Can You Solve 'The Hardest Logic Puzzle In The World'?

[DaleTheWhale]

To Answer: "What exactly is the new information that you gave the dragons?"

You give them the information they need to figure out their own eye colour, you've told them nothing new in regards to information however they now know that at least 1 dragon has green eyes.

They have a rule on the island which states that if a dragon ever finds out that he/she has green eyes, then at precisely midnight on the day of this discovery, he/she must relinquish all dragon powers and transform into a long-tailed sparrow.

Come the 99th day as they're logical creatures they will know that as all 100 dragons heard the 'new information' and that all can see 99 green eyes and none have changed they have each learnt that they have green eyes, thus all changing into a sparrow at midnight.

The dragon can't know they have green eyes until in the example below, N numbers of days have passed to match the amount of dragons.

@cheesyboy, because following the logic, 3 dragons would see 2 other green eyes, the 2 dragons the first dragon can see won't have realised that itself has green eyes until the third day when no dragon has changed, meaning all 3 of them then learn they have green eyes, thus changing at midnight.

Breakdown:

1 dragon with the same information, dragon 1 can be the only one with green eyes and changes on midnight.
2 dragons with the same information, dragon 1 see's dragon 2, however after midnight neither change as they don't realise they have green eyes, day 2 midnight they both then change as they find out they have green eyes from logic.
Continue on till 1 million dragons, as it will be the millionth day they all change.

 

[cheesyboy]

Why would a second and third day make a difference?

Dragon1 can see two green eyed dragons, the other two dragons can (to the knowledge of Dragon1) see one green-eyed dragon each. Under what logic would Dragon1 expect to see either of the others turn into a sparrow?

The outlook is the same for Dragon2 and Dragon3. Day 2 brings no new information.

Edit:
It works for 2 dragons, but I don't see how it would work for 3 or more dragons, as my above breakdown.

 

[Greboth]

I still struggle to see how any of the individual dragons can come to the certain conclusion that they have green eyes.

We know they all have green eyes, but the dragons only know that at least one has green eyes, and that all the other dragons have the same colour eyes.

With 2 dragons, it would work. An individual would see a green-eyed-dragon and, knowing one or more of them must have green eyes, would know that if the other dragon doesn't turn into a sparrow (since he too can see the one green-eyed dragon), then the individual would assume that he himself must also have green eyes.

I don't see how that works with 3, or more, dragons. An individual would see 2 green-eyed-dragons, and would know that each of those two would see the other green eyed dragon. Neither would be expected to turn into a sparrow since they can all see the prescribed "at least one" green-eyed dragon and would, therefore, have no compulsion to assume they, themselves, are green-eyed.

If you have 3 dragons, lets call them A,B and C and they all get hear that one dragon has green eyes.
A knows B and C have so B and C should change midnight end of day 2.
B knows A and C have so A and C should change midnight end of day 2.
C knows A and B have so A and B should change midnight end of day 2.

None change however as none know they have green eyes only that the other two do. Once you get to day 3 the only conclusion is that every dragon must also be able to see 2 dragons with green eyes, therefore every dragon knows that they all, including themselves, have green eyes.

It doesn't matter how many dragons you add, it simply takes longer for them to work out they have green eyes.

 

[cheesyboy]

If you have 3 dragons, lets call them A,B and C and they all get hear that one dragon has green eyes.
A knows B and C have so B and C should change midnight end of day 2.

Why would B and C know they, themselves, have green eyes, though?

None change however as none know they have green eyes only that the other two do. Once you get to day 3 the only conclusion is that every dragon must also be able to see 2 dragons with green eyes, therefore every dragon knows that they all, including themselves, have green eyes.

It doesn't matter how many dragons you add, it simply takes longer for them to work out they have green eyes.

Or that, from the perspective of the individual (who sees two green-eyes), each of the other dragons can see one green-eyed dragon. So none of them can know they are themselves a green-eye

 

[Greboth]

Why would B and C know they, themselves, have green eyes, though?

They don't know, B doesn't change because B expects A and C, C doesn't change because C expects A and B to change.

Same 3 dragons, if only A has green eyes and B & C have blue then after being told one has green eyes it goes;
A - looks at B and C and knows it is himself.
B - looks at A and C and expects A will change and C wont.
C - looks at A and B and expects A will change and B wont.

Result - midnight day 1, dragon A knows it is himself and becomes a sparrow.

If A and B have green eyes then it goes;
Day 1;
A - knows B has green eyes and C doesn't so expects B to change and C wont
B - knows A has green eyes and C doesn't so expects A to change and C wont
C - know A and B have green eyes so expects both to change.

