Books: Math Recreations (Kraitchik) Part 4

I have more old math books than I'll ever read or need. This is just a fact. I would collect them, and sometimes read through parts of them, but never finish any of them. ...

Day 4: Final Thoughts about Triangle Numbers and Squares

Continuing from yesterday's very long post:

We've seen images that two triangular numbers make a rectangle, which has dimensions N X (N + 1). If you take a second one and rotate it 90 degrees and place it next to the first, and then repeat this with a third and a fourth, you get a square with the center missing. That center has an area of 1 square unit. Like this:

We know from yesterday that the formula for finding triangular numbers is T_N = 1/2 (N² + N)

That means that 2T_N = (N² + N)

This is important because that middle piece looks familiar, like part of a quadratic expression. (The above image is already a great hint here.)

If I have 8 times a Triangular number, I have

8T_N = 8 (1/2) (N² + N)
8T_N = 4 (N² + N)
8T_N = 4N² + 4N)
At this point, it looks almost like a perfect square, except that there's something missing:
8T_N + 1 = 4N² + 4N)
8T_N + 1 = (2N + 1)²

So we can see that the formula 8T_N + 1 will always give us a perfect square. Moreover, it will always be an odd square (which makes sense, as it is one more than an even multiple).

The slightly confusing thing (to me) is that, for example, the third triangular number does not lead to the third odd square, but the fourth, because of the + 1 in the formula instead of a - 1.

I guess you can't have everything wrapped in a pretty little bow.


Next up: the book goes into the following topics, but doesn't spend much time on them: Mersenne Numbers and Perfect Numbers, Fermat Numbers, Cyclic Numbers, Automorphic Numbers, Prime Numbers and Multigrades. Then it gets into Cryptarthmetic and other puzzle and games.

There's not much I can add to some of those topics (other than updating data about Mersennes, maybe), and others I've never heard of. So I'll read those and see what I can figure out.