What happens if you de-discretize time in the blue eyes riddle?

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agelessdrifter
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What happens if you de-discretize time in the blue eyes riddle?

Postby agelessdrifter » Sat Mar 04, 2017 11:51 pm UTC

I assume most of the regulars are familiar with the blue eyes problem already, but if not, it's the first result on google for "blue eyes riddle," and it's an xkcd url. I'll talk about the solution below, so don't read past here if you want to solve it yourself.

In the problem, the people on the island disappear from the island at midnight of the day they learn their own eye color, and this is sort of the key to the riddle; each day everyone wakes up and sees who's still on the island, and this information helps them decide their own eye color.

But what if the condition was instead that whoever learned their own eye color disappears instantaneously?

My first thought is that they'd just all be stuck, but then I think that at some point the fact that no one has left the island must convey something. Is there a way to make sense of this problem? Does it have an answer, or does this new condition render the question ill-formed?

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Eebster the Great
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Re: What happens if you de-discretize time in the blue eyes riddle?

Postby Eebster the Great » Sun Mar 05, 2017 2:27 pm UTC

You would need to restate the problem in a continuous form to get a specific answer. There is more than one thing you could mean here because the original puzzle wasn't worded very formally.

A major problem is that if they vanish instantly as you say (and everyone sees this instantly), and moreover, being perfect logicians, they make all deductions instantly, then any step in any finite chain of reasoning must take no time at all, and therefore neither does the entire chain. So if there exists a solution, that solution must only have one group of zero or more people who instantly leave and one group of zero or more people who never leave. There could be no other groups.

However, for an uncountably inifnite number of islanders, this might be interesting.

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Re: What happens if you de-discretize time in the blue eyes riddle?

Postby Tub » Tue Mar 07, 2017 12:03 pm UTC

It's funny, because time is literally the only thing not required to be discrete in the riddle.

The people are discrete.
The information they have is discrete.
Their chains of reasoning are discrete.
The ferry dispenses discrete amounts of information.

The solution hinges on alternating phases of a) new information from the ferry (the amount of islanders who left) and b) everyone updating their knowledge and chains of reasoning based on the information before the next ferry arrives.

You cannot de-discretize phase a). The ferry cannot continuously dispense arbitrarily small amounts of information.
You cannot de-discretize phase b). The chains of reasoning cannot be continuously extended.

You cannot simply collapse the whole process to be instantaneous either, because the solution requires a point in time where exactly 99, but not 100 ferrys have arrived. Imagine, in the original riddle, that 100000 ferrys arrive the first night, simultaneously. Nobody would be able to deduce anything.


Time can be continuous, the riddle only requires alternating phases happening at increasing points in time. With continuous time, it's just that most of the time, nothing interesting happens (just like in real life!).
The ferry, the islanders, the information, the chains of reasoning, they'll remain discrete.

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Re: What happens if you de-discretize time in the blue eyes riddle?

Postby gmalivuk » Tue Mar 07, 2017 8:32 pm UTC

Yes, we (including the OP) know very well that time is not literally discrete in the original puzzle, but nonetheless the original puzzle includes discrete time steps and the OP is asking what happens if that isn't the case.
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Re: What happens if you de-discretize time in the blue eyes riddle?

Postby Caenbe » Fri Mar 10, 2017 12:35 am UTC

Instead of allowing an opportunity to leave every day, allow one every hour. Then every minute, every second, every nanosecond... approaching the continuous case as a limit. The implication is that everyone with blue eyes deduces so and leaves instantaneously. Could this idea be made rigorous?

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Eebster the Great
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Re: What happens if you de-discretize time in the blue eyes riddle?

Postby Eebster the Great » Fri Mar 10, 2017 9:27 am UTC

Caenbe wrote:Instead of allowing an opportunity to leave every day, allow one every hour. Then every minute, every second, every nanosecond... approaching the continuous case as a limit. The implication is that everyone with blue eyes deduces so and leaves instantaneously. Could this idea be made rigorous?

They cannot all leave instantly, since they would not yet have any information. But they can't wait either, because after any finite time, they would already know their eye color. It's just not a coherent puzzle anymore. It's a little like asking "Bob is not infinitely fast but can run at any finite speed. What is the minimum time in which he can run a marathon?" Clearly there is no answer.

