> which BTW isn't a "proof", it only illustrates n=1..7 and hints the generalization to the reader
What you've written isn't a proof, either. All of those ellipses are terribly informal. You should really write:
(\Sigma_{k=1}^{n-1} k + n)² - (\Sigma_{k=1}^{n-1} k)²
...
Personally, I find the "proof without words" both easier to read and more convincing than some algebra. I think I'd have a harder time spotting a mistake in the algebra. Do other people really find it easier to find a mistake in a dozen lines of algebra than in a diagram?
> I think I'd have a harder time spotting a mistake in the algebra. Do other people really find it easier to find a mistake in a dozen lines of algebra than in a diagram?
You're asking about two different issues as if they were one. It is intuitively "easier" to examine the visual presentation. Nevertheless, it is technically much easier to find a mistake in a dozen, or a hundred, lines of algebra because there is a definite, known method of verifying the algebra. With the visual presentation, you're just staring at it and hoping you notice the mistake.
Analogously, it's much easier for people to produce an exhaustive list of things that occur in a known fixed order (say, the 50 US states in alphabetical order) than to produce an exhaustive list of unordered members of some category (say, the 50 states if you never learned them in any particular order). The algebraic proof is like the fixed-order recall; the structure it imposes is, to some extent, error-correcting.
So, visual proofs, which are inherently convincing, are a good tool for persuading someone of their own truth regardless of whether they're actually true or not, and also a good tool for getting people to remember a proof in the future, but they are a terrible tool for formally establishing results. They are best used for communicating results that you have other reasons for believing in, or as an aid to memory or investigation.
What you've written isn't a proof, either. All of those ellipses are terribly informal. You should really write:
Personally, I find the "proof without words" both easier to read and more convincing than some algebra. I think I'd have a harder time spotting a mistake in the algebra. Do other people really find it easier to find a mistake in a dozen lines of algebra than in a diagram?