Twins in a Maze

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EdgarJPublius
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Re: Twins in a Maze

Postby EdgarJPublius » Tue Feb 15, 2011 6:18 pm UTC

No idea, which is why I think the second option (agree to a meeting place) is more likely.

There ought to be a way to imagine a coordinate system that both twins can identify independently of their starting positions. (I'm thinking each twin starts by defining a simple coordinate system which is therefore mirrored by their starting positions, and then combining them using some function or collection of functions that will create a uniform coordinate system)
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Re: Twins in a Maze

Postby t1mm01994 » Thu Feb 17, 2011 5:30 pm UTC

EdgarJPublius wrote:No idea, which is why I think the second option (agree to a meeting place) is more likely.

There ought to be a way to imagine a coordinate system that both twins can identify independently of their starting positions. (I'm thinking each twin starts by defining a simple coordinate system which is therefore mirrored by their starting positions, and then combining them using some function or collection of functions that will create a uniform coordinate system)

You need one of two things here. You either need a random number or a non-rotationally symmetric maze with no uniquely featured points.

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Re: Twins in a Maze

Postby balr » Thu Feb 17, 2011 9:41 pm UTC

The problem states:

Code: Select all

You and your twin know that, being identical in mind and body, each will arrive at the same method as the other


Therefore we will both arrive at the same method, X.

For us to succeed we must break the symmetry, and have one of us implement it as X while the other implements it as X' (ie the inverse of X).

Luckily we do have an infallible way of breaking that symmetry, and it is a way we both know, so it is within the remit of the problem:
Spoiler:
We know one of us is the elder, and one is the younger. So we know that the elder will implement the method as X while the younger will implement X'


Now the problem is to devise a maze that thwarts that solution!

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Re: Twins in a Maze

Postby EdgarJPublius » Fri Feb 18, 2011 5:33 am UTC

t1mm01994 wrote:
EdgarJPublius wrote:No idea, which is why I think the second option (agree to a meeting place) is more likely.

There ought to be a way to imagine a coordinate system that both twins can identify independently of their starting positions. (I'm thinking each twin starts by defining a simple coordinate system which is therefore mirrored by their starting positions, and then combining them using some function or collection of functions that will create a uniform coordinate system)

You need one of two things here. You either need a random number or a non-rotationally symmetric maze with no uniquely featured points.



No, I've been dancing around the solution in my head for a while, but I finally figured it out in the shower a little while ago.
Spoiler:
Each twin can map the rotationally symmetrical maze onto a line and these lines can be overlayed so that at last some points on the line will be the same point of the maze for both twins
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I wrote:Does Space Teddy Roosevelt wrestle Space Bears and fight the Space Spanish-American War with his band of Space-volunteers the Space Rough Riders?

Yes.

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Re: Twins in a Maze

Postby t1mm01994 » Fri Feb 18, 2011 5:50 pm UTC

EdgarJPublius wrote:-snip-
Spoiler:
Each twin can map the rotationally symmetrical maze onto a line and these lines can be overlayed so that at last some points on the line will be the same point of the maze for both twins

Well, sadly, I think not.
Spoiler:
As both persons do exactly the same thing, they will always make the same movements, at any time. So if one of the two "moves the line", so does the other, in the same way, resulting in the same movements, just mirrored. In a circle, with no random device, they will do exactly the same, as they are the same, molecule to molecule, or whatever OP says.

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Re: Twins in a Maze

Postby EdgarJPublius » Fri Feb 18, 2011 8:44 pm UTC

The problem indicates that the twins will arrive at the same method of navigating the maze, not that their movements will always be identical or otherwise mirrored.

Spoiler:
If you break the symmetry of the maze, you also break the symmetry of the twins movements.
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Re: Twins in a Maze

Postby jaap » Fri Feb 18, 2011 11:22 pm UTC

EdgarJPublius wrote:The problem indicates that the twins will arrive at the same method of navigating the maze, not that their movements will always be identical or otherwise mirrored.

If they have the same (deterministic) method of navigating, then given the same input (such as from opposite sides of a rotationally symmetric maze), their movements will be the same.

EdgarJPublius wrote:
Spoiler:
If you break the symmetry of the maze, you also break the symmetry of the twins movements.
I'm sure everyone here agree with that by now, but I don't see how this...
EdgarJPublius wrote:
Spoiler:
Each twin can map the rotationally symmetrical maze onto a line and these lines can be overlayed so that at last some points on the line will be the same point of the maze for both twins
...does that.
Spoiler:
They have not created a global coordinate system, they have each just created a new version of their local coordinate system. To get a shared global coordinate system you need shared information from outside their otherwise identical maze sensory input, e.g. an outside landmark or use of a compass.
If you could overlay the two local coordinate systems in opposite directions, then they would meet because one twin would be going clockwise, the other anti-clockwise. This is not possible for them however. Just saying that it is possible to overlay the two local coordinate systems to make it work, does not mean that the twins each individually have enough information to do so - you need information from both twins to do this, and then to relay that info back to the twins to break their symmetry. As the twins cannot communicate, this cannot work as long as they have identical sensory input.

