Counting Things

I’ve been working on my little thread of posts about sports mathematics. But I’ve also had a rather busy week and I just didn’t have time to finish the next bit of pondering I had regarding baseball scores. Among other things I had the local pinball league’s post-season Split-Flipper Tournament to play in last night. I played lousy, too.

So I hope I may bring your attention to some interesting posts from Baking And Math. Yenergy started, last week, with a post about the Gauss Circle Problem. Carl Friedrich Gauss you may know as the mathematical genius who proved the Fundamental Theorem of Whatever Subfield Of Mathematics You’re Talking About. Circles are those same old things. The problem is quite old, and easy to understand, and not answered yet. Start with a grid of regularly spaced dots. Draw a circle centered on one of the dots. How many dots are inside the circle?

Obviously you can count. What we would like is a formula, though: if this is the radius then that function of the radius is the number of points. We don’t have that, remarkably. Yenergy describes some of that, and some ways to estimate the number of points. This is for the circle and for some other shapes.

Yesterday, Yenergy continued the discussion and got into partitions. Partitions sound boring; they’re about identifying ways to split something up into components. Yet they turn up everywhere. I’m most used to them in statistical mechanics, the study of physics problems where there’s too many things moving to keep track of them all. But it isn’t surprising they turn up in this sort of point-counting problem.

As a bonus Yenergy links to an article examining a famous story about Gauss. This is specifically the famous story about him, as a child, doing a quite long arithmetic problem at a glance. It’s a story that’s passed into legend and I had not known how much of it was legend.