Once upon a time, when nobody had a pocket calculator & only a privileged few had access to a computer, statistical frequency counts were often carried out by hand , even by grown up statisticians. You kept tally with pencil marks, usually in bundles of five using the familiar 5-bar gate notation.
A clerical assistant who once worked with me used to use a system of small squares, with both diagonals included – ie bundles of six. Since he was Turkish-Cypriot I assumed that this was an echo of the Babylonian base-6 number system. Of course we still use this system too in our daily lives, in the way that we count time & the number of degrees in a circle.
If he had added a dot to each corner of the square he could have counted in tens.
The other day somebody on the radio mentioned the Babylonian system & called 6 a perfect number, the sum of 1, 2 and 3; he then went on to say that the next perfect number is 10, the sum of 1, 2, 3 and 4.
This left me a bit confused, but I was then diverted by another thought.
We are often told that primitive man could not count beyond three because languages have words only for one, two, three & many; and under the radio man’s definition 3, the sum of 1 and 2, is the perfect number before 6.
Having realised that he was talking about what I know as triangular numbers, I wonder if this gives us a different way of looking at why 3, 6 and 10 emerged as number bases in common use.
Triangular frames make a very easy way of counting – even without words for the numbers. After all, pharmacists (used to) use them for counting pills.
I cannot remember seeing anything about triangular counting frames in any of the tomes on the history of mathematics which I have read – perhaps I wasn’t just paying attention or didn’t find it interesting enough to remember.
Needless to say there is now plenty of information about tally marks on the web
