Comments on: Gnomes puzzle http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/ Thoughts, opinions and fascinating discoveries by Elliot, a student at USC Sun, 06 Jul 2008 17:47:52 +0000 http://wordpress.org/?v=2.5.1 By: ucle http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/#comment-5390 ucle Fri, 08 Apr 2005 16:50:17 +0000 http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/#comment-5390 <strong>bipakuoeu</strong> wpsdjeqlnrr bipakuoeu

wpsdjeqlnrr

]]> By: Elliot Lee http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/#comment-187 Elliot Lee Sun, 10 Oct 2004 01:47:52 +0000 http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/#comment-187 Here's the actual answer. Since each gnome can see all the hats in front of him and hear all the answers in back of him, here's what he does. If the number of black hats in front of him plus the number of times a gnome in back of him said "black" is even, then he says "white". If the number is odd, he says "black." In other words, the rearmost gnome reports the parity of the hats in front of him, and each subsequent gnome compares the parity report from behind him to the parity he sees in front of him - a sneaky way to both use the information and pass it forward. To prove that 9 gnomes are safe, consider just the front 9 hats. The rearmost gnome reports the parity of the number of black hats. The next gnome compares the parity of the 8 hat sequence in front of him with the parity the last gnome reported. If the parities are the same, he says "White", because he knows that that's the only way the parities could be the same and if the parities are different he says "Black" - in both cases, he both saves his life, and provides the next gnome forward with an update of the parity to use for the 7th case, and so on. Here’s the actual answer.

Since each gnome can see all the hats in front of him and hear all the
answers in back of him, here’s what he does. If the number of black
hats in front of him plus the number of times a gnome in back of him
said “black” is even, then he says “white”. If the number is odd, he says “black.”
In other words, the rearmost gnome reports the parity of the hats in
front of him, and each subsequent gnome compares the parity report
from behind him to the parity he sees in front of him - a sneaky way
to both use the information and pass it forward.

To prove that 9 gnomes are safe, consider just the front 9 hats.
The rearmost gnome reports the parity of the number of black hats.
The next gnome compares the parity of the 8 hat sequence in front of
him with the parity the last gnome reported. If the parities are the
same, he says “White”, because he knows that that’s the only way the
parities could be the same and if the parities are different he says
“Black” - in both cases, he both saves his life, and provides the next
gnome forward with an update of the parity to use for the 7th case,
and so on.

]]> By: Choyak Yakatak http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/#comment-186 Choyak Yakatak Sat, 09 Oct 2004 23:24:05 +0000 http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/#comment-186 The answer seems to be invalid in that it assumes there are 5 white and 5 black hats. In the puzzle it is stated that the amount of black/white could not be equal, so there could be 9 black and 1 white etc. The answer seems to be invalid in that it assumes there are 5 white and 5 black hats. In the puzzle it is stated that the amount of black/white could not be equal, so there could be 9 black and 1 white etc.

]]> By: Elliot Lee http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/#comment-146 Elliot Lee Sun, 26 Sep 2004 01:06:54 +0000 http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/#comment-146 That's what I thought of too. Unfortunately, I cheated and read the answer on how to guarantee the lives of 9 gnomes (can you figure it out?)... but it's still a great puzzle. That’s what I thought of too. Unfortunately, I cheated and read the answer on how to guarantee the lives of 9 gnomes (can you figure it out?)… but it’s still a great puzzle.

]]> By: Nancy Tang http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/#comment-145 Nancy Tang Sat, 25 Sep 2004 10:09:05 +0000 http://www.intelliot.com/blog/archives/2004/09/08/gnomes-puzzle/#comment-145 Hi Elliot. I couldn't post on your guestbook, so I was checking out your blog. I like puzzles like this. I think the solution is for the every other gnome to agree to answer with the color of the gnome before's hat color. That means the 10th gnome would respond with the color of the hat worn by the 9th gnome. That gives the 10th gnome a 50% chance of survival, but the 9th gnome, when it's his turn has a 100% accuracy. Gnome 8 answers with the color of the hat in front of him, giving him a 50% chance, but 7 has 100% accuracy, and so on up the line. You are guaranteed to save 5 gnomes this way. Hi Elliot. I couldn’t post on your guestbook, so I was checking out your blog. I like puzzles like this. I think the solution is for the every other gnome to agree to answer with the color of the gnome before’s hat color. That means the 10th gnome would respond with the color of the hat worn by the 9th gnome. That gives the 10th gnome a 50% chance of survival, but the 9th gnome, when it’s his turn has a 100% accuracy. Gnome 8 answers with the color of the hat in front of him, giving him a 50% chance, but 7 has 100% accuracy, and so on up the line. You are guaranteed to save 5 gnomes this way.

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