1000 logicians are going to play a game against The Adversary. The Adversary has a collection of hats that come in 6 different colours. There are also 6 rooms. The Adversary is going to gather all the logicians together and put a hat on each of their heads, and then the logicians must simultaneously decide what room they are going to enter. After the logicians have each gone to their chosen rooms, each room must have logicians with only one colour of hat. The logicians may strategize before the hats are assigned, find a strategy that guarantees that each room will have a unique hat colour in it. Standard hat rules apply, of course.
You can assume that the logicians know what colours are available. If you like, you can also generalize to N logicians and K hat colours, its not much harder. What is harder is allowing N and K to be infinite, you will need the axiom of choice in this case, but its also an interesting puzzle.
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