Here is a problem that you may have seen before. Suppose you have a chessboard with two opposite corners removed. Can it be tiled with dominoes? The answer is no: each domino will cover one light square and one dark square, and there are different numbers of light squares than dark squares. Now consider the following problem. For which $m$ and $n$ can an $m \times n$ rectangle be tiled by tiles in the following shapes (rotations and reflections are allowed)? $$\begin{array}{ccc}& * & \\ {}* & * & * \\ & * & \end{array} \qquad \qquad \begin{array}{c} * \\ {}* \\ {}* \end{array} \qquad \qquad \begin{array}{cccc} * \\ {}* & * & * & * \\ & & & * \end{array} $$ If $mn$ is divisible by $3$, then it is easy to tile the rectangle with copies of the second tile. If $mn$ is not divisible by $3$, then you will find that tiling the rectangle seems to be impossible, yet there does not seem to be a coloring argument that proves that it is impossible to tile. I will describe a method of proving that a region cannot be tiled by drawing decorations along the edges of the tiles and the region, and explain how this method generalizes coloring arguments.

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