Day 2 - neither A or B changed as they don't know themselves. Therefore logically the conclusion that must be drawn is
A must conclude B didn't change because B can see another Dragon with green eyes so B didn't know that he himself has green eyes.
B must conclude A didn't change because A can see another Dragon with green eyes so A didn't know that he himself has green eyes.
C still knows A and B have green eyes so expects both to change.

As Dragon A and Dragon B know the other can see another Dragon with green eyes and both know it isn't Dragon C it must be themselves that the other can see. So then the end of day 2 both A and B change.

The same example as above works with 3 with green eyes, it just takes a day longer as midnight day 2 A and B don't change. So follows as per my example in my previous post;
A knows B and C have so B and C should change midnight end of day 2.
B knows A and C have so A and C should change midnight end of day 2.
C knows A and B have so A and B should change midnight end of day 2.

As none changed each dragon can conclude that each can see 2 dragons with green eyes. For each to see 2 it must mean they all have green eyes therefore come night 3 they all know and all change.

Edit: 'knows' is bad wording, should be 'expects'

 

[Tefal]

The problem with the dragon one is it realise on each one always being right and them all trusting each other perfectly.

Which is a harder concept to grasp than there being dragons lol

 

[balky12]

People are getting confused on the dragon one.

Nothing would change.

'You tell them all that at least one of them has green eyes'

Every Dragon would see 99 green eyes. Therefore would not be worried about the colour of their eyes.

For those breaking it down into 3 dragons, exactly the same.

3 dragons all with green eyes.

They get told: 'You tell them all that at least one of them has green eyes'

Dragon 1 sees 2 pairs of green eyes.
Dragon 2 sees 2 pairs of green eyes.
Dragon 3 sees 2 pairs of green eyes.

They wouldn't worry about their own eye colour.

 

[balky12]

Same 3 dragons, if only A has green eyes and B & C have blue then after being told one has green eyes it goes;
A - looks at B and C and knows it is himself.
B - looks at A and C and expects A will change and C wont.
C - looks at A and B and expects A will change and B wont.

Erm. All the dragons have green eyes. Where did yo uget blue from??

You visit a remote desert island inhabited by one hundred very friendly dragons, all of whom have green eyes

 

[Greboth]

Erm. All the dragons have green eyes. Where did yo uget blue from??

Cheeseboy said he can knows why it works with 2 but not 3 or more, I was just running through an example.

People are getting confused on the dragon one.

Nothing would change.

'You tell them all that at least one of them has green eyes'

Every Dragon would see 99 green eyes. Therefore would not be worried about the colour of their eyes.

For those breaking it down into 3 dragons, exactly the same.

3 dragons all with green eyes.

They get told: 'You tell them all that at least one of them has green eyes'

Dragon 1 sees 2 pairs of green eyes.
Dragon 2 sees 2 pairs of green eyes.
Dragon 3 sees 2 pairs of green eyes.

They wouldn't worry about their own eye colour.

It is the fact that Dragons don't change when expected that leads to the only logical conclusion that means they all chance. Each dragon expects the other 2 to change midnight day 2, none do thought because none know themselves. Therefore day 3 each dragon must conclude that each dragon can see 2 pairs of green eyes. For that to be true the dragon then knows them he himself has green eyes.

 

[MassiveJim]

If you have 3 dragons, lets call them A,B and C and they all get hear that one dragon has green eyes.
A knows B and C have so B and C should change midnight end of day 2.
B knows A and C have so A and C should change midnight end of day 2.
C knows A and B have so A and B should change midnight end of day 2.

None change however as none know they have green eyes only that the other two do. Once you get to day 3 the only conclusion is that every dragon must also be able to see 2 dragons with green eyes, therefore every dragon knows that they all, including themselves, have green eyes.

It doesn't matter how many dragons you add, it simply takes longer for them to work out they have green eyes.

This theory relies on the dragons realising they have green eyes though.
They surely cannot.
As Dragon 1 of 4 (the idea works for 2 or 3), I can see 3 other dragons with green eyes, so I know that each of those dragons can see at least two other dragon with green eyes, therefore can assume that as they can see two dragons with green eyes, that satisfies the "at least one dragon" rule, that dragon cannot work out that they are "the one" as he can also see another two dragons with green eyes. And no one changes because the dragon can never confirm that they are the one simply because they can see more than 2 dragons with green eyes. And knows that the other dragon would never change as they can see at least 2 other dragons with green eyes.


[EDIT] I think the "logic" behind the 100 day theory is that every dragon can see 99 pairs of green eyes and that after 100 days if no one has changed it must mean everyone has green eyes.
This logic is flawed though, other dragons haven't changed because they don't know they have green eyes, and they know that the other dragons can't know for sure either.

