From the simple 13×21 to an approximation of the value of e. (nl translation)

A month or two ago, well I guess three by now, I saw a video like this on my Facebook reels….

….and it made me wonder about three things.
First of all, how does this work?
Secondly, what is the general formula for let’s say AA x BC? (Where A, B and C can be any whole number between (and including) 1 and 9.
And third: How does this work with for example ABC x DEF? And is there a simple algorithm or pattern to be found when increasing the numbers?

So I took out a pen and a paper and wrote down some nonsense. It took a few hours, but it got me surprisingly far to be honest. It turns out that the first question (unsurprisingly) is easy to answer. The third question I haven’t really gotten to yet.
The second question is what I spent most time on. And to the great surprise of me, myself and especially I (pun intended) I almost found an approximation of the value of e (Euler’s constant), which really surprised me greatly.
I had the feeling I was close to finding it, and ended up just looking it up. It was only the very final step that I missed, and I was stealing myself: If only someone had told me I was that close, I would have found it myself.

Of course, such is the power of hindsight. But still it was a nice surprise. I’m going to write my simple explanation down, because

a) It actually is somewhat related to the previous posts on mathematics.
b) I’d like to see if you can actually really make it mathematically sound, and the process of writing will help, because I’m really not a trained mathematician or so.
c) To learn how on earth you write mathematical formulas in WordPress (the content management system of this blog).
d) Because I think it’s a nice, elegant and intuitive explanation, and who knows might help some suffering students.

So this post is work in progress. I’ll put it public regardless, so that anyone interested can have a peek at the progress that might or might not have been made.

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Okay, so let’s try to be as rigorous here as possible. Let’s first write down the necessary steps.

1. The conventional way of solving AB x CD
2. The ‘lines way’ of solving AB x CD
3. The explanation why the ‘lines way’ works.
4. Finding an algorithm for solving AA x CD
5. Pascal’s Triangle
6. Pascal’s Triangle (extended version — not a necessary step, but probably half worthy of adding)
7. Some formulas behind this
8. Taking a bunch of derivatives
9. The concept of symmetry
10. The value of e: How it was discovered and being denoted
11. An infinite series converging to e.