Skip to main content

Ralsina.Me — Roberto Alsina's website

Hardcore Finger Counting

Ear­li­er to­day, @el­nomote­ta men­tioned in twit­ter that if you count with your fin­gers you are in trou­ble, be­cause at least you have over­flows.

That got me think­ing. Not about whether there is an over­flow, since there is al­ways an over­flow, even if you count by elec­tron quan­tum states us­ing the whole uni­verse, be­cause in­fin­i­ty, dude, but about how high you can fin­ger-­coun­t.

Sure, the naïve an­swer is ten, but that's triv­ial to im­prove. For ex­am­ple, here is a very sim­ple sys­tem to count to 99 but even that is very sim­plis­tic.

If you are a computer nerd, you may think you are clever by now saying \(2^{10}-1\) but really, how unimaginative is that? It's unimaginative enough that it has its own wikipedia page.

One thing I do (and I rec­og­nize it as one of my most an­noy­ing trait­s) is to con­sid­er un­ortho­dox an­swers to ques­tion­s. Be­cause of­ten they will show that the one ask­ing the ques­tion has on­ly a very vague idea of what he is ask­ing, and ex­pos­es a ton of un­ex­pressed as­sump­tion­s.

So, computer geek, \(2^{10}-1\), that is 1023. Congratulations, you have done much worse than the Venerable Bede, who in 710AD described in De Computo vel Loguela per Gestum Digitorum a system to express numbers up to 9999 using both hands.

So, let's think about the un­ex­pressed as­sump­tions here.

Is a finger a bit?

Hell no. A fin­ger is a fin­ger. Sure, it can ex­press a bit, but it can al­so (in some cas­es) ex­press more. For ex­am­ple, I have 6 fin­gers I can bend in­de­pen­dent­ly in more than one place (thum­b, in­dex, pinky).

So, I could use those to have a ternary digit (if you pardon the pun), and count to \(3^6 2^4-1\) (or 11663)

Is finger-counting just about fingers?

If we consider it hand counting instead it's much better. For example, I could hold each hand palm-up or palm-down for 2 extra bits. That's \(3^6 2^6-1\) (or 46655)

Is finger-order relevant?

So, sup­pose I put my left hand to the right of my right hand. Since I can tell which hand is which, be­cause fin­gers are not all the same, I can count that as an ex­tra bit, count­ing to 93311.

How long can I take before I show you the number?

I could say: "if a fin­ger­nail is long, that's a 3 (or a 4) de­pend­ing on which fin­ger". Sure, it will then take me days to ex­press a num­ber, but I just raised the num­ber I can count to, us­ing my fin­ger­s, to a re­al­ly large num­ber I won't both­er cal­cu­lat­ing (2985983)

Do I have to keep my fingers still?

Be­cause with one fin­ger I could tap morse code for any num­ber giv­en pa­tience, a hard sur­face, a re­silient fin­ger and knowl­edge of morse.

Can't I just keep on adding bits?

Of course. I could bite on the back of my left hand and leave a mark. I can use dif­fer­ent hand po­si­tions oth­er than palm up­/­down and straight/crossed. I could tat­too a num­ber on the palm. I could ex­press a URL to a site that con­tains a num­ber. This is be­cause the amount of in­for­ma­tion on a per­son­'s hand is huge.

So, sure, you can count to ten, or 99, or 1023, or 2985983. The trade­off is, the high­er your sys­tem goes, the hard­er it is to read, and the more pre­vi­ous­ly agreed knowl­edge you need be­tween the one ex­press­ing the num­ber and the one read­ing it.

That's why you still count with your fin­gers just to 10. Be­cause it's ob­vi­ous.

Chris Warrick / 2013-10-04 17:41:

The lazy bastard in me uses another system, the senary (or base 6) system, which just so happens to be described by the almighty Wikipedia[0], although it doesn’t get an entire article devoted to itself (just a section). Basically, one hand is devoted to counting sixes, while the other counts ones. So, for example:

00000 00001 1
00000 00010 2
00000 00100 3
00000 01000 4
00000 10000 5
00001 00000 6
00001 00001 7
and so on…

[0] http://en.wikipedia.org/wik...

schettino72 / 2013-10-05 08:08:

Chinese people uses only one hand to count to 10. It is really used in practice, everyone knows that and it is quite convinient...

If you want to count over 10, fisrt you show the first digit than change the hand sign and show the second digit.

http://en.wikipedia.org/wik...

Lucio / 2013-10-05 09:48:

You can also use your thumb standalone (as proposed) and pointing to the other 4 fingers. Making it (7^2)(3^4)(2^4)==63504. (based on the Asian way of pointing to the finger bones)

I would say that morse code does not count, as the idea is that the hand plays the role of memory, not the case in morse code. :)