Geodesic
The Hats game. The power of constructors
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 498 (2020), pp. 26-37

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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. 
@article{ZNSL_2020_498_a2,
     author = {K. P. Kokhas and A. S. Latyshev},
     title = {The {Hats} game. {The} power of constructors},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {26--37},
     publisher = {mathdoc},
     volume = {498},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_498_a2/}
}
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K. P. Kokhas; A. S. Latyshev. The Hats game. The power of constructors. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 498 (2020), pp. 26-37. http://geodesic.mathdoc.fr/item/ZNSL_2020_498_a2/