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S H _ T Clue
PETE TOLD THEM TO FORM A CIRCLE AROUND HIM. ALL THE SQUAD WAS FACING IN AT PETE, READY TO SHOOT, WHEN THEY REALIZED THAT EVERYONE WHO MISSED WOULD LIKELY END UP SHOOTING ANOTHER SQUAD MEMBER. SO NO ONE DARED TO FIRE, KNOWING THE RISK. THUS AT SUNDOWN HE WAS RELEASED THE OPTION TO PULL THE TRIGGER AGAIN WILL GIVE A HIGHER PROBABILITY OF NOT BEING SHOT. THE KEY HINT HERE IS THAT THE BULLETS WERE LOADED ADJACENT TO EACH OTHER. THERE ARE 4 WAYS TO ARRANGE THE REVOLVER WITH CONSECUTIVE BULLETS SO THAT THE FIRST SHOT IS BLANK. THESE ARE THE POSSIBLE SCENARIOS: (XBBXXX) (XXBBXX) (XXXBBX) (XXXXBB) THE OTHER TWO SCENARIOS WOULD HAVE MEANT YOU GOT SHOT ON THE FIRST ATTEMPT. (BBXXXX) OR (BXXXXB). NOW LOOK AT THE SECOND SLOT IN THE 4 POSSIBLE SCENARIOS ABOVE. THE ODDS OF GETTING SHOT ARE 1⁄4 OR 25% (ONLY THE FIRST SCENARIO WOULD GET YOU SHOT). BUT IF YOU RESPIN, THERE ARE 2 BULLETS REMAINING AND A TOTAL OF 6 SLOTS, WHICH GIVES A PROBABILITY OF 2⁄6 OR 33% OF BEING SHOT YOU SHOULD NOT SPIN THE BARREL. THERE ARE 2 CONSECUTIVE BULLETS AND 4 CONSECUTIVE EMPTY SLOTS, SO WE LET IT BE EEEEBB. THE PROBABILITY THAT YOU GET A BULLET AFTER SPINNING THE BARREL IS 2/6 = 1/3. YOU GAINED SOME KNOWLEDGE AFTER THE FIRST SHOT. SINCE YOU DIDN'T HIT YOURSELF YOU KNOW YOU GOT ONE OF THE 4 EMPTY BULLET SLOTS. THE SEQUENCE IS EEEEBB, SO IF YOU GOT THE LAST E FOR THE 1ST SHOT, YOU WILL HIT BY A BULLET IN THE 2ND SHOT. IF YOU GOT ANY OF THE 1ST THREE ES, YOU WILL NOT GET A BULLET IN THE 2ND SHOT. THE CHANCES OF GETTING THE 1ST THREE ES IS 3/4. IT'S SAFER TO NOT SPIN THE BARREL AGAIN AS THE PROBABILITY OF MISSING THE BULLET IS HIGHER 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. SO THAT'S IT, YOU'RE FREE TO GO. IT LOOKS LIKE THESE ALIENS WILL HAVE TO GO HUNGRY, OR FIND SOME LESS LOGICAL ORGANISMS TO ABDUCT