h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

You could set up a proof that there is a number with this property in counting systems with any number base -- for instance in binary you add up all the ones and you get a smaller number consisting of ones and noughts, add them together and you get another, keep on and eventually you will get down to two (10). Similarly if your base was 8, then 7 would have this magic property: since eight is written "10", fourteen would be written "16" (try counting it on your fingers) and so on. All the multiples of 7 would look like 25, 34, 43, 52, 61, 70 etc. Add the digits of the next one (77) and what do you get? Yes, 16 of course!

As one digit goes up another goes down, and they always match.

Thre would be such a proof; but non-mathematicians may well find it lacking the element of "explanation" why it should be so. This is the difference between people who can and people who can't stomach maths.

Some famous mathematician (was it Von Neumann?) said "You don't really get to understand math, you just get used to it".

Sloppy of me -- in binary the magic number is 1 of course; in general, the last number before 10.

I think it was Von Neumann who said it allright.

As for an explanation, here may be one. Writing a number with a series of digits (e.g. 2186) means you write out

2186 = 6x1 + 8x10 + 1x100 + 2x1000.

Now write 1=0+1, 10=9+1, 100=99+1, 1000=999+1 (take as many 9's out of 1, 10, 100, 1000 as possible). The previous equation can be rewritten as:

2186=(6x1 + 8x1 + 1x1 + 2x1) + (6x0 + 8x9 + 1x99 + 2x999).

The right hand parenthesised term (6x0 + 8x9 + 1x99 + 2x999) is a multiple of 9, so it is quite easy to see that 2186 is a multiple of 9 if and only if the left hand parenthesised term (6x1 + 8x1 + 1x1 + 2x1) is a multiple of 9. But now, the latter is none other than the sum of digits of 2186, QED.

(One may even notice that, more generally, a number and its sum-of-digits have the same remainder in the euclidean division by 9.)

So much for the magic/mystic properties that whan you add or multiply numbers whose "eventual sum of digits" is 9, you get another number with the same property.