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Found problems: 1

2022 239 Open Mathematical Olympiad, 6
Tags: hat , 239 , color , combinatorics
$239$ wise men stand in a circle near an opaque baobab. The king put on the head of each of these wise men a hat og one of $16$ colors. Each wise men does nor know the color of his hat and can only see the two nearest wise men on each side around the circle. Without communicating, these wise men must at the same time make a guess about the color of their hat $($i.e, tell one color$)$. These wise men were allowed to consult in advance, while they are afraid of being too lucky. What is the maximum $k$ for which, in any arrangement of hats, they can certainly ensure that no more than $k$ wise men guess the color of their hats$?$