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.
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1 comment:
OK, honest, though...I'm NOT too lazy to do that particular puzzle. OK, maybe I am. But, well, I only got here because I was looking up the Aussie Attack.
What an interesting find, this Aussie Attack, as it seems to be a transposition of not only the Nimzo-Indian, but also the Trompowsky.
Far from a grandmaster myself, one of the folks at chessdotcom had mentioned it in passing, and it's been a bear to track down. (do they have bears in Australia, too?)
Anyway, I do take a bit of fun going at completly off chess topical puzzles on occasion. Thrill of the day for me a few years back was when someone had said I couldn't put eight Pawns on a Chess board in such a manner that they were not on any other Pawn's rank, file or diagonal.
And I started with a fresh, mindless naive grin, and somehow just pushed the Pawns into the correct position first time.
Then I discovered that one could move the Pawns all simultaneously in any direction, and when a Pawn went off the board on one side, just put it back on on the opposite side, and it made the same claim. No shared diagonals, ranks, or files.
So, for fun (that would have to be more fun than reading this, eh?) I kept at it until I found a unique solution that couldn't be moved along in one step increments. It was a unique position, as in, one of a kind. And despite it's being no shared diagonals, files or ranks...it was symmetrically inclined.
Now...how's THAT for a puzzle?
If only I could learn how to play Chess as well as finding such a position...
Practice, practice, practice, I suppose.
MisterBoneman
d=^))
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