I have 3 persons which either wear a white or a black cap. They can only see the color of the other caps, but not their own. White and black caps are eqally likely. As a team, they play a game of guessing their own cap color. If they will win, all of them have to guess correctly their own cap color. Once the game begins, they cannot communicate the color of the other two caps.

Now my interesting question: What is a good strategy for the 3 persons such that with 75% probability all answer correctly? (if the strategy should be made before the caps are donned)

The hard thing is that the players cannot communicate anything with each other once the have the caps.

  • 1
    $\begingroup$ "If they win, all of them have to guess correctly" That's really unclear. It sounds like they are playing some other game, and if they happen to win that game, then they have to guess their hat colour. That's probably not what you meant. Also, how are they guessing? All at the same time? One by one in a dedicated order? One by one but they can freely choose which order as part of their strategy? $\endgroup$ – Arthur Mar 1 at 5:56
  • $\begingroup$ The thing is: one is free how they guess (which order etc). This, one should determine in the strategy. There are really no further rules than I described... $\endgroup$ – JohnD Mar 1 at 6:03
  • $\begingroup$ Hint: every time at least two hats will be the same color and at least one person will see the other two people wearing the same color hat. $\endgroup$ – fleablood Mar 1 at 6:48
  • $\begingroup$ Oh, and please remember to accept my answer. Don't just say "thank you". $\endgroup$ – Parcly Taxel Mar 1 at 9:43
  • $\begingroup$ I thank you AND I accept your answer ;) Thanks a lot! :) $\endgroup$ – JohnD Mar 1 at 9:46

Six of the eight possible cases involve one cap being of the opposite colour to the other two. Therefore, the strategy with a 75% chance of success goes like this (numbering the persons 1, 2, 3):

  • Person 1, then person 2, then person 3 in that order will see the other two person's caps. If they see two caps of the same colour, they will say that their own cap is of the opposite colour. Either one or three people will say this; if all three speak out, they all have caps of the same colour and so lose.
  • If exactly one person spoke out, the other two people then say their cap is of the opposite colour to the person who spoke out first, winning the game for the group.
  • $\begingroup$ Thanks a lot for this answer! One last question (out of curiosity): What strategy woud hold if with 100% certainly at least two persons should answer correctly? $\endgroup$ – JohnD Mar 1 at 9:15
  • $\begingroup$ @JohnD Simple. Just have the first person to see two same-coloured caps say that his cap is of the same colour, whereupon the other two will say that their hat is of the same colour. $\endgroup$ – Parcly Taxel Mar 1 at 9:20
  • $\begingroup$ What if we had 7 people wearing either a white or a black cap? Which strategy can be used here such that all of them guess correctly their own cap color? $\endgroup$ – JohnD Mar 7 at 13:59

Hint: Thinking about the different possibilities of hat distribution, there is one thing which happens $75\%$ of the time. Find a strategy they can follow in that case, and ignore the remaining $25\%$ of cases.


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