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Topic: 3 dots Redux (Read 1786 times) |
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Icarus
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3 dots Redux
« on: Apr 4th, 2005, 7:15pm » |
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This a version of the three hats/three dots riddle that I gave in the most recent thread for that riddle in the easy forum. However, I think this version is both more subtle and harder, so I am repeating it here.
The chief of a tribe wishes to choose a successor. He picks the three most likely candidates and tells them that he will determine the brightest of them by means of a fair contest. The three candidates will be seated in a triangular fashion facing each other, and then blind-folded. The chief will then paint either a red or a green dot on the forehead of each. When he is done, the blindfolds will be removed at the same time, so each can see the dots on the other two, but not their own. They are instructed to raise their hand if they see a green dot. The first who can tell the chief the color of the dot on his own forehead, and give a good explanation of how he knew, will become the new chief.
As the chief readies the blindfolds and paint, one of the candidates speaks up: "I know the color dot you will paint on my forehead." He tells the chief the color, and how he knew it, and the chief declares that he is correct.
What was the color, and how did he know it before the contest even started?
A further explanation: Everyone here is completely honest, and knows that the others are completely honest. And the chief has paint of both colors.
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| « Last Edit: Apr 4th, 2005, 7:24pm by Icarus » |
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markr
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Re: 3 dots Redux
« Reply #1 on: Apr 4th, 2005, 11:28pm » |
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For the contest to be fair, each contestant will receive the same color.
For the contest to be non-trivial (i.e. actually require some intelligence on the part of the winner), that color will be green.
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towr
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Re: 3 dots Redux
« Reply #2 on: Apr 5th, 2005, 1:40am » |
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on Apr 4th, 2005, 11:28pm, markr wrote:For the contest to be fair, each contestant will receive the same color.
For the contest to be non-trivial (i.e. actually require some intelligence on the part of the winner), that color will be green.
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And since the chief knows this, to make it more difficult he'll give everyone a red dot instead.
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| « Last Edit: Apr 5th, 2005, 1:41am by towr » |
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Sjoerd Job Postmus
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Re: 3 dots Redux
« Reply #3 on: Apr 5th, 2005, 3:56am » |
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Ok... let us analyze this:
0 = red, 1 = green
situation 000:
| hidden: | | Nobody raises his hand => None of them see green. Logic dictates that everyone has red. From the cheifs pov, this would be silly to do, so our guy easily eliminates this possibility. |
situation 001:
| hidden: | | Two people raise their hand. Guy three sees zero red dots, but there is at least one. From this follows that he has green. Easy to eliminate, because it's too easy to jump to a conclusion. |
situation 010 and 100: See situation 001.
situation 011:
| hidden: | | All raise their hand. Guy one doesn't know his color. But guy two does. Since he sees one green, and the green guy also sees one green, he must have green. guy three has the same argument. Chief being smart, he won't do this, too easy. We can easily eliminate this. |
situation 101 and 110: See situation 011.
situation 111:
| hidden: | | Guy one thinks: Both raise their hand, because they see eachothers red dot. But, do I have red or green, I don't know! The same argument goes for the other two guys. For them, it's a not that easy. But, after some time, when they have all walked through the options, they realise that if it was situation 011, 101 or 110, the answer would have already been given, so it must be 111. This answer would be the hardest to derive, so the clever chief would choose this situation. |
Smart tribe member knows in advance... he's done the same logic!
On the other hand, because we have <not given>indians</not given>, you can't say whether they have red paint, or no paint at all... So for there to be paint visible, he needs green  ... But that's silly-mans logic, still go for the previous explanation
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rmsgrey
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Re: 3 dots Redux
« Reply #4 on: Apr 5th, 2005, 8:07am » |
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My conclusion is that the chief isn't as smart as his chosen successor - as towr pointed out, markr's solution (probably the intended one) is vulnerable to the chief following the same reasoning and choosing to foil it. A candidate capable of realising this will wait to get further information since it appears the chief's best strategy is to randomly pick an arrangement other than two greens and a red (in some permutation) and apply it - with two greens and a red, the red guy can only deduce his colour by seeing the others deduce theirs so that arrangement is unfair. Of course, there is no strategy the chief can choose that is guaranteed to identify the smartest of any given group of three since very clever and very stupid behave similarly.
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Icarus
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Re: 3 dots Redux
« Reply #5 on: Apr 5th, 2005, 5:27pm » |
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markr had it. I guess I should have left it in Easy after all.
The chief is trying to identify the smartest, not fool them, so towr's rebuttal does not apply. There is no reason for the chief to try and defeat their reasoning. The chief's idea is in accordance with the original puzzle: when each sees the green dots, the first one to realize what the silence of the other two means, will be the smartest.
The candidate has realized that to be fair, they must all be given an equal opportunity to figure out their dot color, and the only way this can be is if they all have the same color. And then he realized that if they all had red, figuring out their own color would be easy, whereas green actually requires some thought. Therefore the chief intends to put green dots on them all.
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By the way: what inspired this was a version of the three hats puzzle that I found as the week's "featured puzzle" on the Grey Labyrinth (another puzzle site) wherein a king was picking a candidate for something. He informs his victims that he has 2 red hats and 3 green ones. When blindfolds are removed one candidate announces his hat is green.
The "solution" was that this candidate saw 2 red hats and therefore knew his hat had to be green! Not only is this version of the riddle trivial, it also left me disgusted at the concept of a king to offers a contest to find the smartest, and then rigs it so that one can win easily while the other two cannot win at all.
This is a large part of why I don't bother to visit the Grey Labyrinth! (The other reason is that even their good puzzles are often of types I don't find interesting.)
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| « Last Edit: Apr 5th, 2005, 5:36pm by Icarus » |
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towr
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Re: 3 dots Redux
« Reply #6 on: Apr 6th, 2005, 12:11am » |
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on Apr 5th, 2005, 5:27pm, Icarus wrote:| The chief is trying to identify the smartest, not fool them, so towr's rebuttal does not apply. |
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Sure it does. Suppose we have two peoploe as smart as the smartest here, and one smarter. Those first two will immediately say, hee chief, you're gonna paint a green dot on my forehead. The other says, he chief those other two probably think you're gonna paint a green dot on each of our foreheads, so you should make it red.
Simple first-order sicilean logic. (Second order makes it green again, etc)
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rony phaeton
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what i think the smartest person would do is just wait for one of the two to raise there hand or see a reaction from there faces when they looked at there forheads. if it were a bad reaction the smartest person would know that he does not have a green dot on his forhead. 
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