This problem was shown to me by my student, Apratim Roy. Though it involves only elementary concepts, I found it rather difficult, and thought the solution was very surprising and elegant.
You are given a standard 8 x 8 checkerboard, with one square removed, and 21 3 x 1 tiles. In other words, there are exactly enough tiles to cover the modified board. Your task is to find a way to do this, without cutting any tile.
(a) Find out what square must be removed for the task to be possible. (4 possible answers).
(b) Describe the tiling. (Many possible answers).
You might guess that the removed square needs to be one of the four corners of the checkerboard, but you would be wrong.