A very heavy armchair needs to be moved, but the only possible movement is to rotate it through 90 degrees about any of its corners. Can it be moved so that it is exactly beside its starting position and facing the same way?
After considering the problem as stated, ask yourself if your answer would be different if it could rotate through angles other than 90 degrees?
Hint: Find a square-tiled floor and cut a piece of paper congruent to one of the tiles. Mark the front/back edges of card to represent the armchair. Now, try various rotations of the card to understand the problem better.
Solution Commentary: Repeat the hint, but this time mark with a "yellow-sticky" those tiles that are reachable according to the directions. Stand back...look for a visual pattern, and then try to explain it mathematically.
When all else fails, think of a checkerboard!