Right. But what would have happened if I had NOT told them that "at least one of you has green eyes"? Do you agree that neither of them would have been able to work out the colour of their eyes? If not, why not?
Can You Solve 'The Hardest Logic Puzzle In The World'?
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Right. But what would have happened if I had NOT told them that "at least one of you has green eyes"? Do you agree that neither of them would have been able to work out the colour of their eyes? If not, why not?
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Oh well in that case you must be right then!
I agree with Jono8 from now on.
If I don't understand it, it mustn't be possible, despite several other people understanding and agreeing with it
Because each dragon does not know that the other dragon knows that there is at least one dragon with green eyes as they never discuss it.
However, with 100 dragons, they all know that the others can see, and have always seen, at least one green eyed dragon.
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I am not answering a mathematical equation. I am answering the proposed question, which is about 100 green eyed dragons
I think this is the problem. You are deriving your arguments from your knowledge that the problem is about 100 green-eyed dragons. From their observation they cannot tell if it is a 99 green eyed dragon situation or a 100 green eyed dragon situation.
Oh well in that case you must be right then!
I agree with Jono8 from now on.
If I don't understand it, it mustn't be possible, despite several other people understanding and agreeing with it
Don't put words in my mouth. I have never said that
I really want to understand the official answer believe me but I just cannot get my head around it.
Also, believing something just because everyone else agrees isn't a good way to live you're live
OK, so now consider the case with three dragons, all with green-eyes, perfectly logical, etc. I tell them "at least one of you has green eyes".
All the dragons can see exactly two sets of green eyes.
They all leave.
Can you see that the conclusions drawn in steps two and three rely on step one? And that without the common knowledge injected into the system that at least one dragon has green eyes, the logical conclusion of step one "any dragon that sees no other green-eyed dragons must have green eyes and will leave" cannot be true?
This is the basis of induction.
- On day one, each dragon thinks: "I don't know the colour of my eyes, but I do know that there is at least one dragon with green eyes. So I know that any dragon who sees only blue-eyed dragons must conclude that their eyes are green, and will leave."
- On day two, each dragon thinks: "From step one, I now know that no dragon saw no other green-eyed dragons, so there are at least two green-eyed dragons. Therefore, if any dragon can see only one other green-eyed dragon, they must also have green-eyes, and will leave."
- On day three, each dragon thinks: "From step two, I now know that no dragon saw only one other green-eyed dragon, so there are at least three green-eyed dragons. Therefore, if any dragon can see two other green-eyed dragons, they must also have green-eyes, and will leave."
All the dragons can see exactly two sets of green eyes.
They all leave.
Can you see that the conclusions drawn in steps two and three rely on step one? And that without the common knowledge injected into the system that at least one dragon has green eyes, the logical conclusion of step one "any dragon that sees no other green-eyed dragons must have green eyes and will leave" cannot be true?
This is the basis of induction.
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so there were these 3 gods.............
but nobody knew who they were... Just A B and C...
This is the issue. The dragons know that this will not be the case (they know the other dragons will NOT see just blue eyes). They know that no one will leave on the first day because they know no other dragon will see all blue eyes!
The point you're missing is that we are creating a logical proof. Because of the statement I make, the fact becomes common knowledge to all the dragons. This allows us to establish the inductive base case (1 GE dragon would leave on day 1). From that case, we can establish that 2 GE dragons leave on the 2nd day, or that 3 GE dragons leave on the 3rd day ... or that 99 GE dragons leave on the 99th day, or that 100 GE dragons leave on the 100th day.
The point isn't what the dragons can see, it's what they can logically conclude, given that they are all perfectly logical and all know each other to be perfectly logical. Any given dragon, upon seeing 99 other pairs of green eyes, knows from inductive proof that if they are all still standing there on the 100th day, then it must be the case that there are 100 green eyed dragons, because if there were fewer than 100, those dragons would already have left.
They are not sitting there on day 63 thinking "well, it's never going to happen that someone sees 62 green-eyed dragons, because I can see 99, so let's forget the whole thing and go eat some dwarves and steal their gold."
Well actually, they might. Perhaps that's why Smaug was so angry.