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Re: Twins in a Maze

Postby charonme » Sun Mar 06, 2011 1:51 pm UTC

Why do we need a maze? Isn't it enough to put them on a circle? I don't believe they can find each other on a circle without a source of randomness.

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Re: Twins in a Maze

Postby Yakk » Sun Mar 06, 2011 2:57 pm UTC

A circle is a maze, if a trivial one. It is rotationally symmetric as well. ;)
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Re: Twins in a Maze

Postby charonme » Sun Mar 06, 2011 4:27 pm UTC

In case of a circle, the symmetry alone is not sufficient for the impossibility of their inability to meet without randomness. The key is they can't go to some special location (eg. a centroid) because it's off all the accessible paths.

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Re: Twins in a Maze

Postby Peewee223 » Fri Apr 29, 2011 5:48 am UTC

Spoiler:
I've never met twins with the same name. In the case of a rotationally symmetric maze, simply go clockwise if my name comes first alphabetically, and counterclockwise if my name comes second alphabetically.

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Re: Twins in a Maze

Postby charonme » Mon May 02, 2011 12:29 pm UTC

I know this might be considered splitting hairs, but
6453893 wrote:being identical in mind and body
would probably mean that
Spoiler:
both having differing knowledge in the form of "my name (or age or any other property) is ... and the name of my other twin is ..." in their minds (or bodies)
would be a violation of this rule

but otherwise I like this solution very much

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Re: Twins in a Maze

Postby Devoninja » Tue May 10, 2011 8:53 pm UTC

Here's a thought.
Spoiler:
i can think of at least 1 randomizing factor which quickly simplifies the problem and is not reliant on external factors.

The older twin performs a comprehensive search while the younger twin waits in place. Rather than hoping for a coin or die, use natural randomness. Also one searching and the other waiting works out better than both randomly searching.

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Re: Twins in a Maze

Postby jestingrabbit » Wed May 11, 2011 1:29 am UTC

Okay, I've edited the OP and I'm going to say it here as well.

Please think of the twins as completely identical. Maybe they're identical robots, or clones, or whatever. Identical, in this context, means completely identical. Don't come up with another difference => solution post, please.
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Re: Twins in a Maze

Postby balr » Wed May 11, 2011 7:02 am UTC

But there is always going to be a symmetry breaker -- if not in the twins themselves (their names or ages as previously discussed) then in their
Spoiler:
initial position or orientation in the maze.


Developing that spoiler further -- consider the example given of a perfectly circular maze and the twins dropped at two random points in the maze. With identical strategies and no symmetry breaker, then yes they'd likely follow each other in the same direction, never catching up or meeting. But....
Spoiler:
....One of them started further North or further East than the other. Or one started facing North while their symmetrically-placed twin started facing South. Given they know their starting cardinal orientation, they can devise a strategy based on that.

Even in a more elaborate maze, initial random-yet-identical explorations will allow each twin to deduce if they started in the Northern half or the Southern half of the maze. Once they know that, the symmetry is broken, and a non-symmetrical strategy can be applied.

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Re: Twins in a Maze

Postby jestingrabbit » Wed May 11, 2011 7:12 am UTC

balr wrote:stuff


Okay, yes, in any real world example of this puzzle, you will likely be able to work something out to break the symmetry. But these are what I call "engineering concerns". They relate to real world implementations of logic puzzle like situations, not the situations and rules that logic puzzles are asking you to think about. Solving a logic puzzle using engineering concerns doesn't solve the logic puzzle, it evades the logic puzzle.

The logic puzzles forum isn't interested in evading puzzles, its interested in describing and solving them.
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Re: Twins in a Maze

Postby charonme » Wed May 11, 2011 10:03 am UTC

balr, your solution does not work if the maze is placed on an infinite featureless plane...
or even on earth if it is placed on a circle of latitude without a view of stars

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Re: Twins in a Maze

Postby balr » Wed May 11, 2011 1:26 pm UTC

I don't want to hijack this thread into a discussion on logical puzzles vs engineering puzzles … so please feel free to cut out this post and its context and park somewhere else.

But many logic puzzles are also engineering puzzles.

