How high can you count on fingers and toes?

[Bartholomew]

BobG's signature is:

If God meant for man to count higher than 1,048,575 he would have given us more fingers and toes.

So this gave me an idea. How high CAN you count on fingers and toes? Starting at 0 and covering every number, how many numbers can you represent distinctly and clearly? For example, on just your hands, with each finger/thumb either being up or down, you can count to 1023 in binary. But you have more than two distinct positions for your fingers. You have two easily bendable joints on each finger (knuckle and first joint), for example, and you might be able to make something out of fingers touching other fingers, or whatever. But I mean just the fingers and toes--like don't use wrist or limb positions to represent numbers.

(Hopefully this wasn't what BobG meant already with his signature)

 

[Gokul43201]

A finger with N joints can be used to represent 2^N numbers, assuming you can bend each joint independently, and you don't count partial bending. I can only make 5 different positions with with 3 jointed fingers .

With my toes, it's almost purely binary. I can sort of get 3 positions with my big toe, but that also ends up bending my next toe.

If you built an M-jointed robot that could bend all joints independently, it could count upto 2^M (starting at 1). This treats bending as a binary state, and doesn't involve other tricks like touching fingers, etc.

 

[futb0l]

Cool LOL

 

[DaveC426913]

"... assuming you can bend each joint independently..."

Which you can't. You can't even bend all your fingers independently. Try bending your pinky without bending your ring finger, or try sticking just your ring finger up (now, no cheating - 'independently' is the operative word here).

 

[Gokul43201]

Hence the robot.

 

[BobG]

The Chinese had a finger counting method where they could count to 99,999 just on their fingers, alone.

You have three links on each finger (although on the thumb, you're kind of down on the base to get the third link. Each finger could also be divided the other direction: one side of the finger towards the outside of the hand (towards the pinky), one side towards the palm, and one side towards the thumb. That gives you nine distinct sections of your finger, or you could use none of them to give you a zero.

The Chinese used the sections of the fingers on one hand as kind of the scorecard. They used the fingers on the opposite hand to point to the appropriate section of the finger as they counted.

If you used your toes as the scorecard and your fingers as the pointers, you could get up to 9,999,999,999. Your back would get very sore, though. If you were a slow conter, one number every second, it would take you nearly 317 years to count so high.

 

[Bartholomew]

Well, BobG, that's one way to count, probably one of the clearest and least-confusing ways. But I say, if you can't get hopelessly mixed up as to what number comes next, you're not doing it right.

I've come up with a few ideas that could probably count very high (haven't worked it out). First, each of the four fingers can bend twice, at the knuckle or the first joint--disregarding the second joint because it's hard to bend. And each finger can either touch the palm or not. The tip of the thumb can touch the first link of any finger and the first two links of the pinky when the fingers are straight, and when any finger is bent the tip of the thumb can touch any part of it. (It's a little difficult to find out how many numbers there are there... but I'm not trying) In addition, the thumb can, without touching any other finger, take on the five positions that the fingers can (namely: straight, 1st joint bent, both joints bent, 2nd joint bent, touching palm). In addition to this, the fingers and thumb can be grouped or not--pressed against its neighbors or not. You have four spaces between the fingers and each can either be a space or be two fingers touching which makes 2^4 or 16 ways to group the fingers. The thumb can take any of the five non-pointing positions while doing this, but it can't point and do it, and the fingers can group while in any position. So how many ways does all this make? (Just one one hand) I'll try to figure it out later but I don't think it will be simple.

(Also, when the fingers are straight or only the knuckles are bent, you have the option of them being crossed or on top of one another, but I'll let that be for now. Another thing I just thought of now is that the thumb can point to the spaces _between_ the fingers and the space to the right of the pinky)