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The king was looking to hire a new scribe. Three bright young lads applied for the job. The king, who was widely known for being very just, made them undergo a test. He took three red and three black hats, and told them that the one who will guess the color if his own hat, will win the honorable title of being the king's scribe. The applicants were blindfolded, a red hat put on the head of each of them, and the three black hats were hidden away.

After the blindfolds were removed, the three applicants stood there for a while in silence, thinking. Each of them could see the hats of the other two, but none of them could see their own hat.

After some time, one of them said that he has a red hat on his head, correctly guessing the solution to the test.

What was the logic behind it?

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  • $\begingroup$ I believe the riddle is wrong. It can't be argued correctly and "honesty" is not the key. The riddle goes like this. There are only 2 black hats, and 3 red hats. The rest is the same. -- Now, the one who is smartest can figure out fastest that he wears a red hat. How? -- Suppose he wears a black hat, then the others would see a black and red hat and wondered what hat they had. They would reason that if they had a black hat, the other one would know it immediately, seeing two black hats. Since nothing happens, none of the other two sees a black hat, so he wears a red hat! $\endgroup$ – Cuc Apr 12 '16 at 6:21
  • $\begingroup$ @Cuc : I know the riddle you mentioned, and it is a completely different riddle. Actually, the riddle you mentioned is the basic version of the hat guessing riddle. I thought it was common knowledge, that's why I didn't post it. $\endgroup$ – vsz Apr 12 '16 at 6:24
  • $\begingroup$ Alright, I guess you have me puzzled. I thought we are missing information, and I noticed the difference. I realize now how the fairness comes in from the King being "just". My bad. $\endgroup$ – Cuc Apr 13 '16 at 2:51
  • $\begingroup$ Why did the "@vsz" (at vsz) disappear in my previous comment? O, I get it, perhaps because the riddle is from vsz . . . Shoots, I am still a newby, I guess. $\endgroup$ – Cuc Apr 13 '16 at 2:52
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The king has to be nondiscriminatory for each person applied to this job so putting one black on one of them and two red on the others would make the game unfair! So The only way to make this game fair is to put the same color on all of them.

The king, who was widely known for being very just

so one of the player might have suspected this possibility and said red!

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  • $\begingroup$ To complete the answer: the unfairness would come from the fact that if you see 1 red and 1 black hat, you have a 50% of guessing correctly, but with seeing two of the same color, the chance of a random guess would be 75%. Thus, the only fair setup is when all the colors are the same. $\endgroup$ – vsz Apr 10 '16 at 17:10
  • $\begingroup$ I fail to see how seeing two hats of the same color improve the chance of a random guess. You don’t know how the hats are picked, you have no reason to suppose the hat you wear was chosen randomly among the remaining hats. It even so occurs that it was not! $\endgroup$ – Édouard Apr 10 '16 at 17:34
  • $\begingroup$ @Édouard I agree with you. If I see two black hats, I could have a black or a red. If I see one of each, I could have a black or a red. If I see two red hats, I could have a black or a red. In all cases, no one else would know any more information than I, so all cases are fair. $\endgroup$ – Trenin Apr 12 '16 at 16:52

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