yeah think that does help...
Can You Solve 'The Hardest Logic Puzzle In The World'?
- Thread starter ikillbears
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yeah think that does help...
I think my post above should help solve this. By asking questions in that format I think you can start to draw enough info from the responses.
Cool.
Associate
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Dragons - There may be no mirrors but water reflects... this is flawed.
Soldato
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I get the dragon one being n days for n dragons, but it's the way it's worded as a 'real' situation that makes it confusing.
You have to just follow strictly the rules that every dragon does not know whether he himself is green eyed, but more importantly knows that every other dragon does not know that they themselves are green eyed.
A dragon will only change when he knows he is green eyed, but since no one discusses it and they can't ascertain that information for themselves, they can only assume they are not green eyed, based on the fact they have not turned into a sparrow yet.
What the human comment of "at least one green eyed dragon" does is that it triggers a logical deduction in the dragons. To be fair, the dragons could have initiated this themselves, but would have to have done it independently and simultaneously, which is improbable. This is probably the weird part that doesn't translate well in this 'real-life' context.
In any case, if you now pick any one dragon, say D1, and apply the above rules and conditions, you'll see that D1 does not know he is green eyed, because there are n-1 green eyed dragons he sees. So he expects that if anyone is to change (based on the humans comment and the resulting logical deductions the dragons are all now making), it will be one of them and not himself. Remember, he must assume that he is not green eyed, otherwise he'd already be a sparrow.
On the second day, the fact that no one changes means that your chosen dragon can conclude that another chosen dragon, say D2, must see the remaining dragons, minus the already chosen dragons (i.e. D1, D2) are green eyed and therefore expects that if anyone is to change it will be one of them and not himself, since he can only assume that he is not green eyed (just like D1 did). Now at this point, you are probably thinking, why does D2 discount D1? Well that is because this whole deduction stems from D1, so naturally, he will discount himself, since he doesn't know he is green eyed yet. Hence, you now have two dragons who don't know they are green eyed, reducing the potential pool of possible candidates to n-2.
On the third day and thereafter, you repeat this process and each day you'll get the resulting list of candidates for green eyes, reduced by one. When you reach the nth day, you'll find that there is only one dragon left, namely Dn and because no one changed at the end of the (n-1)th day, Dn-1 must conclude that since Dn has not changed, then Dn sees another dragon who does have green eyes. But this can't be possible, because all preceding dragons (i.e. D1, ... ,Dn-1) are not green eyed. This is a contradiction, hence it follows that the assumption made by D1 when he took himself as being non-green eyed, must be false. He now must end his days as a sparrow.
Now, you're probably thinking, well what if I chose my dragons differently, each day? This is actually a key part to why it all works out. Each day the dragons can be chosen differently (including the first dragon), but the end result will be the same. If you followed all the possible paths you'd have this branching structure, showing the all the possible combinations of dragons, all coming to the same deduction on the nth day.
You have to just follow strictly the rules that every dragon does not know whether he himself is green eyed, but more importantly knows that every other dragon does not know that they themselves are green eyed.
A dragon will only change when he knows he is green eyed, but since no one discusses it and they can't ascertain that information for themselves, they can only assume they are not green eyed, based on the fact they have not turned into a sparrow yet.
What the human comment of "at least one green eyed dragon" does is that it triggers a logical deduction in the dragons. To be fair, the dragons could have initiated this themselves, but would have to have done it independently and simultaneously, which is improbable. This is probably the weird part that doesn't translate well in this 'real-life' context.
In any case, if you now pick any one dragon, say D1, and apply the above rules and conditions, you'll see that D1 does not know he is green eyed, because there are n-1 green eyed dragons he sees. So he expects that if anyone is to change (based on the humans comment and the resulting logical deductions the dragons are all now making), it will be one of them and not himself. Remember, he must assume that he is not green eyed, otherwise he'd already be a sparrow.
On the second day, the fact that no one changes means that your chosen dragon can conclude that another chosen dragon, say D2, must see the remaining dragons, minus the already chosen dragons (i.e. D1, D2) are green eyed and therefore expects that if anyone is to change it will be one of them and not himself, since he can only assume that he is not green eyed (just like D1 did). Now at this point, you are probably thinking, why does D2 discount D1? Well that is because this whole deduction stems from D1, so naturally, he will discount himself, since he doesn't know he is green eyed yet. Hence, you now have two dragons who don't know they are green eyed, reducing the potential pool of possible candidates to n-2.
On the third day and thereafter, you repeat this process and each day you'll get the resulting list of candidates for green eyes, reduced by one. When you reach the nth day, you'll find that there is only one dragon left, namely Dn and because no one changed at the end of the (n-1)th day, Dn-1 must conclude that since Dn has not changed, then Dn sees another dragon who does have green eyes. But this can't be possible, because all preceding dragons (i.e. D1, ... ,Dn-1) are not green eyed. This is a contradiction, hence it follows that the assumption made by D1 when he took himself as being non-green eyed, must be false. He now must end his days as a sparrow.
