From a known genre of puzzles - "what's my hat?" - appropriate for this group, as it involves information, if not theory - Three men, blindfolded, each wear a hat. H or T is written on each, according to coin tosses. The blindfolds are removed. Each sees the others' labels, but not his own. All must simultaneously, and immediately, guess his own label, or pass. No communication allowed. And no delay, watching the others' behavior. They share a prize if at least one guesses correctly, with no incorrect guesses. They huddle, pre-game. What strategy maximizes their winning chance? -- Rich

# daily quiz

Started by ●March 28, 2018

Posted by ●March 28, 2018

RichD <r_delaney2001@yahoo.com> writes:> From a known genre of puzzles - "what's my hat?" - > appropriate for this group, as it involves information, if not theory - > > > Three men, blindfolded, each wear a hat. H or T is written > on each, according to coin tosses. The blindfolds are removed. > Each sees the others' labels, but not his own. All must > simultaneously, and immediately, guess his own label, or pass. > No communication allowed. And no delay, watching the others' > behavior. > > They share a prize if at least one guesses correctly, > with no incorrect guesses. > > They huddle, pre-game. What strategy maximizes their > winning chance?An interesting problem. I solved it without resorting to a web search. Of course, each player can do no better than 50/50, but the trick is to arrange that all the players fail at the same time. Player 1's strategy: - H H: guess H - H T: pass - T H: pass - T T: guess T Player 2's strategy: H - H: pass T - H: guess T H - T: guess H T - T: pass Player 3's strategy: H H -: pass H T -: guess H T H -: guess T T T -: pass Here are the 8 possible outcomes: H H H: player 1 guesses H, others pass, the group succeeds H H T: player 2 guesses H, others pass, the group succeeds H T H: player 3 guesses H, others pass, the group succeeds H T T: players guess T H H, all failing T H H: players guess H T T, all failing T H T: player 3 guesses T, others pass, the group succeeds T T H: player 2 guesses T, others pass, the group succeeds T T T: player 1 guesses T, others pass, the group succeeds So the group is successful 75% of the time. There are other permutations of this solution. Scott -- Scott Hemphill hemphill@alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear

Posted by ●March 28, 2018

On Wed, 28 Mar 2018 17:55:32 -0400, Scott Hemphill <hemphill@hemphills.net> wrote:>RichD <r_delaney2001@yahoo.com> writes: > >> From a known genre of puzzles - "what's my hat?" - >> appropriate for this group, as it involves information, if not theory - >> >> >> Three men, blindfolded, each wear a hat. H or T is written >> on each, according to coin tosses. The blindfolds are removed. >> Each sees the others' labels, but not his own. All must >> simultaneously, and immediately, guess his own label, or pass. >> No communication allowed. And no delay, watching the others' >> behavior. >> >> They share a prize if at least one guesses correctly, >> with no incorrect guesses. >> >> They huddle, pre-game. What strategy maximizes their >> winning chance? > >An interesting problem. I solved it without resorting to a web search. >Of course, each player can do no better than 50/50, but the trick is to >arrange that all the players fail at the same time. > >Player 1's strategy: > >- H H: guess H >- H T: pass >- T H: pass >- T T: guess T > >Player 2's strategy: > >H - H: pass >T - H: guess T >H - T: guess H >T - T: pass > >Player 3's strategy: > >H H -: pass >H T -: guess H >T H -: guess T >T T -: pass > >Here are the 8 possible outcomes: > >H H H: player 1 guesses H, others pass, the group succeeds >H H T: player 2 guesses H, others pass, the group succeeds >H T H: player 3 guesses H, others pass, the group succeeds >H T T: players guess T H H, all failing >T H H: players guess H T T, all failing >T H T: player 3 guesses T, others pass, the group succeeds >T T H: player 2 guesses T, others pass, the group succeeds >T T T: player 1 guesses T, others pass, the group succeeds > >So the group is successful 75% of the time. There are other >permutations of this solution. > >ScottI'd have everyone agree to guess the same, H or T. That only fails when all three hats display the opposite choice, which would occur in one of the 8 possibilities. So the win rate would be 87.5%.

Posted by ●March 28, 2018

On Wed, 28 Mar 2018 16:39:08 -0700, Mike Paff <paffm@yahoo.com> wrote:>On Wed, 28 Mar 2018 17:55:32 -0400, Scott Hemphill ><hemphill@hemphills.net> wrote: > >>RichD <r_delaney2001@yahoo.com> writes: >> >>> From a known genre of puzzles - "what's my hat?" - >>> appropriate for this group, as it involves information, if not theory - >>> >>> >>> Three men, blindfolded, each wear a hat. H or T is written >>> on each, according to coin tosses. The blindfolds are removed. >>> Each sees the others' labels, but not his own. All must >>> simultaneously, and immediately, guess his own label, or pass. >>> No communication allowed. And no delay, watching the others' >>> behavior. >>> >>> They share a prize if at least one guesses correctly, >>> with no incorrect guesses. >>> >>> They huddle, pre-game. What strategy maximizes their >>> winning chance? >> >>An interesting problem. I solved it without resorting to a web search. >>Of course, each player can do no better than 50/50, but the trick is to >>arrange that all the players fail at the same time. >> >>Player 1's strategy: >> >>- H H: guess H >>- H T: pass >>- T H: pass >>- T T: guess T >> >>Player 2's strategy: >> >>H - H: pass >>T - H: guess T >>H - T: guess H >>T - T: pass >> >>Player 3's strategy: >> >>H H -: pass >>H T -: guess H >>T H -: guess T >>T T -: pass >> >>Here are the 8 possible outcomes: >> >>H H H: player 1 guesses H, others pass, the group succeeds >>H H T: player 2 guesses H, others pass, the group succeeds >>H T H: player 3 guesses H, others pass, the group succeeds >>H T T: players guess T H H, all failing >>T H H: players guess H T T, all failing >>T H T: player 3 guesses T, others pass, the group succeeds >>T T H: player 2 guesses T, others pass, the group succeeds >>T T T: player 1 guesses T, others pass, the group succeeds >> >>So the group is successful 75% of the time. There are other >>permutations of this solution. >> >>Scott > >I'd have everyone agree to guess the same, H or T. That only fails >when all three hats display the opposite choice, which would occur >in one of the 8 possibilities. So the win rate would be 87.5%.Never mind, that's incorrect. I misread the rules.

