Could there still be new hat puzzles?
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Could there still be new hat puzzles?
Here's a hat puzzle inspired by this one.
Seven perfectly rational players stand in a circle.
Everyone gets assigned, randomly and independently, a black hat or a white hat. Nobody can see the color of his own hat but the colors of all other players.
All the players must try to guess the color of their hat simultaneously, without any communication. Let's say they somehow aren't even able to recognize eachother, or they haven't met in person before. They only see hat colors and the relative position of their wearers in the circle.
To win the game, they need a majority (at least 4) of correct guesses.
They can devise a strategy, but it can't be nominative. The player's can't take into account their own identity, the whole strategy has to be exactly the same for all players.
What is the probability that they win the game?
Seven perfectly rational players stand in a circle.
Everyone gets assigned, randomly and independently, a black hat or a white hat. Nobody can see the color of his own hat but the colors of all other players.
All the players must try to guess the color of their hat simultaneously, without any communication. Let's say they somehow aren't even able to recognize eachother, or they haven't met in person before. They only see hat colors and the relative position of their wearers in the circle.
To win the game, they need a majority (at least 4) of correct guesses.
They can devise a strategy, but it can't be nominative. The player's can't take into account their own identity, the whole strategy has to be exactly the same for all players.
What is the probability that they win the game?
Re: Could there still be new hat puzzles?
It is a good start, but you are right, it is possible to get less than half this error rate.HonoreDB wrote:This fails for one of the two most common splits, and doesn't use relative positions, so it's probably not optimal.
Re: Could there still be new hat puzzles?
I'm just starting out the cheap way: generating different strategies and running stochastic simulations.
More news later.
Well, a surprising (to me) improvement gets me up to about 76.4% . I have each player, when 3&3 are seen, count the number of value changes, e.g., WWBWBB has 3 changes in polarity, and WBBBWW has two .
Spoiler:
More news later.
Well, a surprising (to me) improvement gets me up to about 76.4% . I have each player, when 3&3 are seen, count the number of value changes, e.g., WWBWBB has 3 changes in polarity, and WBBBWW has two .
Spoiler:
https://app.box.com/witthoftresume
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Former OTTer
Vote cellocgw for President 2020. #ScienceintheWhiteHouse http://cellocgw.wordpress.com
"The Planck length is 3.81779e33 picas."  keithl
" Earth weighs almost exactly π milliJupiters"  whatif #146, note 7
Re: Could there still be new hat puzzles?
I'm just starting out the cheap way: generating different strategies and running stochastic simulations.
No need to be stochastic (with deterministic strategies, anyway). There are only 128 different ways to place the hats, faster and more accurate to just test each of them once.
Re: Could there still be new hat puzzles?
HonoreDB wrote:I'm just starting out the cheap way: generating different strategies and running stochastic simulations.
No need to be stochastic (with deterministic strategies, anyway). There are only 128 different ways to place the hats, faster and more accurate to just test each of them once.
Yeah, but where's the fun in that? .
FWIW, the execution time for 10 sets of 10000 runs was under a second.
https://app.box.com/witthoftresume
Former OTTer
Vote cellocgw for President 2020. #ScienceintheWhiteHouse http://cellocgw.wordpress.com
"The Planck length is 3.81779e33 picas."  keithl
" Earth weighs almost exactly π milliJupiters"  whatif #146, note 7
Former OTTer
Vote cellocgw for President 2020. #ScienceintheWhiteHouse http://cellocgw.wordpress.com
"The Planck length is 3.81779e33 picas."  keithl
" Earth weighs almost exactly π milliJupiters"  whatif #146, note 7
Re: Could there still be new hat puzzles?
As I said in the OP, "They can devise a strategy, but it can't be nominative. The player's can't take into account their own identity, the whole strategy has to be exactly the same for all players." In other words, there are no prime numbered players. Everybody must follow the same strategy.cellocgw wrote:An absurd strategy where the primenumbered players (2,3,5,7) pick the minority color and the other players pick the majority color
Sorry if my explanations were not clear enough.

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Re: Could there still be new hat puzzles?
Proof of a theoretical maximum, assuming a deterministic strategy: EDIT: Actually, I don't even think we need to assume a deterministic strategy. I'll edit my argument in.
An actual maximum deterministic strategy:
I'm sure there's a more aesthetically pleasing arrangement, but it works, anyway.
Spoiler:
An actual maximum deterministic strategy:
Spoiler:
I'm sure there's a more aesthetically pleasing arrangement, but it works, anyway.
Re: Could there still be new hat puzzles?
Poker wrote:Proof of a theoretical maximum, assuming a deterministic strategy: EDIT: Actually, I don't even think we need to assume a deterministic strategy. I'll edit my argument in.Spoiler:
An actual maximum deterministic strategy:Spoiler:
I'm sure there's a more aesthetically pleasing arrangement, but it works, anyway.
Congrats!
It certainly is more aesthetically pleasing than my own solution, as it is symetric and formulated as a short set of rules.
I just have a table of 64 rules and the situation that triggers a failure with 3 white hats is not the same as with 3 black hats.

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Re: Could there still be new hat puzzles?
Well, I was right  there is a much more elegant maximal strategy. You don't even need to divide it up into cases to use the strategy (though checking it will divide into cases)!
Spoiler:
Re: Could there still be new hat puzzles?
Poker wrote:Well, I was right  there is a much more elegant maximal strategy. You don't even need to divide it up into cases to use the strategy (though checking it will divide into cases)!Spoiler:
That is awesome. May I ask how you came up with this method? Did you extract the pattern from your first answer and tried to figure out how it worked, or did you first came up with the rules?

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Re: Could there still be new hat puzzles?
Yat wrote:Poker wrote:Well, I was right  there is a much more elegant maximal strategy. You don't even need to divide it up into cases to use the strategy (though checking it will divide into cases)!Spoiler:
That is awesome. May I ask how you came up with this method? Did you extract the pattern from your first answer and tried to figure out how it worked, or did you first came up with the rules?
I started by producing every symmetric method (ones where black and white were interchangeable) that would produce the desired result, then I refined my search to look only at those that had nice leftright symmetry (where if the hats were 51 or 33 then reversing the order of the hats would always change the guess, and if the hats were 42 then reversing the order of the hats would never change the guess). At that point I noticed all of the solutions used the 42 case from my original solution, and I realized it might be possible to reduce all four cases to one if I could always use the division of hats from that case. So I checked, and I could.
Re: Could there still be new hat puzzles?
A deliberately obfuscated strategy:
Spoiler:
I NEVER use allcaps.
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