Clyde mentions that the result of the procedure is not amazing because its constraints mean it will eventually settle somewhere, however it is somewhat amazing that it settles on a single number and not a group of numbers... If you apply the technique to numbers in base 9 there at least two possible attractive triplets of numbers you could end up in:
7072 --> 7432 --> 5074 --> 7072...
or
7252 --> 5254 --> 3076 --> 7252...
(there may be more...these are the only ones I've found so far).