



 
Interesting! Yes, I think this is probably known  but it is a very respectable rediscovery. The reasons for it are interesting, and have a lot to do with the theory of repeating decimals. First, note that 1/9 = 0.11111.... This is not a coincidence: in base B, 1/(1B) always equals 0.11111..... Next, note that 11*11 = 121, 111*111 = 12321, etc. (the pattern breaks holds up to 111111111*111111111.) This is not a coincidence either; it comes from the fact that numbers in place value notation act like polynomials until you have to carry. With ones, there isn't a lot of carrying. From these, you expect to see that 1/81 = 0.012345679. It's not quite so clear that it repeats immediately: 1/81 = 0.012345679012345679012345679.... but it does. Thus, multiplying 0123456789 by N is essentially the same as finding N/81, except for the last couple digits. It's common to have patterns like this in repeating decimals  try finding 1/7, 2/7, 3/7... Why does this happen? Well, 1/81 has a loop of period 9. N/81 must also loop with a pure period of 9, or else with period 3 or 1. 1/27 loops with period 3, so if N is divisible by 3, N/81 loops with period 3. (Your "cardinal" numbers are just those not divisible by 3  if a number is divisible by 3 so is the sum of its digits!) This will certainly bust the pattern early on. If N is not divisible by 3, N/81 will have 9 of the digits (not always the same 9) in its circulant. This means that most of the digits of 012345679 * N will be different; and most of these these agree with the digits of 0123456789 * N. Note that (if N is not divisible by 3) the first six (seven?) digits are always different! Finally, we note that 0+1+2+3+4+5+6+7+8+9 is divisible by 9. Thus 0123456789*N is divisible by 9, so its digits must also add to a multiple of 9. But if all but the last digit are different, the last one must be the missing one! Thus there are only a couple decimal places where the pattern can break down. A final thought  a lot of this will work in other number bases. Good Hunting!  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 