In the puzzle as given, no dragons will change on day 2. Why are you hung up on "which" dragon changes? It doesn't matter; they all follow the same process.
Can You Solve 'The Hardest Logic Puzzle In The World'?
- Thread starter ikillbears
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In the puzzle as given, no dragons will change on day 2. Why are you hung up on "which" dragon changes? It doesn't matter; they all follow the same process.
But all the dragons have seen that all the other dragons have green eyes. If someone said to the whole group "at least one of you has green eyes" they would all think to themselves "yup, have known that all my life".
They all knew that all the other dragons knew there was at least one of them with green eyes. I just cannot see what it changes, other than "oh gets them thinking about it" which is stupid.
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Because I believe it does matter. The dragons must change at midnight after they become certain that they have green eyes. What logic do you have that says only one dragon can become sure before midnight on day two ?
Because you are thinking in real-world terms about the situation that exists on the 100th day. You need to be thinking about the base case for an inductive proof, the case where there is only one dragon. In that case, without a stated piece of common knowledge (somebody has green eyes), a single dragon would never change, and so the inductive proof cannot follow. With the single piece of common knowledge in place, it can.
Yes, this is a logical construct which doesn't bear much resemblance to a real-world situation; such is the nature of logic puzzles!
There is no logic that says "only one dragon can become sure before midnight on day two". None of the dragons are sure until the 100th day, then they all become sure at the same time, due to ("perfectly logical") inductive reasoning.
But with one and 2 dragons I understand as they did not have that knowledge before and they did not know that the other one knew that at least one had green eyes.
With 100, they ALL knew that they ALL knew there was at least one dragon with green eyes.
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This is great fun 
This is great fun
I want understand the apparent correct answer but I cannot get past he fact that all the dragons already knew that they all knew there was at least one green eyed dragon. The statement didn't make it "common knowledge". They all already knew that there was at least one green eyed dragon AND they all already knew that everyone else knew that as well.
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The parking one is.
Q: How many women have parked in the bays number 86 - 91?
A: None. The only car shown has reversed in correctly.
Observationally that's true; we, the impartial observer to all of this, don't see any new information added to the system when 100 dragons are all looking at each others' green eyes, and somebody says "at least one of you has green eyes".
BUT, by making the statement common knowledge to the system (try to think of this abstractly), you enable the dragons to say this:
This is the basis of the inductive reasoning which allows them to conclude on the 100th day that since nobody has left in the preceding 99 days, that all 100 dragons must have green eyes.
Without the "at least one..." statement being made, the base case instead becomes this:
Nothing can follow from this; the initial theoretical dragon can never act, as he does not have enough information to do so. Therefore no proof can follow for 2 dragons, 3 dragons... 100 dragons.
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Imagine 4 Dragons instead of 100.
Dragon 1 can see 3 sets of Green eyes
Dragon 1 knows that Dragon 2 can see a minimum of 2 sets of Green eyes
Dragon 1 knows that Dragon 2 knows that Dragon 3 can see a minimum of 1 set of green eyes.
Dragon 1 knows that Dragon 2 knows that Dragon 3 knows that Dragon 4 can see a minimum of 0 sets of green eyes.
Therefore Dragon 1 not have complete confidence that common knowledge exists between all and so no solution can form.
By saying that 'at least 1 set of green eyes' exists the last step in the chain can be removed and the solution can form.
Nope, with 4, all the dragons know that all the dragons can see a minimum of 2 sets of green eyes.
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Because as others have already stated it is badly worded.
and as others have stated, stop applying the "real-world" side of the problem to the mathematical equation you are actually being asked about.
Read further up the thread someone has kindly provided the correct mathematical answer to this.
Because that is the base step required to prove by induction that X green-eyed dragons can all be certain that their eyes are green on day X. Perfectly logical, remember? Forget the real-world stuff, this is a logic problem.
Do you understand the situation for smaller numbers of dragons? One green-eyed dragon would leave on day 1, two green-eyed dragons would leave on day 2, etc.
I am not answering a mathematical equation. I am answering the proposed question, which is about 100 green eyed dragons
Yes understand for the smaller numbers.
With 100, our statement that at least 1 has green eyes changes nothing. What "base step" is it?
Are you denying that they didn't all know that at least one of them was green eyed and also denying that they couldn't logically work out that each of them knew that as well?
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Before the statement is made, they have NEVER had to logically think about whether they have green eyes.
The important part is the rule they follow, I've stated this at least 3 of my replies.
Once a dragon realises that they have green eyes, they must then change. Dragon 1 knowing that Dragon 2 has green eyes is already common knowledge between them all, however doesn't mean they will change because each dragon DOES NOT KNOW it's own eye colour.
Once stated: "At least one of you has green eyes", this is NOT new information however is enough for the dragon to realise that if no dragons change within 99 days that they must also have green eyes.
At that point of the 'question/riddle' the dragon(s) has then realised they have green eyes, following their rule they then change on day 100.
You don't need to look at this mathematically at all, understanding the rule and the implication of saying "At least one of you has green eyes" has, concludes each of them will change when they can deduce that they have green eyes, which after 99 days & no dragon have changed is figured out.
In Short - It take a Dragon 99 days to realise that it's own eye colour is indeed green because no other dragon has changed during this period. The statement made means that during 100 days at least 1 dragon should have changed to a sparrow, however on day 99 when no other dragon has changed, the only dragon left that matches the statement is yourself, you then realise you have green eyes and change.
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I don't see how you can understand for the smaller numbers but not the larger ones - it's the same answer. Let's imagine there are two dragons on the island, both with green eyes. I tell them "at least one of you has green eyes". The perfectly logical dragons then both change on day two. Do you agree? Do you understand why?
Yes. Both dragons think " if I have blue eyes, the other dragon would see this and deduce that he has green eyes and would leave on the first day". As neither leave on the first day, both dragons deduce that they have green eyes and leave at midnight on the second day.
For 100 dragons, I don't see how they can ever make this deduction.
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I'd give up Mr Reaper lol