The Hats game. The power of constructors

We analyze the following general variant of the deterministic Hats game. Several sages wearing colored hats occupy the vertices of a graph. Each sage can have a hat of one of $k$ colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors.
We present an example of a planar graph for which the sages win for $k=14$. We also give an easy proof of the theorem about the Hats game on “windmill” graphs.