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Jo(e) has given bloggers the go-ahead to google to solve, so I’m putting the answer

under the fold, courtesy of reader Breviloquence.

“Before they are lined up, the Gnomes all agree that whoever is at the back of the line (Picked first) is pretty screwed. So that Gnome, they all agree, will call out a color to tell them how the rest of them look. If there is an even number of Gnomes wearing red hats, he will call out “Red!” and if there is an odd number of Gnomes wearing red hats, he will call out “Blue!” The Evil Dictator comes to the first Gnome (Who is standing at the back of the line). I will use the first picture on the blog as the example set, so you can reference it visually.

That first Gnome looks down the line and sees that there are four red hats. He calls out “Red!” and the Evil Dictator… Blows his little Gnomey brains out. Th next Gnome in line, wincing and shuddering, looks down the line and sees four red hats.

Now armed with the knowledge that there are, including himself, an even number of red hats, he calls out “Blue!” and… Survives.

The next Gnome in line looks down the line and sees three red hats. Confused for a moment, he then realizes that because the first Gnome saw an even number of hats and he sees an odd number (And the only other Gnome who could have been that last hat was not it), he must be wearing a red hat. So he calls out “Red!” and lives.

The fourth Gnome in line looks down the line and sees two red hats. Thinking about it, he realizes that since the first Gnome saw an even number of hats, he also sees an even number of hats and one red hat has already been picked out, he must also be wearing a red hat. He calls out “Red!” and lives.

Th fifth Gnome in line looks down the line and sees two red hats. Knowing there must be an even number of red hats and two Gnoms behind him are wearing red hats, he must be blue. He calls out “Blue!” and lives.

The sixth Gnome in line looks down the line and sees only one red hat left. Knowing that there must be an even number of red hats, he deduces that he must be wearing a red hat. So he calls out “Red!” and lives.

The seventh Gnome in line looks down the line and sees that same one red hat left. Knowing that there must be an even number of red hats and that there are three Gnomes in red hats behind him, he deduces that he must be wearing a blue hat. He calls out “Blue!” and lives.

The eighth Gnome in line looks down the line and sees… No more red hats! Uh-oh. That means he must be wearing a red hat, since there are an even number of red hats total and three Gnoms behind him are wearing red hats. He calls out “Red!” and lives.

The ninth Gnome in line looks at the back of the last Gnome and thinks hard. He doesn’t see any more red hats. There were an even number of red hats, and four other Gnomes called out “Red!” and lived so far… He must be wearing a blue hat. He calls out “Blue!” and lives.

The tenth Gnome in line sighs. He can’t see anyone else. But he knows there must be an even number of Gnomes wearing red hats in the line, including himself. He also knows four other Gnomes have called out “Red!” and lived. He must be wearing a blue hat. He calls out “Blue!” and lives.

So by possibly sacrificing himself, the first Gnome in line saves all the other Gnomes. If the first Gnome had been wearing a red hat, he would have lived as well. The important thing to remember here is that no matter what color hat the first Gnome is wearing, he doesn’t count when the other Gnomes are trying to figure out what color hat they have on, because no one can see what color hat he is wearing. He has a straight 50/50 chance to live no matter what.

Incidentally, six Gnomes were killed in the formulating of this explanation. Don’t tell PETG.”

Isn’t that the best? Thanks, Breviloquence!

**5 Comments so far**

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Comment by nmJuly 13, 2007 @ 11:08 amYour solution is correct, but your example is incorrect. You say that the first Gnome sees 4 red hats, which means that there are 5 Gnomes total (1 + 4 = 5). But your example ends up with 10 Gnomes total in line. Also, if the first Gnome sees 4 red hats, the second Gnome should see 3 red hats, which is an odd number, leading him to say “Red!”, not “Blue!” as you stated.

Comment by BrianCJuly 18, 2007 @ 12:27 pmOh nevermind, pardon my ignorance. Your example is correct. There are 10 gnomes in line, at least 4 wearing red and at least 5 wearing blue. I guess I should think about it some more before posting. 🙂

Comment by BrianCJuly 18, 2007 @ 12:31 pmBut doesn’t this assume that the gnomes have hat colors that alternate R B R B R B etc? What if the evil leader put their hats on as R R R R R B B R B B B R B?

Comment by WolAugust 1, 2007 @ 4:17 pmAs I understand it, the gnomes must simply figure out even/odd from the hats they see in front of them (say, 4 blue, 3 red). Each gnome can then deduce what color he’s wearing from that information. It really wouldn’t matter if the hats he can see were BBBBRRR, BRBBRBR, or BRBRBRB — the order of the color is not significant so much as being able to count them accurately.

Comment by bridgettAugust 1, 2007 @ 6:35 pm