 

[cheesyboy]

They don't know, B doesn't change because B expects A and C, C doesn't change because C expects A and B to change.

Same 3 dragons, if only A has green eyes and B & C have blue then after being told one has green eyes it goes;
A - looks at B and C and knows it is himself.
B - looks at A and C and expects A will change and C wont.
C - looks at A and B and expects A will change and B wont.

Result - midnight day 1, dragon A knows it is himself and becomes a sparrow.

If A and B have green eyes then it goes;
Day 1;
A - knows B has green eyes and C doesn't so expects B to change and C wont
B - knows A has green eyes and C doesn't so expects A to change and C wont
C - know A and B have green eyes so expects both to change.

Day 2 - neither A or B changed as they don't know themselves. Therefore logically the conclusion that must be drawn is
A must conclude B didn't change because B can see another Dragon with green eyes so B didn't know that he himself has green eyes.
B must conclude A didn't change because A can see another Dragon with green eyes so A didn't know that he himself has green eyes.
C still knows A and B have green eyes so expects both to change.

As Dragon A and Dragon B know the other can see another Dragon with green eyes and both know it isn't Dragon C it must be themselves that the other can see. So then the end of day 2 both A and B change.

The same example as above works with 3 with green eyes, it just takes a day longer as midnight day 2 A and B don't change. So follows as per my example in my previous post;
A knows B and C have so B and C should change midnight end of day 2.
B knows A and C have so A and C should change midnight end of day 2.
C knows A and B have so A and B should change midnight end of day 2.

As none changed each dragon can conclude that each can see 2 dragons with green eyes. For each to see 2 it must mean they all have green eyes therefore come night 3 they all know and all change.

Edit: 'knows' is bad wording, should be 'expects'

But your 2nd example doesn't work when applied to the scenario we have, because Dragon C is unable to come to come to a conclusion on its own eyes until the other dragons have changed into sparrows. And they can't do that if they both see 2 green-eyed dragons, because they know that each of the others can see at least one green-eyed dragon, which satisfied the "at least one of you has green eyes" without the individual having to accept that he himself has green eyes.

 

[Gruber]

People are getting confused on the dragon one.

Nothing would change.

'You tell them all that at least one of them has green eyes'

Every Dragon would see 99 green eyes. Therefore would not be worried about the colour of their eyes.

For those breaking it down into 3 dragons, exactly the same.

3 dragons all with green eyes.

They get told: 'You tell them all that at least one of them has green eyes'

Dragon 1 sees 2 pairs of green eyes.
Dragon 2 sees 2 pairs of green eyes.
Dragon 3 sees 2 pairs of green eyes.

They wouldn't worry about their own eye colour.

This. The dragons can work it out if there's only 2 of them. When there is 3 or more they don't have enough information to conclude anything and carry on living in ignorant bliss.

 

[Greboth]

This theory relies on the dragons realising they have green eyes though.
They surely cannot.
As Dragon 1 of 4 (the idea works for 2 or 3), I can see 3 other dragons with green eyes, so I know that each of those dragons can see at least two other dragon with green eyes, therefore can assume that as they can see two dragons with green eyes, that satisfies the "at least one dragon" rule, that dragon cannot work out that they are "the one" as he can also see another two dragons with green eyes. And no one changes because the dragon can never confirm that they are the one simply because they can see more than 2 dragons with green eyes. And knows that the other dragon would never change as they can see at least 2 other dragons with green eyes.

So you as Dragon 1 think I wont change because I haven't got green eyes and but I know the other 3 have green eyes and expect all 3 to change. Dragon 2, 3 and 4 all come to the same conclusion though so don't change. Therefore, again as Dragon 1, you must conclude that you were wrong in the beginning and you also have green eyes. The day at which Dragon 1 realises this 2, 3 and 4 also do too.

Edit: I give up trying to explain this

 

[Rroff]

You can keep the rolleyes. You said nothing would change, which is wrong. Now you're saying the answer on the site is wrong and not logical, I suspect (although I haven't read it) it is both logical and correct. Please, do keep pushing your point though, you clearly have infallible logic yourself.

The answer on the site is both logical (highly so) and "correct" but IMO a little contrived - there could arguably be more than one sound line of reasoning applied to the situation but they are looking for a specific one as the correct answer.

 

[cheesyboy]

So you as Dragon 1 think I wont change because I haven't got green eyes and but I know the other 3 have green eyes and expect all 3 to change. Dragon 2, 3 and 4 all come to the same conclusion though so don't change. Therefore, again as Dragon 1, you must conclude that you were wrong in the beginning and you also have green eyes. The day at which Dragon 1 realises this 2, 3 and 4 also do too.

