They're not testing anything at 1 day... they know there are either 99 or 100 with green eyes. They don't communicate about their eye colour the human has given a starting point - they only know their own eye colour from observing what happens on the 99th day. As there is no communication there is no other way of knowing.
Can You Solve 'The Hardest Logic Puzzle In The World'?
- Thread starter ikillbears
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They're not testing anything at 1 day... they know there are either 99 or 100 with green eyes. They don't communicate about their eye colour the human has given a starting point - they only know their own eye colour from observing what happens on the 99th day. As there is no communication there is no other way of knowing.
Jono does this help at all?
I know they all know that every other one can see at least 98, but there is more to it than that. They need to all consider the full chain of perspectives, which as above means they have a lot more doubt than may initially be apparent.
I know they all know that every other one can see at least 98, but there is more to it than that. They need to all consider the full chain of perspectives, which as above means they have a lot more doubt than may initially be apparent.
Do you agree that if the man had said there are at least 98 green, they would have all changed on day 3?
Now if we say that every dragon has always known that every other dragon knows there are at least 98 green, do you see why there is confusion?
My above post tries to explain why the two situations are subtly different. And why the man's comment does actually change all of their knowledge.
I know you understand the problem, I just don't think you've helped explain away the possible confusion.
I'm not sure if my post helps either, but repeating the same points obviously isn't helping to highlight the flaw in what might otherwise appear to be a logical thought process.
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I understand what you are saying, but why doesn't the actual cognitive perception of seeing 99 green eyed dragons stop them after d2.
"But d1 thinks, if I do have blue eyes, then d1 thinks d2 could think he also has blue eyes. So d1 thinks there is a chance d2 thinks that d3 will be looking at a maximum of 97 green"
OK, but surely d1 knows this is impossible? Why is he even entertaining that hypothesis? d1 also knows that everyone else knows this would be impossible. This my difficulty with the whole thing. For this to work, the dragons have to hypothesise about situations and permeations that they know do not exist.
yup - and likewise if he'd said 100 they'd have all changed on day 1 after he'd left or if 99 they'd have changed on day 2...
yup... but I can also see that the confusion is perhaps easier to clear up when looking at smaller numbers of dragons
I was repeating the same question actually... the principle is the same - if there is a shortcut then at what number does the shortcut come into effect? - at what number does the solution supposedly breakdown and this shortcut by 'deduction' come into play?? The other poster seemingly didn't want to attempt that - doing so might illustrate that there isn't a shortcut... If you notice I've now switched to suggesting that he considers the short cut and how it would apply to increasing numbers of blue eyed dragons amongst the 100.
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*Edit.. speaking rubbish gonna delete*
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Double post
They're not contemplating scenarios that they know can't exist.
As you've said:
d1 thinking that d3 thinks there could only be 97 green is an invalid scenario.
d1 thinking that d2 thinks that d3 thinks there could only be 97 is not an invalid scenario. Even though d1 knows d2 and d3 know there are at least 98 green. And also d1 and d2 both individually know that d3 knows there are at least 98.
It's hard to get your head round. I'll try to see if I can explain better.
As you've said:
d1 thinking that d3 thinks there could only be 97 green is an invalid scenario.
d1 thinking that d2 thinks that d3 thinks there could only be 97 is not an invalid scenario. Even though d1 knows d2 and d3 know there are at least 98 green. And also d1 and d2 both individually know that d3 knows there are at least 98.
It's hard to get your head round. I'll try to see if I can explain better.
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Nope, they know the others can see at least 98 green eyed dragons, they've always known this... but there is no communication between them - their only new information is the starting point - they therefore know that at the 99th day if all the others change they have blue eyes otherwise they have green.
How would the shortcut apply to 95 green eyed dragons and 5 blue eyed? How about 96:4 or 94:6?
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No not quite. They all know the others see at least 98. But they also know there is a chain of thought which means they all think another dragon looking at them, might see blue and then follow the thought process I tried to highlight.
I know it seems to go against the fact they all know there are at least 98 green, but it doesn't.
If I can explain better I will.
The 'shortcut' is the assumption that A and B are identical, where:
A = each dragon knows that all other dragons know there are at least 98 green.
B = man says there are at least 98 green.
In both situations all dragons know that all other dragons know there are at least 98. But in situation B there is subtly more information given.
It is the difference between A and B that I am trying to make clear.
