tag:blogger.com,1999:blog-6988729.post8590522216632133841..comments2024-03-27T11:42:47.601-04:00Comments on Another Monkey: Outlaws, hats, and the new interview paradigmD.B. Echohttp://www.blogger.com/profile/01797128570217627410noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-6988729.post-57284817487675619862011-05-18T16:52:58.377-04:002011-05-18T16:52:58.377-04:00A more elegant statement of the solution:
If 2 an...A more elegant statement of the solution:<br /><br />If 2 and 3 are both wearing white, 1 would know he was wearing black. He doesn't guess correctly. He either guessed white and was wrong, or black and was wrong.<br /><br />Now 2 knows the same thing: if 1 and 3 are both white, then he is black. But he doesn't guess that, so that's not the case. <br /><br />2 also knows that either all three of them are wearing black (= 2 and 3 are wearing black) or he and 3 are wearing different colors (= 2 and 3 not both wearing white at the same time.)<br /><br />So if 3 is wearing white, 2 can guess black and go free. HE DOESN'T. So 3 is NOT wearing white ( = 3 is wearing black.)<br /><br />The solution only requires that 3 know what color his hat is, and he knows that it isn't white.<br /><br />NOTE: I don't think 3 needs to know what colors 1 and 2's hats were to arrive at this conclusion - he just has to know that they both guessed wrong.D.B. Echohttps://www.blogger.com/profile/01797128570217627410noreply@blogger.comtag:blogger.com,1999:blog-6988729.post-19291868654112833442011-05-18T16:22:43.125-04:002011-05-18T16:22:43.125-04:00OK, I've worked out the solution, and my error...OK, I've worked out the solution, and my error. Parts that have been revised from my comments on Bill's blog are in all-caps.<br /><br />1 could have given two answers: white or black. If he saw two white, he would know that his hat was black. If he said "black" and was wrong, then his hat was white, and the other two were A (2w 3b), B (2b 3w), or C (2b 3b). If he said "white" and was wrong, then his hat was black, and the other two could be A (2w 3b), B (2b 3w), or C (2b 3b). (The same three possibilities for 2 and 3 either way.) <br /><br />2 now knows that he and 3 are either both wearing black hats, or are wearing hats of different colors. If 1 and 3 have white hats, then he would know that his hat was black. IF ONLY 3 HAS A WHITE HAT, 2 KNOWS HIS HAT MUST BE BLACK, OR ELSE 1 WOULD HAVE GUESSED CORRECTLY. SO IN EITHER OF THESE CASES HE CAN GUESS "BLACK" AND BE CORRECT. But we know he gets this wrong, so NEITHER OF THESE CAN be the case. So the possibilities are D (1w 3b), or F(1b 3b). (POSSIBILITY E - 1B 3W - IS ELIMINATED BY THE "BOTH 2 AND 3 ARE BLACK OR DIFFERENT COLORS" RULE.) SO 2 EITHER GUESSED WHITE AND WAS WRONG, BECAUSE ALL THREE ARE WEARING BLACK, OR GUESSED BLACK AND WAS WRONG.<br /><br />3 is blind. He can't see anything. At this point he knows what the color of 1's hat is (the opposite of what he guessed). HE ALSO KNOWS THAT IF HE HAD A WHITE HAT, 2 WOULD HAVE CORRECTLY GUESSED THAT 2 HAD A BLACK HAT. IF 2 GUESSED THAT 2 HAD A WHITE HAT AND WAS WRONG, THAT MEANS 3 HAS A BLACK HAT - OTHERWISE 1 WOULD HAVE SEEN THAT THEY BOTH HAD WHITE HATS. He knows that either he and 2 are both wearing black, or are wearing different colors. He also knows that both he and 1 are wearing black or are wearing different colors. So if 1 and 2 are white, he is black. If 1 is black and 2 is white, he is black. If 1 is white and 2 is black, he is black. <br /><br />IF 2 AND 3 WERE WHITE, 1 COULD HAVE CORRECTLY GUESSED THAT HIS HAT WAS BLACK. SO 2 AND 3 ARE BOTH BLACK OR ARE DIFFERENT COLORS. IF 3 WAS WHITE, 2 COULD HAVE GUESSED THE COLOR OF HIS OWN HAT WAS BLACK - EITHER 1 AND 3 WOULD BOTH BE WHITE SO 2 WOULD HAVE TO BE BLACK, OR 1 WOULD BE BLACK AND 3 WOULD BE WHITE, AND BY THE "2 AND 3 DIFFERENT COLORS" RULE 2 WOULD BE BLACK. (OR 1, 2, AND 3 COULD ALL BE BLACK, AND 2 INCORRECTLY CHOSE WHITE.) SINCE NONE OF THESE HAPPENED, 3 KNOWS HIS HAT IS NOT WHITE.D.B. Echohttps://www.blogger.com/profile/01797128570217627410noreply@blogger.com