ten gnomes for the chop

democrates

Ten gnomes are about to be executed. Although they don't like the idea, they are each selfless and want to do anything (even by sacrificing themselves) in order to help their fellow gnomes. They are told what will happen to them: They will each be lined up single file, so that they are each facing the gnome in front of them. Each of them will be given a red or a white hat on their head, and starting from the back of the line (the gnome who can see everyone else) they will ask each in turn to state his own hat color, 'red' or 'white', if he says anything else he will be executed. If he can state it correctly (he cannot see his own hat, only those in front of him), he is allowed to live. Knowing what is going to happen to them, they are allowed to devise a strategy beforehand. How many peolpe can they guarantee to save, and what strategy will ensure this?
(There are NOT five reds and five whites necessarily, there could be any mix including them all being the same colour)

kittenkiller

The guy at the back of the line has a 50/50 chance, all the rest have reason to survive.
They could just whisper the colour of the gnome in front of them's hat to them.
But no one can see the colour of the gnome at the back's hat!

democrates

The whisper counts as their guess, if it's the colour of the hat in front, that gives each one a 50/50 chance of it being the same as their own.

Your approaching it the same way I did, there is a better answer, even better than the official one given on the page I got it from.

Blisterman

The first guy calls out the colour of the hat in front of them. That guy then calls out his own hat colour. The next guy does calls out the colour of the hat in front of him, and so on, So they are guarenteed to save at least half of them.

Inspector Gadget

Maybe I'm being a dozy git when I say this, but if I'm reading this situation right, and working off a principle of maximum selflessness ad absurdum (and that all of the gnomes latch on to the same idea) - that is, naming the colour of the hat on the gnome in front, the actual survivability depends on how the colours "cluster".

If we assume that each gnome will call out the colour of the hat on the gnome in front of him/her, only the last gnome is guaranteed to survive (as he/she'll be inclined to save him/herself as there's no-one else to be selfless for), each of the others have a 50% chance of being right.

The effective survivability of this strategy (for the eight in the middle, as the first guy has a fixed 50% chance and the last guy should be a lock) depends on how the colours cluster. For each group of two or more that doesn't include the first gnome, the person who warns the first gnome in the cluster (unless he/she's also the first gnome in the line, in which case his/her chances increase to 50-50) and the last person in the cluster are dead (unless he/she's also the last gnome in the line), and the rest survive. As such, one gnome is guaranteed to survive a sequence of two same colours, two survivors from a group of three, and so on.

Either that or they do a who-wants-to-be-a-millionaire and develop harsh coughs

Gadget

Hagar

Each gnome takes off his hat and looks at it before declaring its colour.
Nobody said they couldn't.

Hagar

Somebody say something. Im very worried about these poor gnomes. I didn't sleep a wink last night fretting
.

democrates

If anyone wants to look, the official solution is here
I came up with another solution which is open to being called cheating, but there's nothing in the rules against it.
Anyone want me to spill?

Hagar

Yeah go ahead, you have piqued my curiousity.

Inspector Gadget

democrates wrote:

If anyone wants to look, the official solution is here
I came up with another solution which is open to being called cheating, but there's nothing in the rules against it.
Anyone want me to spill?

I like the original solution, but I'm all ears for an alternative?

Gadget

Illkillya

That was one of the original boards.ie puzzles... there used to be a link to 'puzzles' on top of the page iirc, it had good diagrams for explanation.

democrates

They decide to use two tones of voice when calling out the color of their own hat to signal the color of the hat in front. EG if they say their own hats color in a low tone, it means the hat in front is white, if they use a high tone, the hat in front is red.

The first gnome has a 50/50 chance of getting his own right, the rest survive by noting whether the previous gnome used a high or low tone.

Inspector Gadget

democrates wrote:

They decide to use two tones of voice when calling out the color of their own hat to signal the color of the hat in front. EG if they say their own hats color in a low tone, it means the hat in front is white, if they use a high tone, the hat in front is red.

The first gnome has a 50/50 chance of getting his own right, the rest survive by noting whether the previous gnome used a high or low tone.

This is just a variation of the "cough" trick I mentioned, isn't it? (Okay, I didn't flesh it out, but it's reasonably obvious)

The biggest problem is getting someone to volunteer to go first...

Gadget

democrates

Agreed now that you mention it, my senility seems to be,
eh, was i starting a thread or replying?

Capt'n Midnight

When the executioner goes to ask the colour of the hat. The gnome looks him straight in the eyes, to see his reflection, then stalls for time until M.A.L.A.G. arrive.