There are six plates next to each other in a row on a table. Bob grabs two plates and moves each plate one position to the left or to the right (if there is one plate or more at the target position then he puts the plate on top). Repeating this procedure, can he stack the six plates into a single pile? (Bob can move one plate to the right and one plate to the left or both plates in the same direction, but only the top plates).
We can imagine the six plates to be at six consecutive integer points on the real number line, and colour all the integers alternately black and white (odd black, even white).
Each move of the form you've described preserves the parities of both the number of plates on black integers and the number on white integers - either you're moving B->W twice, or W->B twice, or one of each. But the initial position has three plates on black integers and three on white, while the desired final position has all six plates on a single colour. Thus the problem is impossible.
The key, as with many such puzzles, is to find an invariant.