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S _ _ Clue
THE KEY IS THAT THE PERSON AT THE BACK OF THE LINE WHO CAN SEE EVERYONE ELSE'S HATS CAN USE THE WORDS "BLACK" OR "WHITE" TO COMMUNICATE SOME CODED INFORMATION. SO WHAT MEANING CAN BE ASSIGNED TO THOSE WORDS THAT WILL ALLOW EVERYONE ELSE TO DEDUCE THEIR HAT COLORS? IT CAN'T BE THE TOTAL NUMBER OF BLACK OR WHITE HATS. THERE ARE MORE THAN TWO POSSIBLE VALUES, BUT WHAT DOES HAVE TWO POSSIBLE VALUES IS THAT NUMBER'S PARITY, THAT IS WHETHER IT'S ODD OR EVEN. SO THE SOLUTION IS TO AGREE THAT WHOEVER GOES FIRST WILL, FOR EXAMPLE, SAY "BLACK" IF HE SEES AN ODD NUMBER OF BLACK HATS AND "WHITE" IF HE SEES AN EVEN NUMBER OF BLACK HATS. LET'S SEE HOW IT WOULD PLAY OUT IF THE HATS WERE DISTRIBUTED LIKE THIS. THE TALLEST CAPTIVE SEES THREE BLACK HATS IN FRONT OF HIM, SO HE SAYS "BLACK," TELLING EVERYONE ELSE HE SEES AN ODD NUMBER OF BLACK HATS. HE GETS HIS OWN HAT COLOR WRONG, BUT THAT'S OKAY SINCE YOU'RE COLLECTIVELY ALLOWED TO HAVE ONE WRONG ANSWER. PRISONER TWO ALSO SEES AN ODD NUMBER OF BLACK HATS, SO SHE KNOWS HERS IS WHITE, AND ANSWERS CORRECTLY. PRISONER THREE SEES AN EVEN NUMBER OF BLACK HATS, SO HE KNOWS THAT HIS MUST BE ONE OF THE BLACK HATS THE FIRST TWO PRISONERS SAW. PRISONER FOUR HEARS THAT AND KNOWS THAT SHE SHOULD BE LOOKING FOR AN EVEN NUMBER OF BLACK HATS SINCE ONE WAS BEHIND HER. BUT SHE ONLY SEES ONE, SO SHE DEDUCES THAT HER HAT IS ALSO BLACK. PRISONERS FIVE THROUGH NINE ARE EACH LOOKING FOR AN ODD NUMBER OF BLACK HATS, WHICH THEY SEE, SO THEY FIGURE OUT THAT THEIR HATS ARE WHITE. NOW IT ALL COMES DOWN TO YOU AT THE FRONT OF THE LINE. IF THE NINTH PRISONER SAW AN ODD NUMBER OF BLACK HATS, THAT CAN ONLY MEAN ONE THING. YOU'LL FIND THAT THIS STRATEGY WORKS FOR ANY POSSIBLE ARRANGEMENT OF THE HATS. THE FIRST PRISONER HAS A 50% CHANCE OF GIVING A WRONG ANSWER ABOUT HIS OWN HAT, BUT THE PARITY INFORMATION HE CONVEYS ALLOWS EVERYONE ELSE TO GUESS THEIRS WITH ABSOLUTE CERTAINTY. EACH BEGINS BY EXPECTING TO SEE AN ODD OR EVEN NUMBER OF HATS OF THE SPECIFIED COLOR. IF WHAT THEY COUNT DOESN'T MATCH, THAT MEANS THEIR OWN HAT IS THAT COLOR. AND EVERY TIME THIS HAPPENS, THE NEXT PERSON IN LINE WILL SWITCH THE PARITY THEY EXPECT TO SEE MIKE COOPER IS THE CULPRIT. WE CAN EXCLUDE MARTIN FROM OUR INVESTIGATIONS AS HE WAS THE VICTIM. TERRY SINGER POSSESSES AN ORANGE CAR. THE THIEF POSSESSES A BLUE CAR, THUS TERRY SINGER CANNOT BE THE CULPRIT. THE SUSPECT WHO OWNS A BLUE CAR WAS WEARING PURPLE SHOES. THE SUSPECT WHO WEIGHS 190 POUNDS WAS WEARING PURPLE SHOES. THE SUSPECT WHO WEIGHS 190 POUNDS IS NOT THE ONE WHO HAS BLACK HAIR. THEREFORE, ONE SUSPECT WAS WEARING PURPLE SHOES, POSSESSES A BLUE CAR, WEIGHS 190 POUNDS, AND IS NOT THE ONE WHO HAS BLACK HAIR. BUT JIM CONDON WAS WEARING BROWN SHOES, SYLVIAN BOGARD HAS BLACK HAIR, JONAH JAMESON WEIGHS 210 POUNDS, AND