Question:  What do       $  $  $  $       and       o  o  o  o   have in common?

Answer:  4.

This may seem obvious, but in fact, it is not. My friend Nik spent a year in Japan teaching at a Tokio university. A year is long enough to learn some basic japanese. But this language has some peculiarities. Usually one of the first things a beginner learns in a foreign language is the number system. But not so in japanese — there counting is not scheduled for the beginner lessons but only for the advanced level!

Basically there are two ways to count 1, 2, 3, ... , i.e., ichi, ni, san, yon, go,... or hito, futa, mi, yo, .... But then there are a lot of exceptions and alternatives. For example 4 can be yon or shi, and 9 kyu or ku, both are possible. All depends on WHAT you want to count:

• long thin objects (pencil, umbrella, wine bottle):   ippon, nibon, sanbon, ...
• flat objects (ticket, sheet of paper):   ichi-mai, nimai, sanmai, ...
• floors of a building:   ikkai, nikai, sankai, ...
• months:   ichi-gatsu, ni-gatsu, san-gatsu, ...
• days in a month:   tsuitachi, futsuka, mikka, yokka, ...
• persons:   hitari, futari, san-nin, yon-nin, ...
• liquids (glasses of beer):   hitotsu, futatsu, mittsu, yotsu, ...

Maybe this is a relic of former times when men didn't have yet an abstract concept of the notion of a numbers. Counting different things the same way is not a facility that children invent by their own, on the contrary they have to learn it. It is, along with the invention of the wheel, a cultural achievement; it is sufficient that once a single human got that idea, it then spreads out and enriches humanity.

History of mathematics is full of such simple but essential ideas, for whose discoveries we had to wait thousands of years — and nowadays they can be learned by every child...