Basically, the way I see it, is that 7 is a prime number. And the movements of 3 cards, or whatever. Form a group of order 7. But by Lagrange's theorem this group is cyclic. Hence the operation of the elements 7 times will return back the identity element, the selected card.
EDIT: It does not work for 1 (identity element).
REDIT: Then you can use "magician's choice" and say "Ah, the first card is it!". Even better.
RE-REDIT: It should work with any prime number. So try it with 5, 11, 13, it works!
Some things involving cards are truly remarkable. There are a number of very interesting properties the cards must obey when they are riffle shuffled. Very mathematical, that I use as a tool. I never tried to prove them they seem complicated, and also seem group theory related.