Counting On Your Fingers...

    This is probably how you and most other people
have first been taught mathematics. Such a tool is
powerful in maths considering that you won't lose
your fingers in a hurry, but you will have only a
limited number of fingers, which very soon your
needs will outgrow. So unless you're looking at
very, very painful surgery, you're not going to
have all the fingers you'll ever need.
    It was in 4th grade that this idea struck me that
it might be possible to count past the usual number
of fingers on your hand. I ran it through my parents,
who showed me the system of using the joints on
your fingers to count further than usual. This did not
satisfy as it was impractical for counting quickly and
looked very silly. Then the answer came to me in the
most unlikely of places: my Computer Textbook. We
were learning about number systems and so I found
how binary could be used to represent even larger
numbers on my hand:

If you number your fingers on the right hand going
from the thumb to the little finger in this order:
1, 2, 4, 8 and 16 (powers of two), like this-

then you should be able to represent any number from
0-31(2^5-1) using only your fingers.
    Lets test - how about 31?- well, just raise all your fingers, 1+2+4+8+16=31(11111)
                     hmmm.....  28?- your last three fingers, 16+8+4=28(11100)
                     well then, try 17- little finger and thumb, 16+1=17(10001)

    Like that you can number all your fingers and if you
memorise all the values, you can actually represent all
the numbers from 0-1023(2^10-1) on only your fingers.

closed

BUT WHAT ABOUT HIGHER BASES?

    Okay, here we run into a sort of problem because in
binary, you only need 2 inputs- finger open or finger closed.
    For higher bases like 20, you will need to evolve VERY
flexible fingers.
    But we can allow for a maximum base of 3 as the human
finger can have 3 different positions- closed, half-open or
fully open-(right)

    Now, after you number your fingers 1, 3, 9, 27 and 81,
you have to take this into account- if a finger is half-open,

half-open

it represents whatever number is on that finger, but if it is
fully open, then it represents twice the number on that particular finger because in ternary (base 3), there are 3 digits- 0, 1 and 2.
    So let's see how much you can do with 5 fingers,
it's 3^5-1 which is 242. With 10, you can get 59,048.


    But what practical use is this? I gave some thought to this,
and I've got it. Imagine you make a code language where in
a message, you divide the code up into 3 letter parts and assign

fully open

a numeral to each part as in a=1, km=299 and tuc=14,089.
(This is not random. I've used the a=1, b=2, c=3 system)
So sending a message becomes 3 times as fast.
The total number of three-letter sets is exactly 18278, so using only
two hands you can represent any three letter set in the language.
You will need an amazing memory though and an equally
amazing computational prowess.