Are you positive that's true for all bases?

The statement as juanpaulo gives it has no hidden dependence on bases. It is all spelled out by the conditions. The symbols a b c d stand for any values, not the "actual digits" 0-9, which are also merely placeholders for values with no "true meaning" of their own.

He is using both digits and the nubers that sequence of digits represents in that equation.

In base 10, 7641 - 1467 = 6174.

In base 11, 7641 - 1467 = 4808.

Finding tricks that work in base 10 is interesting, but it's a much more fun to look for patterns that work across several number systems. A simple example is for any base > 3, (base - 1) times x where base > x > 1; you get a 2 digit number [x - 1],[base - x]

No, they are not. The symbols in the equation are simply numbers, and "9 ≥ a ≥ b ≥ c ≥ d ≥ 0" constrains them to 0-9.

You are mistaking the map for the territory, sort of. The positional aspects of the number system get encoded the equations and then the mere "numbers" don't matter anymore.

I think we are talking past each other. In base ten if you start with 1234 and cycle though asdf - fdsa > new number you hit a single stable number 6174 .

In base 11 using 6174 you cycle:

4808, 87A3, 7094, 9272, 3098, 9452, 7094 (cycle)

I was thinking of a http://en.wikipedia.org/wiki/Logistic_map for this equation abcd - dcba > might be interesting across different bases.

Yes they are. Where he meant 1000a+100b+10c+d he wrote abcd.