Beach Day by Rodger Bliss
May 2024

Three Cowboys in Trouble

cowboysThree Cowboys were taken captive by a hostile Indian tribe. According to tribe’s custom they had to pass an intelligence test, or die.
The chieftain showed 5 headbands – 2 red and 3 white. The 3 cowboys were blindfolded and positioned one after another, face to back. The chief put a headband on each of their heads, hid two remaining headbands, and removed their blindfolds. So the third man could see the headbands on the two men in front of him, the second man could see the headband on the first, and the first could not see any headbands at all.

According to the rules any one of the three cowboys could speak first and try to guess his headband color. And if he guessed correctly – they passed the test and could go free, if not – they failed. It so happened that all 3 cowboys were prominent logicians from a nearby academy. So after a few moments of silence, the first man in the line said: “My headband is …”.
What color was his head band? Why?

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3 guesses to Three Cowboys in Trouble

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  • brent

    It was white.

    If 1 & 2 were both red, #3 would have immediately known he was white and said so (the other two would know they were white at that time, and all would be saved)
    If 1 was red and 2 was white, #3 would be silent (he didn’t know what he was) but #2 seeing #1 was red would know he (#2) was white. (#1 would then know he was white also) Poor #3.
    Since neither #2 or #3 knew what they were, #1 must have been white.

    Given #1 is white, and says it. #3 should now guess at his colour. If #2 is also white, #3 has a 2/3 chance of being red, and should say “I’m most likely red”. He is 67% likely to be correct and also #2 will be able to infer his own colour (white).
    If #2 is red, then #3 knows he has a 2/3 chance of being white and should declare that. Again #3 has a 67% chance of saving his own life, and will give #2 the information he needs to save his own.

  • brent

    Ah, this part of my first sentence: “(the other two would know they were white at that time, and all would be saved)” should have said red instead of white. Other than that, I think I said everything the way I meant to…!?!

  • Brent you are today’s winner!

    The first one (he did not see any head bands) thought this way: The last one is silent, which means, he does not know, ergo at least one of head bands he sees is white. The one in the middle is silent too even though he knows what I already mentioned. If I had a red head band, the second one would have known that he had a white head band. However, nobody says anything, so my head band is not red – my head band is white.