TERRY SINGER POSSESSES AN ORANGE CAR. THEREFORE, THE ONLY SUSPECT WHO COULD BE THIS SUSPECT IS MIKE COOPER. AS MIKE COOPER, OWNS A BLUE CAR AND THE THIEF OWNS A BLUE CAR, HE IS DEFINITELY THE THIEF 45 MINUTES. METHOD 1 LET US TALK ABOUT 24 HOURS AS IT WILL SIMPLIFY THE PROBLEM. IN 24 HOURS: THE COLD TAP WILL BE ABLE TO FILL THE BATHTUB 80 TIMES. THE HOT TAP WILL BE ABLE TO FILL THE BATHTUB 96 TIMES. THE DRAIN WILL BE ABLE TO DRAIN THE BATHTUB 144 TIMES. THUS IN 24 HOURS, THE BATHTUB WILL BE FILLED 80 + 96 - 144 = 32 TIMES (IF BOTH TAPS ARE TURNED ON WITH THE PLUG LEFT OUT) THUS IT WILL TAKE 45 MINUTES TO FILL THE TUB ONCE. METHOD 2 THE BATHTUB FILLING RATE IS: THE COLD TAP WILL FILL THE BATHTUB AT 1/18 TUB/MINUTE THE HOT TAP WILL FILL THE BATHTUB AT 1/15 TUB/MINUTE THE DRAIN WILL DRAIN THE BATHTUB AT 1/10 TUB/MINUTE THE NETT BATHTUB FILL RATE IS: 1/18 + 1/15 - 1/10 = 1/45 TUB/MINUTE THE WATER WILL FILL 1/45 OF THE TUB IN 1 MINUTE, SO THE TUB WILL BE FULL IN 45 MINUTES LARRY IS THE PRESIDENT. SUPPOSE MATT'S FIRST STATEMENT IS TRUE. THIS WOULD MEAN THAT HE IS NOT A TYPE B PERSON (WHO ALWAYS LIE), AND LARRY'S SECOND STATEMENT IS TRUE. THIS WOULD MEAN THAT LARRY IS NOT A TYPE B PERSON, AND MATT'S SECOND STATEMENT IS TRUE. SINCE NEITHER LARRY NOR MATT IS THE PRESIDENT, NICK MUST BE PRESIDENT, MAKING BOTH HIS STATEMENTS FALSE. SINCE NICK'S FIRST STATEMENT IS FALSE, AND SINCE HE'S A TYPE B PERSON WHILE MATT IS A TYPE A PERSON, LARRY MUST BE A TYPE C PERSON, MAKING HIS FIRST STATEMENT FALSE. BUT SINCE THE PRESIDENT BELONGS TO A DIFFERENT RACE FROM EACH OF THE OTHER TWO, LARRY'S FIRST STATEMENT MUST BE TRUE. THIS MEANS THAT MATT'S FIRST STATEMENT IS FALSE. SUPPOSE MATT IS PRESIDENT. THIS WOULD MEAN THAT LARRY'S SECOND STATEMENT IS FALSE AND MATT'S AND NICK'S SECOND STATEMENTS ARE TRUE. THEN, SINCE MATT IS BOTH THE PRESIDENT AND A TYPE C PERSON, AND SINCE LARRY MUST ALSO BE A TYPE C PERSON IF HIS FIRST STATEMENT IS TRUE (WHICH WOULD MAKE HIS FIRST STATEMENT FALSE), LARRY'S FIRST STATEMENT MUST BE FALSE, MAKING HIM A TYPE B PERSON. THEN IF NICK'S FIRST STATEMENT IS TRUE, HE IS A TYPE A PERSON, WHICH MEANS THAT ALL THREE ARE DIFFERENT RACES AND MAKES NICK'S FIRST STATEMENT FALSE, BUT IF HIS FIRST STATEMENT IS FALSE, HE IS A TYPE C PERSON, WHICH MEANS THAT HE AND MATT BOTH BELONG TO A RACE THAT LARRY DOES NOT AND MAKES NICK'S FIRST STATEMENT TRUE. SO MATT CANNOT BE PRESIDENT. SUPPOSE NICK IS PRESIDENT. THIS WOULD MEAN THAT LARRY'S AND MATT'S SECOND STATEMENTS ARE BOTH TRUE, BUT NICK'S IS FALSE. THEN, SINCE MATT'S FIRST STATEMENT AND NICK'S SECOND STATEMENT ARE BOTH FALSE, NICK'S FIRST STATEMENT MUST BE TRUE. THEN, SINCE MATT AND NICK ARE BOTH TYPE C PEOPLE, LARRY'S FIRST STATEMENT MUST BE FALSE AND ALL THREE MUST BE TYPE C PEOPLE, MAKING NICK'S FIRST STATEMENT FALSE. SO NICK CANNOT BE THE PRESIDENT EITHER, LEAVING ONLY LARRY