The main thing is the lack of communication - the only information they've got individually to deduce whether they've got green or blue eyes is whether everyone else changes on the 99th day or not
I'll attempt a different approach:
if there is a shortcut* where they can all change on day 3 then surely that shortcut would apply to 99 dragons or 98, 97, 96 dragons etc..etc..?
how about 99 dragons and one blue eyed dragon - what is stopping the blue eyed dragon from changing on day 3 if this shortcut applies? How about 98 dragons and 2 blue eyed ones?
There is not sufficient information for them to individually deduce their eye colour after 3 days - for example the blue eyed dragon with 99 green eyed dragons is going to go through exactly the same thought process as a green eyed dragon with 99 green eyed dragons....
(*edit - for clarity- I'm not referring to the visitor saying there is at least 98 but rather Jono's proposal that the solution could be shortened)
I'll attempt a different approach:
if there is a shortcut* where they can all change on day 3 then surely that shortcut would apply to 99 dragons or 98, 97, 96 dragons etc..etc..?
how about 99 dragons and one blue eyed dragon - what is stopping the blue eyed dragon from changing on day 3 if this shortcut applies? How about 98 dragons and 2 blue eyed ones?
There is not sufficient information for them to individually deduce their eye colour after 3 days - for example the blue eyed dragon with 99 green eyed dragons is going to go through exactly the same thought process as a green eyed dragon with 99 green eyed dragons....
(*edit - for clarity- I'm not referring to the visitor saying there is at least 98 but rather Jono's proposal that the solution could be shortened)
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His proposal that it can be shortened is because he thinks they already have the same knowledge as they would if the man had told them there were at least 98 green.
You've agreed that if the man told them this then they'd all change on day 3. I'm trying to explain why there is a difference between them being told there are at least 98 green and the initial starting condition where they all already know that everyone else knows there are at least 98, even before the man says anything.
You've agreed that if the man told them this then they'd all change on day 3. I'm trying to explain why there is a difference between them being told there are at least 98 green and the initial starting condition where they all already know that everyone else knows there are at least 98, even before the man says anything.
yup and it boils down to them not being able to communicate... all they have essentially been introduced to is the starting point
I just thought it was easier to show that such a shortcut doesn't work starting from one by showing that it would also have to apply to 99, 98 etc..etc..
I just thought it was easier to show that such a shortcut doesn't work starting from one by showing that it would also have to apply to 99, 98 etc..etc..
I'm not sure you understand his confusion.
But at this point I'm going to point out that if we change the question slightly to:
100 green eyed dragons are dropped onto an island and all immediately look each other in the eyes. Each dragon knows that there are at least 99 green eyed dragons. Each dragon also knows that any other dragon knows there are at least 98 green eyed dragons. As soon as any dragon deduces their eyes are green they must change into a sparrow at midnight.
What happens?
In this situation they still will not change, despite what people have previously said about needing to know that each dragon learns the information about each other simultaneously. (I realise I had said this myself earlier).
It is nothing to do with learning the information simultaneously, it is simply the fact that when the man speaks, he actually gives them more information than they previously had.
So the most important thing to understand is that before the man speaks, every dragon knows there are 99 other green. They also know that every other dragon knows that there are at least 98 green.
But they do not know that there cannot be a dragon who thinks another dragon thinks another dragon thinks another dragon think ..... another dragon thinks there could be 100 blue dragons.
Despite this seeming to contradict the above situation where every individual dragon knows there are already 99 other green dragons (it doesn't).
This is the reason nothing can start before the man speaks. Him saying that there is at least 1 green adds totally new information (despite the fact it would seem obvious that each dragon knew this without him saying anything).
I don't know if this is confusing some people, but I wanted to clear up the reason why the induction process hadn't started before the man spoke. Even if the question had been phrased to say they all knew each dragon looked all other dragons in the eye simultaneously at time 0.
Just wanted to try and make sure that wasn't adding to anyone's confusion.
But at this point I'm going to point out that if we change the question slightly to:
100 green eyed dragons are dropped onto an island and all immediately look each other in the eyes. Each dragon knows that there are at least 99 green eyed dragons. Each dragon also knows that any other dragon knows there are at least 98 green eyed dragons. As soon as any dragon deduces their eyes are green they must change into a sparrow at midnight.
What happens?
In this situation they still will not change, despite what people have previously said about needing to know that each dragon learns the information about each other simultaneously. (I realise I had said this myself earlier).
It is nothing to do with learning the information simultaneously, it is simply the fact that when the man speaks, he actually gives them more information than they previously had.
So the most important thing to understand is that before the man speaks, every dragon knows there are 99 other green. They also know that every other dragon knows that there are at least 98 green.
But they do not know that there cannot be a dragon who thinks another dragon thinks another dragon thinks another dragon think ..... another dragon thinks there could be 100 blue dragons.