Edit: I give up trying to explain this

Dragon1 wouldn't expect the others to change, though, as he knows logically that they can't themselves know if they are green-eyed.

 

[DaleTheWhale]

Where people are getting confused is:

The dragons don't change until they know their own eye colour & that the information you have given to them is not something they didn't know about others, but as they're logical they then realise that 1 dragon has to have green eyes, thus should change at midnight however does not, come the 99th day, each dragon realises that ITSELF has green eyes because the others have yet to change, and all 100 of them change at the same time.

 

[balky12]

So you as Dragon 1 think I wont change because I haven't got green eyes and but I know the other 3 have green eyes and expect all 3 to change. Dragon 2, 3 and 4 all come to the same conclusion though so don't change. Therefore, again as Dragon 1, you must conclude that you were wrong in the beginning and you also have green eyes. The day at which Dragon 1 realises this 2, 3 and 4 also do too.

Edit: I give up trying to explain this

Say there are 3 dragons.

Dragon 1: Sees two pairs of green eyes and thinks It's not me I won't change. But he knows Dragon 2 and Dragon 3 can see each others green eyes. So why would they change? It doesn't conclude that Dragon 1 has geen eyes.

Remember the clue is atleast one of you has green eyes.

Dragon 2: Sees two pairs of green eyes and thinks It's not me I won't change. But he knows Dragon 1 and Dragon 3 can see each others green eyes. So why would they change? It doesn't conclude that Dragon 2 has geen eyes.

Dragon 3: Sees two pairs of green eyes and thinks It's not me I won't change. But he knows Dragon 1 and Dragon 2 can see each others green eyes. So why would they change? It doesn't conclude that Dragon 3 has geen eyes.

They have to conclude that they themselves have green eyes. They would just live in ignorance.

 

[cheesyboy]

Why would 100 dys make a difference?

What new information can logically be gained on day 2, 3 , 4 etc?

As far as I see, day one, you see a load of green-eyed dragons, and know that the other dragons all see a load of green-eyed dragons. You wouldn't expect anyone to change into a sparrow, because they are in the same boat as you: you don't know your own eye-colour, and you can see at least one green-eyed dragon.

Day 2 changes nothing. day 100 changes nothing

 

[Greboth]

Say there are 3 dragons.

Dragon 1: Sees two pairs of green eyes and thinks It's not me I won't change. But he knows Dragon 2 and Dragon 3 can see each others green eyes. So why would they change? It doesn't conclude that Dragon 1 has geen eyes.

Remember the clue is atleast one of you has green eyes.

Dragon 2: Sees two pairs of green eyes and thinks It's not me I won't change. But he knows Dragon 1 and Dragon 3 can see each others green eyes. So why would they change? It doesn't conclude that Dragon 2 has geen eyes.

Dragon 3: Sees two pairs of green eyes and thinks It's not me I won't change. But he knows Dragon 1 and Dragon 2 can see each others green eyes. So why would they change? It doesn't conclude that Dragon 3 has geen eyes.

The question isn't 'Why would they change?' it is 'Why wouldn't they change?'. The only reason each dragon wouldn't change is because each can see the same number of green eyes so assumes they don't have green themselves. For this to happen all of them must have green eyes.

 

[DaleTheWhale]

Dragon1 wouldn't expect the others to change, though, as he knows logically that they can't themselves know if they are green-eyed.

3 Dragon Scenario:

After 2 days no dragons would have changed, so each would then think to themselves I must have green eyes because I have been given the information of AT LEAST ONE DRAGON HAS GREEN EYES, thus meaning they change at midnight.

Each dragon can see the other green eyes

Day #1 - Dragon 1 expects Dragon 2 and 3 to change, they do not.
Day #1 - Dragon 2 expects Dragon 1 and 3 to change, they do not.
Day #1 - Dragon 3 expects Dragon 1 and 2 to change, they do not.

Day #2 - Dragon 1 expects Dragon 2 and 3 to change, they do not.
Day #2 - Dragon 2 expects Dragon 1 and 3 to change, they do not.
Day #2 - Dragon 3 expects Dragon 1 and 2 to change, they do not.

Day #3 - Dragon 1 then realises well at least 1 dragon has to have green eyes and change, that must be me.
Day #3 - Dragon 2 then realises well at least 1 dragon has to have green eyes and change, that must be me.
Day #3 - Dragon 3 then realises well at least 1 dragon has to have green eyes and change, that must be me.

I cannot put this any simpler.