Here’s an easy starter: I saw this in Martin Gardner’s The Incredible Dr. Matrix. Take any four Fibonacci numbers. For example: .
Divide them like so:
Now reverse the order of all the numbers:
It’s the same answer!
In the book, Dr. Matrix is a bit of a charlatan. Gardner reveals in the appendix that this trick actually works for any four numbers.
Okay, here’s a very interesting fact that you may have noticed about Fibonacci numbers: A lot of them begin with ‘1’ when written in decimal. If you investigate further, say with a computer program, you will find that over 30% of Fibonacci numbers begin with a ‘1’, and less than one in twenty begin with a ‘9’. What magic is this?
You may already know of this phenomenon, but if you don’t, let me introduce you to Benford’s Law.
As promised, it’s got nothing to do with Fibonacci numbers in particular, but with certain statistical distributions. The populations of US cities follow the same pattern. Notably, “naturally occurring” items on expense accounts tend to follow this law, but fraudulent items don’t. That’s one way the IRS catches you. Now you know.
Since we’re generating Fibonacci numbers on the computer, here’s something you may have noticed when printing out Fibonacci numbers on an old 80-character console:
Look about two-thirds of the way down this image, and you’ll notice that the commas between the numbers seem to be arranged in C-shaped patterns. What’s with that?
A little later, it happens again, a little more clearly:
And even more well-defined:
And we even see some multi-line cases:
These curves are actually parabolas. And no, it’s not some deep Fibonacci-related thing. The same phenomenon occurs in any sequence which approximates exponential growth. Straight-up powers of two will do it. The “Minecraft sequence” (in which each term is approximately the previous term) will do it. And the Fibonacci series will do it, with its growth of about .
If you print powers of in this way, you realize that it’s just got to do with a linearly-growing length of each number.