Consider, for example, the whole genre of coin piles that have to be partitioned using a limited number of observations on scales or balances.....

....They can be pushed to subtler engineering solutions by, one-by-one, striking out assumed physical givens (no scales; in space so no gravity; under inconstant acceleration and rotation so balances won't work; other shenanigans to eliminate density comparisons).

....Eventually, the cleverer solvers will be pushed to find a solution using only the axioms of set theory, or they'll have to conclude that some engineering given is needed for a solution to be possible.

Whether this puzzle should be treated as a pure logic one or not has been decided by our esteemed moderator. Logic only is the rule.

Otherwise, I would rule charonme's requirement ("...infinite featureless plane...") out as incompatible with the original stipulation ("You and your identical twin are dropped …") (or, at least, I'd want a demonstration of how such a drop is possible first). And I'd deal with the other sticking point ("placed on a circle of latitude without a view of stars") in various symmetry-breaking ways (from latency in pinging NIST time servers through to angle of solar neutrino flow).

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Re: Twins in a Maze

Postby charonme » Wed May 11, 2011 4:18 pm UTC

balr wrote:angle of solar neutrino flow
haha I love this one

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Re: Twins in a Maze

Postby Trebla » Fri May 13, 2011 12:26 pm UTC

The simplest maze (in my mind) seems to be the surface of a sphere (or perhaps the edge of a circle as mentioned before, contriving the absence of absolutes such as 'north' as being 'indeterminable'). The twins are placed on opposite points with the same orientation (to keep it symmetrical, say they're both on an equator facing east, one at 90E, one at 90W).

Assuming the twins know the rules of the game and are completely identical (robots made more sense than twins), it seems to me they'd have to devise an algorithm based on movements relative to absolute features of the environment (e.g., I'll go north - no matter where they are, if they both go north they'll meet at the pole). If we say that's an "engineering" solution rather than a "logical" solution (e.g., if this sphere isn't rotating and has no concept of north, perhaps it's contained in a featureless atmosphere providing uniform light) and their decisions can only decide movements relative to their position and orientation, then each twin will necessarily stay opposite the other.

Now, if only someone could develop a proof of that...
Edit: Or a dispute of such a proof.

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Re: Twins in a Maze

Postby balr » Sat May 14, 2011 10:19 am UTC

If we're on a featureless sphere orbiting a featureless star, then
Spoiler:
I'd fly to the L2 Lagrange point -- as would my twin robot; after we've used our unlimited time to develop space flight...If I can see the sky, then it's part of the maze by the original definition, that that's a logical solution; even if a somewhat impractical engineering one.


If a featureless sphere and no visible star, then
Spoiler:
I'd drill to the center of the sphere -- as would my twin after we've amassed the drilling equipment; again logical but impractical.


If you ban both those approaches then I'd
Spoiler:
map the whole maze complex and then proceed to its centre of mass. If that centre is below ground, then some practcial drilling equipment would be needed. If it is in an inaccessible part of the maze, then I'd need some very modest demolition equipment .... well within the range of a robot with unending energy supplies


And, if you won't let me do that -- just tell me it's a featurless, inescapable, undiggable, sphere, then
Spoiler:
I'd spread out to eventually cover up to 50% of its surface; at some point my expanding body would connect with my twin's.


And if all that is banned then
Spoiler:
I'd develop hyper-sensitive sensors and detect the faint vibrations or gravity disturbances cause by my twin's existance, and head towards those.

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Re: Twins in a Maze

Postby jestingrabbit » Sat May 14, 2011 7:07 pm UTC

balr wrote:If a featureless sphere and no visible star, then
Spoiler:
I'd drill to the center of the sphere -- as would my twin after we've amassed the drilling equipment; again logical but impractical.


This is pretty much exactly why I think that this sort of thinking is full of fail. You don't have to evade the puzzle on a sphere, you can just solve it.

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Re: Twins in a Maze

Postby Dr. Willpower » Mon May 16, 2011 4:32 pm UTC

Spoiler:
I think the question comes down to how complex their methods can be for finding each other. If they have to use a simple method (i.e. go right first, left second), they'll probably never find each other on the symmetrical maze. If they can make a map of the maze then I think it would be easily possible for them to introduce a random element into their strategies (well it depends, human brains have a degree of randomness to them, the question really is how identical are they, robots might arrive at the same random solution [if they're really identical]).

There are a few possibilities:
1. The maze is finite, but on a sphere (or some other shape which can facilitate endless non-trivial maze loops).
They'd try to map it and wander around until they eventually found each other (with infinite time).
2. The maze is finite, but with a geometric shape in the center around which they must walk to find each other.
They either can or cannot use random decisions. If not, they'll probably never find each other (the simplest case is a circle around which they will always walk the same direction, thus never getting any closer to each other).
3. The maze is finite, there is a central point.
They go to the central point and begin re-exploring to find the new opening for the exit.

