Logic Problem #1 - Gnomes and Party Hats
There are ten gnomes who have gotten themselves into a fair bit of trouble. They are trapped in the dungeon of an evil king who likes to play games with his prisoners. The king tells them the first game he will play with them, it goes like this:
The king lines the gnomes up in a single-file row. Meaning that the tenth gnome sees the back of the person in front of him, and there is no gnome behind the tenth gnome. The ninth gnome has the tenth gnome behind him and the eighth gnome directly in front of him, and so on. Finally, the first gnome has the second gnome directly behind him, and there is no one in front of the first gnome.
For some reason or another, the king has a bag full of black and white party hats. The king randomly reaches into his bag and places a party hat on each of the gnomes' heads. This means that the tenth gnome can see everyone's hat except his own, the ninth gnome can see everyone's hat except his own and the tenth gnome's hat, and so on. The first gnome can see no one's hat. Now, it is important to note that there is not necessarily the same number of black party hats as white party hats.
Once he has placed party hats on all the gnomes, the king will then takes out his gun and put it to the temple of the tenth gnome. The king will ask the gnome: "What color is your party hat?" If the gnome answers correctly, he lives and gets freed from the dungeon. If he does not, he dies. The king continues up the line in this progression until he reaches the last gnome.
Despite his evilness, the king has a silver lining in his heart. So, he has given the gnomes a chance to live. His offer is this:
"Before I place the party hats on you, you are allowed to meet as a group and discuss a strategy to save as many of each other as possible."
Imagine that you are one of these gnomes. What strategy would you develop? How many gnomes can you guarantee to save?
When it is your turn to say the color of your hat you must ONLY say "white" or "black." If you say anything else, the king will shoot you and all of the remaining gnomes.
This puzzle was told to me by a legend of the Bates Math Department, Eric Towne.
I know the answer, but only because I too know the Bates Mathematics Legend, Eric Towne
So I won't give it away
The easiest way to save everybody would be for each gnome to take the party hat of the gnome in front and place it on their respective heads. This way, gnomes 2-10 would know their hat's color.
Once gnome number 10 finishes his turn, we know he "lives", and is is "freed". Gnome 10 doesn't need the party hat anymore, so he gives it to gnome 1 on his way out. All gnome 1 has to do is listen to what gnome 10 says.