As if by magic, some maths appeared

I've not really done any programming lately, so this week there's no Minecraft. Instead there's some maths. Hurrah! Hey, where are you—

I was reading Richard Wiseman's Friday puzzle. It's very short and not at all hard, so you might want to go and solve it and come back. Go on, I'll wait.

Okay, you're back? I'll try not to state the answer outright, but it will probably become pretty obvious very quickly. Right, so this got me thinking about sums of series. The case in the question is sufficiently small that you don't even need to work out the formula for the sum of sequence in question, but I was thinking about it nonetheless.

Now, I knew the formula for the sum of an arithmetic sequence, and I knew that you could easily enough prove it by induction, but this is a very unsatisfying form of proof. It doesn't feel like something you could have figured out on your own. Imagine just picking random equations one at a time and testing them until you find one that you can prove correct with induction! I thought about it a little and came up with what I thought was a pretty compelling geometric proof, at least for the simple case where you start at 1 and increment by 1. (I have no idea if a proper mathematician would call this a proof. I am certainly not a proper mathematician.)

Is it obvious? On the left in the green we have a square with area 1, a rectangle with area 2, a rectangle with area 3, etc. The area of the whole shape must be the sum of those areas. By dividing up the space differently, we can easily see how to produce the closed form.

Great, I thought. Now lets do the same for the sum of squares. It must be just as obvious in 3D. Right?

Hmmm. Well, it gives the right answer, and it's satisfying to figure it out by hand, but it lacks the obviousness of the diagram for the arithmetic sequence sum. Perhaps there's a simpler way to do it?

As an aside, I think I'm getting the hang of Inkscape, although there are still some areas I find I much prefer Visio. Inkscape has some pretty painful and inflexible handling of arrowheads, for example. They're always black, regardless of the line colour, and then there's a special menu action just to make them change to match the line colour. The selection of arrowheads available is limited and weird, and most of them aren't resizable. I'm not sure if that's because SVG forces arrowheads to behave in strange and unintuitive ways, or that's just the way that Inkscape chooses to work.

Maybe next week there will be more programming. We shall see.

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