I'm leaning towards a no, depending on whether or not they can be random.
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Re: Twins in a Maze

Postby blah28722 » Mon May 16, 2011 8:52 pm UTC

If time is truly a non-issue, and randomness is possible:

Spoiler:
Have both twins travel in a semi-random fashion. For example, spinning in a circle for a varying amount of time and proceeding in a straight line, then repeating the procedure when an obstacle is reached.

The two will meet *eventually*.

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Re: Twins in a Maze

Postby Trebla » Wed May 18, 2011 11:45 am UTC

blah28722 wrote:If time is truly a non-issue, and randomness is possible:

Spoiler:
Have both twins travel in a semi-random fashion. For example, spinning in a circle for a varying amount of time and proceeding in a straight line, then repeating the procedure when an obstacle is reached.

The two will meet *eventually*.


But they won't
Spoiler:
Every "random" decision one makes, the other will make the identical random decision.

If I "randomly" decide to spin, by the identical nature of our scenario, my twin will do the same. And further, he'll randomly choose to stop at the exact same time. This is easily demonstrable in code. Take two random number generators and give them the same seed. Call next() on them and you should get the same sequence of output. Note that in this case the generator either needs to be ignorant of the environment, or at least any aspect that isn't the same across multiple runs (such as system time).


Honestly, if we're taking identical to be completely identical, then even with communication they'd be unable to meet....

Spoiler:
Imagine a large transparent cylinder with the twins on opposite sides. It would be like looking in a reversed mirror. Every time you walk left, your image would simultaneously move right (his left). If you held up a sign saying "you go left, I'll go right" he would simultaneously do the same thing! If you choose to obey his sign (and go left), he'll choose to obey yours (and go to his left).


I'm reasonably convinced at this point that the only logical answer is to devise a way to break symmetry thereby giving the twins different experiences and making them no longer identical.

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Re: Twins in a Maze

Postby t1mm01994 » Wed May 18, 2011 8:07 pm UTC

Trebla wrote: -snip-
I'm reasonably convinced at this point that the only logical answer is to devise a way to break symmetry thereby giving the twins different experiences and making them no longer identical.

this.

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Re: Twins in a Maze

Postby charonme » Sat May 21, 2011 8:10 am UTC

Trebla wrote:Take two random number generators and give them the same seed. Call next() on them and you should get the same sequence of output.
That's the reason they're not called "random" generators. They are pseudorandom generators.

I like your observation that even with communication they wouldn't be able to meet

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Re: Twins in a Maze

Postby dedalus » Sat May 21, 2011 10:00 am UTC

Chain their left hand to a circular ring with diameter = 2D+1cm, where D is the maximum stretch possible of either twin. They can leave when they touch.
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Re: Twins in a Maze

Postby Kayomaro » Tue May 24, 2011 2:16 pm UTC

dedalus wrote:Chain their left hand to a circular ring with diameter = 2D+1cm, where D is the maximum stretch possible of either twin. They can leave when they touch.


That would work, except that the twins are rotating in the same direction at the same speed. So they will never touch in this fashion. We're assuming that they're on a sphere without any landmarks, right? Well then there's no real reason that they would be in the same hemisphere. So what about the Coriolis effect? That could allowed them to turn in opposite directions, and yes I'm aware that it applies to phenomena larger than a person on a chain, but it's really the best thing I can come up with. If the twins can only use what's available to them, which so far I understand to be themselves, the knowledge that the other twin will come up with the same solution, and a featureless sphere. So regardless of chains or digging tools or whatever if both of them start at the poles, which isn't impossible because of the randomness of their placement, and then proceed to walk in the appropriate direction in a very shallow spiral (clockwise in the northern hemisphere and counterclockwise in the southern if I remember correctly.) towards the equator, they should meet at some point on it. Both twins are technically doing the same actions, merely taking advantage of a natural phenomena that allows them to move opposite directions relative to one another. No randomness and they follow the same plan.

Thoughts on my 2 cents worth?

