Wednesday, March 11, 2009

Count on this

Let's switch things up a bit and just do something for fun.  How high can you count on your fingers? Ten, you say? Amateurs. I'm going to teach you how to count much, much higher... without even taking your shoes off.

However, you're going to have to learn to count in binary first. Again, this is a very basic topic to computer science students, but I hear that there are plenty of non-programmers who read this blog. For their benefit, I'm going to give a primer. If you already know binary, feel free to skip down to the next bold header.

How to count in binary

When you count by tens, the digits can be represented as wheels on the odometer of your car. The right-most wheel goes:
  • 1, 2, 3, 4, 5, 6, 7, 8, 9…
Then what happens? There is no "10" printed on the wheel. So where do we go next? The rightmost wheel flips from "9" back to "0", and as that happens, it also pushes the next wheel up by 1. So you have a 1 and a 0, and that's ten.
  • 11, 12, 13, 14, 15, 16, 17, 18, 19…
Then the first wheel again flips over from 9 to 0, which in turn advances the second wheel from 1 to 2. Twenty. Right? Don't tell me this is hard, my son Ben understands this and he's in first grade.

So anyway, things are pretty much the same from here on out, until we get up to:
  • 97, 98, 99…
And now? Well, the first wheel flips to zero, which in turn advances the second wheel. But the second wheel is already maxed out at nine, so it flips to zero, which also advances the third wheel, so now that wheel reads "1". 1, 0, 0. A hundred. Got it?

Okay, that's base 10. Binary is a similar system, except that it's got a severe shortage of numerals. Whereas we were using all the numbers from zero to nine, in binary you only have two possible values: one, and zero. Like a light switch: on, or off. Circuit open, or circuit closed. That's how a computer works. But with just ones and zeroes, you can represent every positive integer if you have the patience.

Let's imagine we have an odometer like before, but this one has very small wheels. They flip from 0 to 1. Then they flip from 1 back to 0. When they flip to 0, they advance the next wheel. Okay?

So your odometer reads:
  • 0: 00000
Advance the rightmost counter by one:
  • 1: 00001
Advance the counter again, and it flips to zero while dragging the second wheel along:
  • 2: 00010
So that's two. Ten is two, in binary. Makes perfect sense, right? Advance the right hand wheel one more.
  • 3: 00011
Next time we advance, both of the first two wheels will advance the next one over.
  • 4: 00100
One hundred is four. I think you can take it from here.

In base ten, every digit represents a value that is ten times the previous digit. So the first digit is worth one, the second digit is worth ten, the third digit is worth a hundred, the fourth is worth a thousand, and so on.

Similarly, in base two, every digit represents a value that is two times the previous digit.
  • 1 -> 1
  • 10 -> 2
  • 100 -> 4
  • 1000 -> 8
  • 10000 -> 16
Makes sense, right? You can write out any string of ones and zeroes and figure out what number they represent in binary by adding up the digits. For example,
  • 10110 = 16 + 4 + 2 = 22
Okay, here's the fun part.

How to count to 1,023 on your fingers

  • Hold out your hands, palms upward. Now make them fists. No fingers are extended. That's zero.
  • Extend your right thumb. The right-most digit is now set to "on". That's one.
  • Close up your thumb, but extend your right index finger. That's two.
  • Extend your thumb again, and also leave your index finger out. Three.
  • Close up your thumb and index fingers, but then extend your middle finger. That's four. It is also an obscene gesture to most people, so if you're in public, make sure your hands are under a desk or something.

That's really all there is to it. The highest you can go with this system is 1111111111 in binary, which is 1,023 in regular counting (210 - 1).

Other numbers that form obscene gestures to Americans: 128; 132

Numbers that may be obscene to Europeans: 6; 384; 390. They may also be interpreted as peace signs or a Richard Nixon impression.

What can you do with this? Not much. You can kill about ten to twenty minutes. You can also probably perform a cheap mind-reading trick if you have an accomplice who can also do it. ("I'm thinking of a number from one to a thousand." "583. I certainly wasn't paying attention to the funny way you're holding your hands.") And finally, you can waste a perfectly good afternoon writing a silly blog post.

By the way... if you DID take your shoes off, you could count to 1,048,575.


  1. I use it sometimes when I have to count, for example, how many Lamps are there in a blueprint. It's fun.

  2. I used to play second oboe in an orchestra, and often had to count rests of, say, a hundred and nine bars. I did it in binary on my fingers.

    If you can count on your toes, then you have a lot more motor control than I do: I can only make the big toes move independently from the others.

  3. "By the way... if you DID take your shoes off, you could count to 1,048,575."
    You must have very dexterous toes.