Counting Powers of Two - A Neat Pattern

A little while ago I was going for a walk and searching for something to occupy my mind for a while. Something that would take my mind away from other problems and give it a simple, solvable, repetitive activity to do for a while. I hit upon counting up by powers of two.

Why? I’m a programmer, and powers of two are important and prevalent numbers in this field. By this point 128, 256 and 512 seem like nice round numbers to me, and I wouldn’t mind feeling the same way about 65,536 and 131,072. Also, counting powers of two has the virtue of getting hard before getting boring.

As I counted I noticed a neat pattern starting at the higher numbers. I thought it was pretty cool and maybe even a bit easier than doubling the numbers in the standard fashion (once you’re out of memorized territory anyway).

From what I’ve seen of this pattern it only works at and above 1024 (2¹⁰) so that’s what I’ll focus on.

The Pattern

Take the number 1024 and split it into two parts, 1000 and 24. Keep on doing this:

1000 and 24
2000 and 48
4000 and 96
8000 and 192

There’s an interesting relationship between the thousands digit and the rest (well, there’s a few), do you see any yet?

What if I represent it like this:

1000 + (25 - 1)
2000 + (50 - 2)
4000 + (100 - 4)
8000 + (200 - 8)

You can just keep on going, doubling both numbers and subtracting the thousands digit from the other number. It is a pretty quick mental operation to double 200 and subtract 16. This isn’t the only way to express this relationship though:

1000 + (24 * 1)
2000 + (24 * 2)
4000 + (24 * 4)
8000 + (24 * 8)

It’s a simpler expression of the relationship but lacks the calculation shortcut the first one has. So what does these look like if you generalize them one step more…?

The Formulas

Where x represents the thousands digits and is equal to 1 or greater (multiple digits work fine but substituting 0 in doesn’t end up making any sense):

(x * 1000) + ((25 * x) - x)

This can be simplified to:

(x * 1000) + (24 * x)

An interesting benefit is that now you can answer the question “Okay, 64kB… 64 thousand and how many bytes?”

Well using the first formula it is (25 * 64) - 64 bytes, but that’s easier to do with rounder numbers. The number you’re multiplying 25 by will always be easily divisible by 4 ^{note 1} and 100 is easier to multiply by than 25, so let’s multiply 25 by 4 and divide 64 by 4 to get 100 * 16. It’s easy to instantly see that’s 1,600. Now the subtraction: 1,600 - 64 = 1,536. The answer is 64 thousand and 1,536 bytes, or 65,536 bytes. For those like me who were wondering, that’s 2¹⁶!

In Closing

I feel like this formula is only partway to somewhere - sure it’s a neat pattern but it is a bit awkward. I’d love to hear about any more cool patterns or neat mathematical tricks, if you know any and would like to share!

note 1 - Unless it's smaller than 4 in which case it doesn't matter, don't get too pedantic on me :)