The Paradox of the Muddy Children

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Rhombic
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The Paradox of the Muddy Children

Postby Rhombic » Sat Feb 21, 2015 3:48 pm UTC

Three children return home after playing outside, and their father tells them that at least one of them has a muddy face. He repeats the phrase “Step forward if you have a muddy face” until all and only the children with muddy faces have stepped forward.

If there’s only one child with a muddy face, then she’ll step forward immediately — she can see that no other children have muddy faces, so her father must be talking about her. Each of the other children will see her muddy face and stand fast, since they have no way of knowing whether their own faces are muddy.

If there are two children with muddy faces, then no one will step forward after the first request, since each might think the father is addressing the other one. But when no one steps forward after the first request, each will realize that there must be two children with muddy faces, and that she herself must be one of them. So both will step forward after the second request, and the rest will stand fast.

A pattern emerges: If there are n children with muddy faces, then n will step forward after the nth request.

But now imagine a scenario in which more than one of the children has a muddy face, but the father does not tell them that at least one of them has a muddy face. Now no one steps forward after the first request, for the same reason as before. But no one steps forward at the second request either, because the fact that no one stepped forward after the first request no longer means that there is more than one child with a muddy face.

This is perplexing. In the second scenario all the children can see that at least one of them has a muddy face, so it seems needless for the father to tell them so. But without his statement the argument never gets going; despite his repeated requests, no child will ever step forward. What’s missing?

[Source: Futility Closet]

Is this really a paradox or is there any definite answer?

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Xenomortis
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Re: The Paradox of the Muddy Children

Postby Xenomortis » Sat Feb 21, 2015 3:58 pm UTC

Replace "muddy face" with "blue eyes".
Replace "father" with "guru".
Pretend they're on an island.
Read this thread: viewtopic.php?f=3&t=3

There's no paradox, the concept you're looking for is Common Knowledge.
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Rhombic
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Joined: Wed Jan 01, 2014 11:42 pm UTC

Re: The Paradox of the Muddy Children

Postby Rhombic » Sat Feb 21, 2015 10:37 pm UTC

Xenomortis wrote:Replace "muddy face" with "blue eyes".
Replace "father" with "guru".
Pretend they're on an island.
Read this thread: viewtopic.php?f=3&t=3

There's no paradox, the concept you're looking for is Common Knowledge.


This:
But now imagine a scenario in which more than one of the children has a muddy face, but the father does not tell them that at least one of them has a muddy face. Now no one steps forward after the first request, for the same reason as before. But no one steps forward at the second request either, because the fact that no one stepped forward after the first request no longer means that there is more than one child with a muddy face.

This is perplexing. In the second scenario all the children can see that at least one of them has a muddy face, so it seems needless for the father to tell them so. But without his statement the argument never gets going; despite his repeated requests, no child will ever step forward. What’s missing?

Xias
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Re: The Paradox of the Muddy Children

Postby Xias » Sun Feb 22, 2015 12:24 am UTC

Rhombic wrote:
Xenomortis wrote:Replace "muddy face" with "blue eyes".
Replace "father" with "guru".
Pretend they're on an island.
Read this thread: viewtopic.php?f=3&t=3

There's no paradox, the concept you're looking for is Common Knowledge.


This:
But now imagine a scenario in which more than one of the children has a muddy face, but the father does not tell them that at least one of them has a muddy face. Now no one steps forward after the first request, for the same reason as before. But no one steps forward at the second request either, because the fact that no one stepped forward after the first request no longer means that there is more than one child with a muddy face.

This is perplexing. In the second scenario all the children can see that at least one of them has a muddy face, so it seems needless for the father to tell them so. But without his statement the argument never gets going; despite his repeated requests, no child will ever step forward. What’s missing?



The connection that Xenomortis was making was with the fact that no islanders can leave until the Guru speaks, and the reason the Guru speaking changes anything at all has been discussed at length on these forums already.

Basically:

Spoiler:
Call the kids A, B, and C, and let's say that all of them are muddy. A can see that B and C are muddy, and so A's knowledge is that there are either 2 or 3 muddy children. B sees the same but with A and C; C sees the same with A and B. So the following statement is known by everyone:

"There are at least 2 muddy children."

Consider what A knows that B knows. When A imagines what B sees, B can either see 1 muddy child (if A is not muddy) or 2 muddy children (if A is muddy). So A knows that "B knows that there are either 1, 2, or 3 muddy children." A can do the same with C; B can do the same with A and C; C can do the same with A and B. So the following statement is known by everyone:

"Everyone knows that there is at least 1 muddy child."

While everyone knows that 1 is not a possibility, not everyone knows that everyone knows that.

Consider what A knows that B knows that C knows. A knows that B at least sees one muddy child, but it's possible that A is not muddy, so B could be considering the possibility that B is not muddy, and therefore C would see no muddy children. So from B's perspective within A's perspective, C could be considering the possibility that there are 0 muddy children. Again, this applies to every knowledge chain of three children, so the following statement is known by everyone:

"Everyone knows that everyone knows that there are at least 0 muddy children."

And that's where things break down. Since within the nested hypotheticals of what A knows that B knows that C knows, 0 muddy children is a possibility until the father removes that possibility.

To take it from another angle: You explain the solution by induction, which requires the children to understand that if there was only one muddy child, he would step forward when the father speaks because he would know that it must be him. This is how two muddy children would step forward at step 2, and how three would step forward at step 3. However, what if there is only one muddy child, and the father doesn't tell them that at least one of them is muddy? As far as that one child knows, he could be muddy, or he could not be muddy, and the father saying "If you are muddy, step forward" isn't enough to deduce that he has a muddy face. So he never steps forward. Since a single child would never step forward, at what point would two children decide to step forward? There is no beginning to the induction process.

twinsen
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Re: The Paradox of the Muddy Children

Postby twinsen » Mon Feb 23, 2015 10:02 am UTC

It isnt a paradox at all.

In 10 people, 3 mud face will know that at least 2 are mud, and 7 will know that at least 3 are mud. In hive mind you see "2 or more"
In 10 people, 2 mud face will know that at least 1 are mud, and 8 will know that at least 2 are mud. In hive mind you see "1 or more"
In 10 people, 1 mud face will know that at least 0 are mud, and 9 will know that at least 1 are mud. In hive mind you see "0 or more"
In 10 people, 0 mud face will know that at least 0 are mud, and 10 will know that at least 0 are mud. In hive mind you see "0 or more"

In general, setting the hive mind to 1 or more, removes the cases of indistinct information.


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