I want to make your heads a'splode
Jun. 29th, 2005 05:25 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
flexagonWAKE UP, SLEEPY NEURONS, IT'S BRAINTEASER TIME.
This one is fun to think about, yet has one of the stupidest setups I've ever heard: n people are buried in the sand up to their necks, forming a queue, and each of them is wearing either a red or a blue hat. Each of them can see the hats of all the people in front of them (from which you can infer that they're buried on a slope, and a concave slope at that... but I digress) but no person can see his or her own hat or the hats of the people behind them. You don't know anything about the distribution of red or blue hats... they could be all blue, all red, or chosen by an intelligent and malevolent force.
Now, they get to do something. Starting with the person in the back of the queue, each person can say "red" or "blue" once. No tricks here. The guy in back says a word, then the person in front of him, then the person in front of hat person, etc.
Here is the trick: these people get to strategize before docilely allowing themselves to be buried in the sand. If the idea is for the maximum number of people to say the color of the hat on their own head, what is their strategy and how well can they do? They are not allowed to encode more than one bit of information in their word using tone, rhythm or any other method, as was my first idea: "blooooo-oo-oo-uh-OO-ooh-oo". Strictly one bit per utterance. :)
To guarantee getting 50% right, the people could work in pairs. One person simply says the color of the hat right in front of them, and then the next person says that color. That way, you get a minimum of 50%, and if the distribution is random you get 75%. Not bad for an initial solution.
I find this much easier to think about if each person either has a hat or doesn't, as opposed to having a red or a blue hat. The people can then say "yes" or "no". Mathematically, it's still the same puzzle, but now there's something obvious to focus on.
Lastly, we can assume that nobody panicks from being buried in the sand, nobody is too hungry to think straight, nobody ever messes up the strategy in their head or simply blurts out the wrong thing, nobody's glasses fall off so that they can't see the hats in front of them, there are no shills secretly taking payments from the malevolent intelligence in return for lying, etc. Also, there are no equations in this one, so I don't think I've made any dumb typos this time.
Edit: someone got it. So don't read the comments if you don't want the answer.
This one is fun to think about, yet has one of the stupidest setups I've ever heard: n people are buried in the sand up to their necks, forming a queue, and each of them is wearing either a red or a blue hat. Each of them can see the hats of all the people in front of them (from which you can infer that they're buried on a slope, and a concave slope at that... but I digress) but no person can see his or her own hat or the hats of the people behind them. You don't know anything about the distribution of red or blue hats... they could be all blue, all red, or chosen by an intelligent and malevolent force.
Now, they get to do something. Starting with the person in the back of the queue, each person can say "red" or "blue" once. No tricks here. The guy in back says a word, then the person in front of him, then the person in front of hat person, etc.
Here is the trick: these people get to strategize before docilely allowing themselves to be buried in the sand. If the idea is for the maximum number of people to say the color of the hat on their own head, what is their strategy and how well can they do? They are not allowed to encode more than one bit of information in their word using tone, rhythm or any other method, as was my first idea: "blooooo-oo-oo-uh-OO-ooh-oo". Strictly one bit per utterance. :)
To guarantee getting 50% right, the people could work in pairs. One person simply says the color of the hat right in front of them, and then the next person says that color. That way, you get a minimum of 50%, and if the distribution is random you get 75%. Not bad for an initial solution.
I find this much easier to think about if each person either has a hat or doesn't, as opposed to having a red or a blue hat. The people can then say "yes" or "no". Mathematically, it's still the same puzzle, but now there's something obvious to focus on.
Lastly, we can assume that nobody panicks from being buried in the sand, nobody is too hungry to think straight, nobody ever messes up the strategy in their head or simply blurts out the wrong thing, nobody's glasses fall off so that they can't see the hats in front of them, there are no shills secretly taking payments from the malevolent intelligence in return for lying, etc. Also, there are no equations in this one, so I don't think I've made any dumb typos this time.
Edit: someone got it. So don't read the comments if you don't want the answer.
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
no subject
Date: 2005-06-30 05:05 am (UTC)The person in the back sees the hat of the person in front of them, they call out that color. The strategy is that everyone in front of them will call out that color as the color of their own hat, until someone has a new color in front of them, at which point they call out the new color.
Worst case scenario: no one gets any right because every hat alternates colors
Best case scenario: All the people get their hats right because they are the same color.
That's my initial speculation...might lead to something more useful.