NL-NWO 2009 Problem 5

We number a hundred blank cards on both sides with the numbers $1$ to $100$ . The cards are then stacked in order, with the card with the number $1$ on top. The order of the cards is changed step by step as follows: at the $1$ st step the top card is turned around, and is put back on top of the stack (nothing changes, of course), at the $2$ nd step the topmost $2$ cards are turned around, and put back on top of the stack, up to the $100$ th step, in which the entire stack of $100$ cards is turned around. At the $101$ st step, again only the top card is turned around, at the $102$ nd step, the top most $2$ cards are turned around, and so on. Show that after a finite number of steps, the cards return to their original positions.