Posted by ●March 29, 2018

Scott Hemphill <hemphill@hemphills.net> writes:> RichD <r_delaney2001@yahoo.com> writes: > >> From a known genre of puzzles - "what's my hat?" - >> appropriate for this group, as it involves information, if not theory - >> >> >> Three men, blindfolded, each wear a hat. H or T is written >> on each, according to coin tosses. The blindfolds are removed. >> Each sees the others' labels, but not his own. All must >> simultaneously, and immediately, guess his own label, or pass. >> No communication allowed. And no delay, watching the others' >> behavior. >> >> They share a prize if at least one guesses correctly, >> with no incorrect guesses. >> >> They huddle, pre-game. What strategy maximizes their >> winning chance? > > An interesting problem. I solved it without resorting to a web search. > Of course, each player can do no better than 50/50, but the trick is to > arrange that all the players fail at the same time. > > Player 1's strategy: > > - H H: guess H > - H T: pass > - T H: pass > - T T: guess T > > Player 2's strategy: > > H - H: pass > T - H: guess T > H - T: guess H > T - T: pass > > Player 3's strategy: > > H H -: pass > H T -: guess H > T H -: guess T > T T -: pass > > Here are the 8 possible outcomes: > > H H H: player 1 guesses H, others pass, the group succeeds > H H T: player 2 guesses H, others pass, the group succeeds > H T H: player 3 guesses H, others pass, the group succeeds > H T T: players guess T H H, all failing > T H H: players guess H T T, all failing > T H T: player 3 guesses T, others pass, the group succeeds > T T H: player 2 guesses T, others pass, the group succeeds > T T T: player 1 guesses T, others pass, the group succeeds > > So the group is successful 75% of the time. There are other > permutations of this solution.I wrote a computer program to enumerate the solutions. My solution has three permutations, which are obtained when when you rotate the roles of player1 -> player2 -> player3 -> player1. There is also one other solution, where all the players use the same strategy: If the other two players both have H, guess T. If the other two players both have T, guess H. Otherwise, pass. The 8 possible outcomes are: H H H: players all guess T, failing H H T: player 3 guesses T, others pass H T H: player 2 guesses T, others pass H T T: player 1 guesses H, others pass T H H: player 1 guesses T, others pass T H T: player 2 guesses H, others pass T T H: player 3 guesses H, others pass T T T: players all guess H, failing Scott -- Scott Hemphill hemphill@alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear

Posted by ●March 29, 2018

MCP <paffm@yahoo.com> writes:> On Wed, 28 Mar 2018 16:39:08 -0700, Mike Paff <paffm@yahoo.com> wrote: > >>On Wed, 28 Mar 2018 17:55:32 -0400, Scott Hemphill >><hemphill@hemphills.net> wrote: >> >>>RichD <r_delaney2001@yahoo.com> writes: >>> >>>> From a known genre of puzzles - "what's my hat?" - >>>> appropriate for this group, as it involves information, if not theory - >>>> >>>> >>>> Three men, blindfolded, each wear a hat. H or T is written >>>> on each, according to coin tosses. The blindfolds are removed. >>>> Each sees the others' labels, but not his own. All must >>>> simultaneously, and immediately, guess his own label, or pass. >>>> No communication allowed. And no delay, watching the others' >>>> behavior. >>>> >>>> They share a prize if at least one guesses correctly, >>>> with no incorrect guesses. >>>> >>>> They huddle, pre-game. What strategy maximizes their >>>> winning chance? >>> >>>An interesting problem. I solved it without resorting to a web search. >>>Of course, each player can do no better than 50/50, but the trick is to >>>arrange that all the players fail at the same time. >>> >>>Player 1's strategy: >>> >>>- H H: guess H >>>- H T: pass >>>- T H: pass >>>- T T: guess T >>> >>>Player 2's strategy: >>> >>>H - H: pass >>>T - H: guess T >>>H - T: guess H >>>T - T: pass >>> >>>Player 3's strategy: >>> >>>H H -: pass >>>H T -: guess H >>>T H -: guess T >>>T T -: pass >>> >>>Here are the 8 possible outcomes: >>> >>>H H H: player 1 guesses H, others pass, the group succeeds >>>H H T: player 2 guesses H, others pass, the group succeeds >>>H T H: player 3 guesses H, others pass, the group succeeds >>>H T T: players guess T H H, all failing >>>T H H: players guess H T T, all failing >>>T H T: player 3 guesses T, others pass, the group succeeds >>>T T H: player 2 guesses T, others pass, the group succeeds >>>T T T: player 1 guesses T, others pass, the group succeeds >>> >>>So the group is successful 75% of the time. There are other >>>permutations of this solution. >>> >>>Scott >> >>I'd have everyone agree to guess the same, H or T. That only fails >>when all three hats display the opposite choice, which would occur >>in one of the 8 possibilities. So the win rate would be 87.5%. > > Never mind, that's incorrect. I misread the rules.There's another, more interesting modification to the rules. Suppose that the players make their guesses in turn, so that each player can use the preceding player's guesses to inform their own. Then you can achieve 87.5%. Scott -- Scott Hemphill hemphill@alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear