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# game theory ...tough

## Question

Three explorers are getting kidnapped by an evil tribe deep in the jungle, and their life now depends on

their ability to correctly answer the following challenge the evil tribe's chief has for them. The

explorers are tied to three trees facing each other and are presented with the chief's 5 wives: 3

brunettes, and 2 blondes. The three explorers are now blindfolded, and the chief picks three of his five

wives to stand behind the trees the explorers are tied to, one behind each tree. The remaining two wives

disappear from view.

Then the blindfolds come off, and each explorer is able to see the two wives that are standing behind his two colleagues, but not the one behind himself.

Each explorer now has up to 10 minutes time to think, after which each has to correctly answer what hair color the wife has that is standing behind him, or lose his life.

When you, as one of the three explorers, gain your sight after the blindfold comes off, you see

two brunettes standing behind your two colleagues. And now your life depends on figuring out who is behind you: a blonde or a brunette?

This is not a trick question (e.g., they cannot talk or otherwise cheat), and it's not about probabilities. There is one--and only one--logical solution to this

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Reasoning from a similar problem:

a) If no one answers immediately brunette, it means there is at most one blonde. (That one who sees 2 blondes answers immediately brunette.)

b) If the others see a blonde and a brunette, they will answer simultaneously brunette. (There are not 2 blondes.)

c) From b), everyone can deduce the others see two brunettes.

We have just to assume that all three think at the same speed.

Edited by harey

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I think harey has it.

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i don't think so. we can't make his assumption that everyone thinks at the same speed.

besides if i saw a blonde and brunette, how could i possibly say brunette? i have no idea who is behind me.

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i don't think so. we can't make his assumption that everyone thinks at the same speed.

besides if i saw a blonde and brunette, how could i possibly say brunette? i have no idea who is behind me.

The guy with the brunette would immediately say brunette if you had the other blonde.

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okay let's work through it slowly.

let's say there are 1 blonde and 2 brunette.

two guys see 1 blonde 1 brunette, and the other guy sees two brunette.

and the other possibility is all 3 brunette for the guy seeing two brunette.

if I’m the guy that sees two brunette, i can't know what’s behind me. nor do I see how the other two could.

the only conclusion we can safely reach is that no one sees two blonde.

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Scenario 1: Two blondes and one brunette are present.

Scenario 2: One blonde and two brunettes are present.

Scenario 3: Zero blondes and three brunettes are present.

Scenario 1: Whoever has the brunette behind him will see two blondes, and will say brunette very quickly.

Scenario 2: The two people with brunettes behind them will see one blonde and one brunette behind the other people. They will realize that they must be dealing with scenario 1 or scenario 2. They wait a couple of minutes, and since no one said that he has a brunette behind him very quickly, they will know that they're not dealing with scenario 1. They therefore know they're dealing with scenario 2, and since they can see the one blonde behind someone else they will know that they have a brunette behind themselves.

Scenario 3: Hopefully if we were dealing with scenario 2 then someone would have spoken up after a couple of minutes, so if you reach the end of the 10 minutes then guess this by default.

Here's the tricky part: The OP doesn't ask whether the first player who answers can be sure that he's correct, it asks whether you can be sure that you're correct if you see two brunettes. If you wait a while and then both of the other guys correctly answer brunette, then are you safe or in danger? Remember, you didn't get a chance to discuss strategy with the others before this all started.

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This riddle reminds me very much of one that my dad (who just recently passed away) always used to tell me. You can read his telling of it here, on his blog ("I've Got a Blue Spot" was originally going to be the title of a book he was going to write...but turned into the start of a blog instead...the name coming from the riddle):

http://www.ivegotabluespot.com/archives/18

Edited by Pickett

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Phil actually has a point. It is hard to actually draw any solid conclusions without making the unrealistic assumption that all of the explorers are intelligent and think at the same speed.

A more complete version of the same puzzle is actually posted on the official site, http://brainden.com/logic-puzzles.htm, under "Masters of Logic Puzzles (Dots)" - although the story is more like "Head Bands", another similar puzzle also listed on the same page.

Edited by gavinksong

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