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A beautiful puzzle, and I know the solution. I read it on an Italian riddle forum.

An evil witch kidnaps infinite (countable) dwarves. The witch tells the dwarves that the next day she will put them all facing the same direction and uniformly at random will place on each dwarf's head a hat that can be either white or black. So that the first one sees all the other dwarves. The second all but the first, etc, i.e. each dwarf will then see only the color of the dwarves' hats in front of him.

Each dwarf will have to guess their hat color, whoever guesses is free whoever misses dies. The dwarves have one night to be able to discuss a strategy. The main problem is that the dwarves are deaf! So they will not hear anything the other dwarves say. Can you come up with a strategy that can save all but finitely many dwarves?

Edit: My question is very different from Infinitely many dwarves wearing hats of 2 colours because if the dwarves are not deaf you can find a strategy for which you can save all the dwarves except the first one. However, it is true that the answer given (by theosza) in the other question works perfectly to my question. The difference is the fact that the answer of theosza is not the optimal solution to Infinitely many dwarves wearing hats of 2 colours, and the optimal solution of Infinitely many dwarves wearing hats of 2 colours is not applicable to the case where the dwarves are deaf.

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    $\begingroup$ Seems similar to puzzling.stackexchange.com/questions/7819/… $\endgroup$ Nov 19, 2022 at 17:14
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    $\begingroup$ Similar but not the same! Since here the dwarves are deaf, the same strategy used for the other one doesn't work here. $\endgroup$
    – 3m0o
    Nov 19, 2022 at 17:15
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    $\begingroup$ The dwarves being deaf shouldn't impact the strategy in the other answer. They don't need to hear what the others say, they can still just follow the representative of the equivalence class and only finitely many will die. $\endgroup$
    – hexomino
    Nov 19, 2022 at 19:38
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    $\begingroup$ @3m0o Note that your original question still does not ask for an optimal solution but that you come up with a strategy to "save all but finitely many dwarves" which theosza's answer does. $\endgroup$
    – hexomino
    Dec 19, 2023 at 11:06
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    $\begingroup$ Also, if there is a better solution to the other question, wouldn't it be sensible to post it as an answer there? $\endgroup$
    – hexomino
    Dec 19, 2023 at 11:08

1 Answer 1

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A simple strategy to save all but one of the infinite dwarves would be

They are all responsible only for the dwarf directly in front of themselves. If the hat worn by the dwarf ahead of them is black, tap the left shoulder (or vice versa), if the hat is white, tap the opposite shoulder. By doing this, and assuming the other dwarves are trustworthy, each dwarf other than the first will go free, and the first would still have a 50-50 shot of guessing correctly and going free himself.

Perhaps this would be better as a comment but unfortunately I am currently unable to add comments so this is what I must do

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  • $\begingroup$ You right I didn't mention it in the problem, but it was assumed that they cannot comunicate with each other in any ways after arriving to a startegy the day before. $\endgroup$
    – 3m0o
    Dec 22, 2023 at 17:10

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