Delightful Puzzles

Now I realise that many chessers are mathematically inclined and, naturally, enjoy puzzles. So I thought I would post this. Appropriately, this page full of puzzles happens to include this, too:

Tiling a Chessboard with Trominoes 
Show that a chessboard of size 2^n by 2^n can be tiled with L-shaped figures of 3 squares, such that only one square remains uncovered. In fact, the uncovered square may be any square — for every choice, there exists a tiling. In fact, the puzzle may be extended to 3D: Eight unit cubes make a cube with edge length two. We will call such a cube with one unit cube removed a "piece". A cube with edge length 2^n consists of (2^n)3 unit cubes. Prove that if one unit cube is removed from T, then the remaining solid can be decomposed into pieces.

For more puzzles, check out Delightful Puzzles.