I offer Randall Monroe's write up of the solution:
https://xkcd.com/solution.html
Or a paper on common knowledge from Yale University:
http://cowles.econ.yale.edu/~gean/art/p0882.pdf
(Section 2 "Puzzles about reasoning based on the reasoning of others", starting on page 3.)
The point isn't what the dragons can see, it's what they can logically conclude, given that they are all perfectly logical and all know each other to be perfectly logical. Any given dragon, upon seeing 99 other pairs of green eyes, knows from inductive proof that if they are all still standing there on the 100th day, then it must be the case that there are 100 green eyed dragons, because if there were fewer than 100, those dragons would already have left.
They are not sitting there on day 63 thinking "well, it's never going to happen that someone sees 62 green-eyed dragons, because I can see 99, so let's forget the whole thing and go eat some dwarves and steal their gold."
Well actually, they might. Perhaps that's why Smaug was so angry.
I offer Randall Monroe's write up of the solution:
https://xkcd.com/solution.html
Or a paper on common knowledge from Yale University:
http://cowles.econ.yale.edu/~gean/art/p0882.pdf
(Section 2 "Puzzles about reasoning based on the reasoning of others", starting on page 3.)
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But you said all the dragons on the first day think "So I know that any dragon who sees only blue-eyed dragons must conclude that their eyes are green, and will leave."
The dragons already know this will never happen. They know that no one can see all blue eyed dragons. So why are they thinking this and expecting someone to leave on the first day? They know no one will leave on the first day because they know no one can see all blue eyed dragons. They are after all, perfectly logical.
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Jono8, have you read the official answer?
You are right that they know it won't happen. They are not expecting anyone to leave on the first day because, as you correctly state, they are all able to see other green-eyed dragons.
Now FORGET about what they can see and think about the LOGIC of the puzzle.
The point is that they ARE certain about what WOULD have happened, and from that certainty can draw conclusions about what certainly IS and IS NOT true. By induction - in other words showing as I did above that if step one holds, then step two, step three, etc. will hold - the dragons know that if they see X dragons, then all green-eyed dragons will leave on day X+1.
That's right but it doesn't actually matter - the logic is still true. AJK is just trying to explain (and doing a good job, imho
) how the initial statement one day one gives information that starts a process of induction that resolves only in 100 days.In fact, because, as you point out, each dragon can see 99 green eyed dragons, then each does indeed know that nobody will get up and leave on day 2, 45, 68 etc etc. Each individual dragon is actually waiting for only one of two possible outcomes: If he himself has non-green eyes, the other 99 will get up and leave on day 99 having concluded their own eyes are green. Or, if nobody leaves on day 99, he will conclude his eyes are also green and will leave with the rest on day 100.
EDIT: beaten again
I understand the logic behind induction and that is of course correct.
My difficulty with the whole thing is why that statement would start them on the path to that logical induction.
I suppose my argument is that if they were totally logical they would never think "So I know that any dragon who sees only blue-eyed dragons" as they know that not to be the case. I also still don't understand why that statement starts them off on the path to that deduction.
As I have said, they all knew that, they all knew that at least one has green eyes so going by the questions logic, they should have started and tested the 100 day induction theory the first time they were all together and could all see everyone else has green eyes.
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They don't know the colour of their own eyes, for all they know they could be red, you have to remember that all the dragons are thinking the same thing..."I can see X number of green eyes, so in X days they should all leave, if however, on X+1 they don't all leave, I too must have green eyes, so must leave then".
I think part of it that gets confusing for some is they see it from one dragon's perspective, you have to remember that all dragons think the same logical thought process though.
I know. I understand the induction. I still don't understand what the statement that at least one of them has green eyes changes.
They all know this and they all know that all the others know this as well.
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But they also all assume that they themselves have non-green eyes so when you introduce the message you've started a count down to genocide. With the knowledge that one has green eyes, they all think "well that one will leave", however as there are X number of them they can see, their thought is that think "in X days they will all leave as logically they will all have deduced that they themselves have green eyes and I do not".
Once day X passes the only logical conclusion is that they too have green eyes and so on X+1 they all go and you've caused genocide.
