TITLE: An Olympiad Problem (tiling a rectangle with the L-tetromino)
QUESTION [5 upvotes]: An L block that is 3 unit blocks high and 2 unit blocks wide . It is true that if an n by m rectangle can be covered by such L blocks with out overlap that we would require an even amount of L blocks,  why?

REPLY [6 votes]: This can be solved in one fell swoop, with just one coloring.
The total area of the rectangle is even (and even a multiple of 4). Then we can assume without loss of generality that the width is even. Now let us paint all cells in the first column black, in the second column - white, in the third - black, and so on. Clearly, exactly half of the cells will be black, and the other half white.
Notice that each L-block either covers 3 black cells and 1 white one, or 3 white cells and 1 black one. In the former case let's call this L-block mostly black, in the latter - mostly white. It is easy to see that the number of mostly black L-blocks must be the same as the number of mostly white L-blocks, because the total amounts of black and white cells are equal to each other. It follows that the total amount of L-blocks is even.