This is the 400th installment of The game of sciencewhich means almost eight years of uninterrupted publication in the pages of Matterso it seems opportune to dedicate a few minutes of attention to these two very special and closely related numbers, 8 and 400.
For moviegoers, the number 400 immediately reminds us of François Truffaut’s masterpiece, the four hundred blowswhose title refers to the French phrase faire les quatre cents cups, which means something like “do the thousand and one”. But why four hundred and not a hundred or a thousand or another more common round number?
As for 8, this article could be filled with the mere description of its properties: it is the first perfect cube (not counting the trivial cases of 0 and 1, which are the cubes of themselves), it is a Fibonacci number, a cake number, a practical number, a Leyland number (in future installments we will deal with all of them) and many other things (infinite, if we knock it down). I invite my shrewd readers to look for other properties of this very special number, and also to deduce how this curious sequence continues that skips 7 so that its seventh term is 8:
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30…
(For once and without setting a precedent, I will not give the solution next week but in the next paragraph, so if you accept the challenge, do not continue reading until you have found the next terms in the sequence.)
The following terms are 32, 36, 40, 45, 48, 50, 54, 60…, since it is the sequence of regular numbers, that is, whose only prime factors are 2, 3 and/or 5. Therefore, 400 is a regular number, since 400 = 2⁴ x 5².
Regular numbers are also called “soft 5” because 5 is the largest prime that can appear in their factorization.
Since 2 x 3 x 5 = 30, all regular numbers are divisors of some power of 30, and hence also of some power of 60, so they are of great importance in relation to the Babylonian sexagesimal numeral system. (which we still use in our clocks and when measuring angles). And computer scientists may know them as Hamming numbers (named after the American mathematician Richard Hamming and his algorithms for generating regular numbers).
On the other hand, the number 400 is the sum of the first four powers of 7: 400 = 1 + 7 + 49 + 343. Can you write 400 in base 7 in less than 7 seconds?
Ten fingers or twenty?
In the Mayan numeral system, a vigesimal positional system, 400 (20 x 20) was the value of the digits of the third level: in the first were the units, in the second the scores and in the third the four hundred. It has been said that the Mayans probably developed this system because, when sitting cross-legged and barefoot (what in yoga is called the Tailor’s Pose), they had 20 appendages to count with their fingers. But the other Indians (the real ones, the ones from India) also used to sit like this and go barefoot, and they invented the decimal positional system.
Why did they only have their fingers, while also having their toes at their disposal? Could it be that the decimal system, which is the one that has ended up being imposed throughout the world, has some advantage over the vigesimal? What can that advantage be?
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