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Re: Twins in a Maze

Postby charonme » Thu May 26, 2011 6:15 am UTC

Kayomaro wrote:
dedalus wrote:Chain their left hand...
That would work, except...
I too thought first that this was supposed to be an attempt at a solution, but then I realized it's a clever reformulation of the puzzle. It's even more mind boggling because the twins are now so close, they see each other where they are, can communicate and all that is needed is that one of them move a few inches, yet they still can't touch

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Re: Twins in a Maze

Postby Kayomaro » Mon May 30, 2011 3:31 pm UTC

charonme wrote:
Kayomaro wrote:
dedalus wrote:Chain their left hand...
That would work, except...
I too thought first that this was supposed to be an attempt at a solution, but then I realized it's a clever reformulation of the puzzle. It's even more mind boggling because the twins are now so close, they see each other where they are, can communicate and all that is needed is that one of them move a few inches, yet they still can't touch



I honestly have no idea whether you're talking about my solution, or dedalus'. Either way, it's really unclear why the twins wouldn't be able to touch. Could you clarify, please?

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Re: Twins in a Maze

Postby Yakk » Mon May 30, 2011 7:37 pm UTC

Kayomaro wrote:(clockwise in the northern hemisphere and counterclockwise in the southern if I remember correctly.)

What is this "north" and "south" you speak of?
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Re: Twins in a Maze

Postby charonme » Tue May 31, 2011 7:03 pm UTC

Kayomaro wrote:Could you clarify, please?
ok, go back to dedalus' post, but don't think about is as a "solution". It's not a solution, it's a different puzzle. If you can find an answer to his new puzzle, you'll find a solution to the original "twins in a maze" puzzle.

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Re: Twins in a Maze

Postby Age of Fable » Fri Jul 08, 2011 8:11 am UTC

Spoiler:
You could come up with a formula that's effectively random ('go left if an even number of seconds have passed since I was dropped in the maze...' or 'take the next unused digit of pi...') Given infinite time to explore a finite space, you'll meet up eventually.
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Re: Twins in a Maze

Postby mward » Thu Aug 18, 2011 11:16 am UTC

Spoiler:
The puzzle statement specifically mentions "twins" rather than "clones": so we may assume that my twin and I both know which of us is older.

Now, when we were little, our mother told us that if we got separated from her we should stay still and not run around looking for her: she would look for us.

Our mother is older than us: so the rule is that the younger twin should stay still while the older twin systematically searches the maze.

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Re: Twins in a Maze

Postby ElCarl » Wed Aug 24, 2011 9:48 pm UTC

mward wrote:
Spoiler:
The puzzle statement specifically mentions "twins" rather than "clones": so we may assume that my twin and I both know which of us is older.

Now, when we were little, our mother told us that if we got separated from her we should stay still and not run around looking for her: she would look for us.

Our mother is older than us: so the rule is that the younger twin should stay still while the older twin systematically searches the maze.


jestingrabbit has already pointed out that we don't need any more difference>solution posts... We know that if you introduce differences then you can make a probability 1 solution...

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Re: Twins in a Maze

Postby Trebla » Mon Sep 12, 2011 12:13 pm UTC

charonme wrote:
Trebla wrote:Take two random number generators and give them the same seed. Call next() on them and you should get the same sequence of output.
That's the reason they're not called "random" generators. They are pseudorandom generators.


That was meant to be part of my point, true randomness doesn't appear to be possible without a break in the symmetry. We reduce the twins to perfectly logical robots (this makes it easier to imagine them being completely identical) and you need an external factor to provide a difference in the seed to their pseudo-random decisions. Thank you for clarifying, I should have just used the proper terminology, I guess.

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Re: Twins in a Maze

Postby blademan9999 » Thu Sep 29, 2011 9:54 am UTC

They observe the spins of subatomic particles or something like that.
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Re: Twins in a Maze

Postby wanderingnoob » Sun Nov 13, 2011 10:04 pm UTC

Multiple people have referred to some need to break symmetry in order to solve this in the absolute worst case (wandering around a circle or similar).

My knowledge on this topic is somewhat slim (a very disheartening disclaimer, I know), but for anyone else more educated, does it seem likely that breaking P symmetry (parity symmetry) in some way could be used to unite the twins, possibly requiring the stronger CP symmetry break?

http://en.wikipedia.org/wiki/Parity_(physics)#Parity_violation

One problem I see is that any experiment on the scale where the CP break plays a factor tends to be quantum in nature anyway, so it would imply access to true random numbers, rather spoiling the puzzle. Well, one problem apart from it maybe just not working...

mfb
Posts: 950
Joined: Thu Jan 08, 2009 7:48 pm UTC

Re: Twins in a Maze

Postby mfb » Mon Nov 14, 2011 1:41 pm UTC

They both have a unique "up" direction (from gravity), parity breaking would just give them a way to distinguish between "going clockwise" and "going counterclockwise". But as humans are not invariant under parity transformations ("left" and "right" are not the same), we can do the same without particle physics.
And it does not help them.


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