Now, you're probably thinking, well what if I chose my dragons differently, each day? This is actually a key part to why it all works out. Each day the dragons can be chosen differently (including the first dragon), but the end result will be the same. If you followed all the possible paths you'd have this branching structure, showing the all the possible combinations of dragons, all coming to the same deduction on the nth day.
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I think this would only work with 2 dragons as with 3 (as said before) they would all think the other dragons can see at least 1 with green eyes so know its not them and wouldnt turn. None of them would then turn as they all can see one with green eyes but no one has told them that they too have green eyes. A would think its b and c and a would think b and c would see one green eyed dragon so they wouldnt know its them either unless the dragon told them so and it wouldnt as they hadnt so far.
And if there was 99 green eyes dragons and one with different colour that one would think it has green eyes too with that logic so it just doesnt make sense.
And if there was 99 green eyes dragons and one with different colour that one would think it has green eyes too with that logic so it just doesnt make sense.
Christ! Word search fail!
don't worry you're gonna be a lawyer logic isn't something you need
Remember, he must assume that he is not green eyed, otherwise he'd already be a sparrow.
You are going to have to explain that. Why would he have to assume he doesn't have green eyes and why would he already be a sparrow if he doesnt?
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Soldato
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we're still talking about the dragons!
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The dragons are all in tacit denial before the human arrives. Each one believes that they themselves have non-green eyes but they can see that every other dragon has green eyes. In order to maintain this falsehood they don't have mirrors and they don't discuss eye colour. In this way no dragon can be 100% sure that they have green eyes and thus do not need to submit for transmogrification.
The human breaks the 'no discussion of eye colour' rule and removes this self-deception. Logically, now, each dragon is forced to test the hypothesis that they themselves have non-green eyes by watching which dragons submit to be changed on the first night.
When no dragon changes, they ALL must deduce that they have green eyes because they can look at any other dragon (of the 99) and ask themselves, 'why didn't that one change ?' to which the only answer is 'because he can see I have green eyes'. And they all transform on the second night.
The human breaks the 'no discussion of eye colour' rule and removes this self-deception. Logically, now, each dragon is forced to test the hypothesis that they themselves have non-green eyes by watching which dragons submit to be changed on the first night.
When no dragon changes, they ALL must deduce that they have green eyes because they can look at any other dragon (of the 99) and ask themselves, 'why didn't that one change ?' to which the only answer is 'because he can see I have green eyes'. And they all transform on the second night.
What I cannot fathom is that with 100 dragons, what would make the dragons think that any of them would change on the first night?
Soldato
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Because they know that at least one of them needs to change (the human said so) and are obliged to deduce which one of them it should be.
But on the first day, how can any of them deduce which one of them it is?
Lots of you are trying to apply real-world thinking to this puzzle, which isn't going to work. The puzzle is solved via inductive reasoning (find solution for a base case 1, prove that the solution holds for n+1, therefore solution holds for any value); the key point is that the initial solution (for case when there is only 1 dragon) only has a solution when the common knowledge is introduced that "at least 1 dragon has green eyes".
You cannot solve this problem by starting at step 100; that's impossible. The dragons are perfectly logical (as stated in the question), so you must be too.
You cannot solve this problem by starting at step 100; that's impossible. The dragons are perfectly logical (as stated in the question), so you must be too.
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They can't, so their self-deception survives. But the very fact that none of them changed on the first day tips the logical balance on the second day. The only way the deception could have survived the first day is if they all have green eyes. If only one dragon had green eyes that dragon would see 99 pairs of non-green eyes. But they all see 99 pairs of green eyes. Which means they all swallow
on he second day.
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In your solution how do they decide which dragon changes first ?
We have to assume they don't want to be swallows (hence the 'no discussion of eye colour' rule).If they are perfectly logical, surely they would have all turned into sparrows the first time they saw everyone else. They all already knew that every other dragon knew that there was at least one dragon with green eyes.
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They might have had colour-blindness ? The statement of the puzzle is absolutely bobbins.
In your solution how do they decide which dragon changes first ?
What do you mean "changes first"? A dragon changes when he is logically certain that he has green eyes. End of. You can't start introducing wants and needs into the situation, because then the dragons would not be acting "perfectly logically".
Evidently that doesn't make sense. You meet 99 other people with green eyes, you don't logically conclude that you have them too. Some information and process must exist in order for you to reach that conclusion. The information that the visitor provides allows for the inductive base case to hold true (for 1 dragon), which it otherwise would not.
For those interested, here's the XKCD version of the puzzle (slightly different wording but exactly the same situation) and Randall's solution.
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Do you see some specific logic that will mean that only one specific dragon will change on day two ? if so, which dragon is it and why ?