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Re: What happens if you de-discretize time in the blue eyes riddle?

Postby rmsgrey » Sat Mar 11, 2017 3:55 am UTC

Caenbe wrote:Instead of allowing an opportunity to leave every day, allow one every hour. Then every minute, every second, every nanosecond... approaching the continuous case as a limit. The implication is that everyone with blue eyes deduces so and leaves instantaneously. Could this idea be made rigorous?


In general, the limit of a property is not the property of the limit.

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Re: What happens if you de-discretize time in the blue eyes riddle?

Postby Caenbe » Sat Mar 11, 2017 6:19 pm UTC

rmsgrey wrote:In general, the limit of a property is not the property of the limit.

Not if it's undefined, it's not. Why not define it to be so?
I said "deduce" in my post, and I realize now that that was a big mistake, because that's not what I meant at all.

I agree with Eebster; there is no chain of deductions that makes sense. Not wanting this to be the end of it, I tried something analogous to analytic continuation: just look at the islanders' behavior, and figure out how it would extend naturally to the new problem. Maybe it would lead to a new form of logic that would make sense of this new behavior. This sort of thing has happened before, after all.

Those were my thoughts at the time, and I'm not sure if they're a little misguided or very misguided. If there actually is a "new logic" to be found here, it isn't going to be based on my limiting process, because there's a generalization of the original problem where it breaks down.

Get rid of the ferry, and give everyone a jetpack instead. Each jetpack has a timer that periodically allows its owner to leave. Specifically, islander i can leave at m*T_i for any integer m. For limit purposes, the n-tuple of all the islanders' T's can be mapped to a point in R+^n. Similarly, the behavior just consists of the n-tuple of L_i's when the islanders leave, one or more of which can be undefined if they never leave.

Let's think about 2 islanders, both with blue eyes.
If T_1>T_2, their behavior is L=(T_1, never). This just follows from their deductions. You can check it for yourself.
If T_1=T_2, just like in the original puzzle, L=(2*T_1, 2*T_2).
So, if T approaches (0,0), the limit of L depends on the direction it comes from in R+^2. Overall, the limit does not exist.
It would be interesting to work this out when n>2. But that's getting off-topic, and anyway, I'm pretty sure the limit still wouldn't exist.

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Re: What happens if you de-discretize time in the blue eyes riddle?

Postby rmsgrey » Sun Mar 12, 2017 1:12 am UTC

Caenbe wrote:
rmsgrey wrote:In general, the limit of a property is not the property of the limit.

Not if it's undefined, it's not. Why not define it to be so?

Because it tends to lead to weirder consequences than just saying you can't do it. If you want, you could define 1/0 to be equal to 3 - nothing's stopping you - but I'd regard it as an unwise choice because, while it lets you divide 1 by 0, it screws up algebra that relates to that result - multiply both sides by 0 and you get 1=0...

So we try to assign definitions that give things the properties and interrelationships we expect from analogy with familiar mathematics rather than just picking up one property. When we can't extend things consistently, it's best to leave them undefined...

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Re: What happens if you de-discretize time in the blue eyes riddle?

Postby Caenbe » Sun Mar 12, 2017 5:07 pm UTC

rmsgrey wrote:
Caenbe wrote:
rmsgrey wrote:In general, the limit of a property is not the property of the limit.

Not if it's undefined, it's not. Why not define it to be so?

Because it tends to lead to weirder consequences than just saying you can't do it. If you want, you could define 1/0 to be equal to 3 - nothing's stopping you - but I'd regard it as an unwise choice because, while it lets you divide 1 by 0, it screws up algebra that relates to that result - multiply both sides by 0 and you get 1=0...

So we try to assign definitions that give things the properties and interrelationships we expect from analogy with familiar mathematics rather than just picking up one property. When we can't extend things consistently, it's best to leave them undefined...

I understand if you didn't read my whole post. I wanted to write all my thoughts down, and it ended up pretty confusing. I should have taken extra care with the first sentence too, but I liked its snappiness too much. I basically agree with you.


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