTBoS to not be excited about maths. Sometimes, you do get some people that really do take it to the next level and that’s where beautiful things can emerge. So, a big thanks to Rich and Dan for not only sharing your incredible simulations and creations, but also for just considering something I found interesting to be worth spending some time thinking about and playing around with.