Despite this seeming to contradict the above situation where every individual dragon knows there are already 99 other green dragons (it doesn't).
This is the reason nothing can start before the man speaks. Him saying that there is at least 1 green adds totally new information (despite the fact it would seem obvious that each dragon knew this without him saying anything).
I don't know if this is confusing some people, but I wanted to clear up the reason why the induction process hadn't started before the man spoke. Even if the question had been phrased to say they all knew each dragon looked all other dragons in the eye simultaneously at time 0.
Just wanted to try and make sure that wasn't adding to anyone's confusion.
I can see why someone might initially assume they can skip the first x days - really the only person who does fully understand his own confusion is going to be himself - a better way to examining it if you thought there was a flaw in the solution is to start with a lower number (Think I've mentioned that before
) The alternative way to explore it is showing the shortcut/skipping idea is flawed and results in a contradiction - which I've very roughly outlined above.
Tbh... I genuinely don't understand why, if someone was confused about the solution and incorrectly thought there was a shortcut that they wouldn't try exploring that shortcut with a smaller number as doing so can both quite quickly highlight the flaws with it and better illustrate the nested hypotheticals.
But you're assuming that he will see the 'flaw' in the 'shortcut' at lower numbers. I'm not convinced that it does help to understand this particular flaw.
Take 4 green eyed dragons (again being told at least 1 is green):
Everyone accepts that the following logic could be used to arrive at a solution:
Day 1 - any dragon which sees 3 blue, knows he is the only green, so changes.
Day 2 - any dragon which sees 2 blue, knows that none have yet changed, so he must be one of 2 greens, so he changes.
Day 3 - any dragon which sees 1 blue, knows that none have yet changed, so he must be one of 3 greens, so he changes.
Day 4 - all dragons now know they are all green.
I don't think anyone is disputing this at all.
But the 'flawed' thought process which is still causing confusion is:
1) That they all individually know there are at least 3 green - this is correct
2) They all also know that every other dragon knows there are at least 2 green - this is also correct
The man then tells them that at least 1 is green (but with flawed logic we deduce that they already knew this, because in fact they all knew that the others all knew there were at least 2 green).
Using this flawed logic, incorrectly leads to the following two assumptions:
1) That the man telling them that there is at least one green adds nothing new to their information (they already all knew there was at least 1 green).
2) That the dragons starting condition (by observation alone) is identical to the man having told them there are at least two green eyed dragons.
Applying the above 'flawed' logic, we now say that if the man had told them there were at least 2 green eyed dragons, then the following would happen:
Day 1 - any dragon which sees 2 blue, knows that he is one of the 2 greens, so he changes.
Day 2 - any dragon which sees 1 blue, knows that none have yet changed, so he must be one of 3 greens, so he changes.
Day 3 - all dragons now know they are all green.
The 'shortcut' still applies even with a smaller sample set.
I don't think it helps understand the 'flaw' in the 'logic' which is being applied.
Take 4 green eyed dragons (again being told at least 1 is green):
Everyone accepts that the following logic could be used to arrive at a solution:
Day 1 - any dragon which sees 3 blue, knows he is the only green, so changes.
Day 2 - any dragon which sees 2 blue, knows that none have yet changed, so he must be one of 2 greens, so he changes.
Day 3 - any dragon which sees 1 blue, knows that none have yet changed, so he must be one of 3 greens, so he changes.
Day 4 - all dragons now know they are all green.
I don't think anyone is disputing this at all.
But the 'flawed' thought process which is still causing confusion is:
1) That they all individually know there are at least 3 green - this is correct
2) They all also know that every other dragon knows there are at least 2 green - this is also correct
The man then tells them that at least 1 is green (but with flawed logic we deduce that they already knew this, because in fact they all knew that the others all knew there were at least 2 green).
Using this flawed logic, incorrectly leads to the following two assumptions:
1) That the man telling them that there is at least one green adds nothing new to their information (they already all knew there was at least 1 green).
2) That the dragons starting condition (by observation alone) is identical to the man having told them there are at least two green eyed dragons.
Applying the above 'flawed' logic, we now say that if the man had told them there were at least 2 green eyed dragons, then the following would happen:
Day 1 - any dragon which sees 2 blue, knows that he is one of the 2 greens, so he changes.
Day 2 - any dragon which sees 1 blue, knows that none have yet changed, so he must be one of 3 greens, so he changes.
Day 3 - all dragons now know they are all green.
The 'shortcut' still applies even with a smaller sample set.
I don't think it helps understand the 'flaw' in the 'logic' which is being applied.
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