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        riddles >> hard >> A collaboration game for three
        
(Message started by: JocK on May 7^th, 2006, 2:28am)

Title: A collaboration game for three
Post by JocK on May 7^th, 2006, 2:28am

You and two of your friends are invited to play the following game:

After allowing ample time for the three of you to make any preparations, the team is taken apart and each person is put in a separate room. The rooms are all soundproof with metal cladded walls and no windows. They are also far apart. Once in the rooms no communication is possible between the three players.

At a certain time, each player is simultaneously asked to make a selection between two choices. There are two sets of choices:

1) "Make a selection between +1 and -1", and:

2) "Make a selection between +2 and -2".

Each player must stick to the two alternatives provided to her/him. The sets of alternatives for each player are selected randomly, but with one constraint: according to the rules of the game, either all players are asked to make a choice between +1 and -1, or only one player is asked to make this choice (and the two others are asked to chose between +2 and -2). Hence, there are four different sets of choices that can be given to the three players. Each of the four have equal likelihood of occurring.

The team wins if the product of the three selections made is either -1 or +4.

What strategy should the team adopt to maximise the chances of winning? Given that any preparations are allowed (no matter how tedious or how costly), how close can you approach a likelihood of 1 of winning?

No cheating! Hint: [hide]I am quite sure you will be surprised by the answer..![/hide]

Title: Re: A collaboration game for three
Post by Icarus on May 7^th, 2006, 11:19am

No surprises. Solving this one will be a cakewalk. 8)

Title: Re: A collaboration game for three
Post by JocK on May 7^th, 2006, 12:15pm

on 05/07/06 at 11:19:48, Icarus wrote:

No surprises.

But you should be surprised! :o
(Did you notice this methodology can also be used to solve the Hundred Prisoners and a Lightbulb riddle?)

It's a strange world we live in, isn't it?

Title: Re: A collaboration game for three
Post by Icarus on May 8^th, 2006, 3:13pm

on 05/07/06 at 12:15:25, JocK wrote:

(Did you notice this methodology can also be used to solve the Hundred Prisoners and a Lightbulb riddle?)

I don't think so. First of all, that riddle does not allow unlimited resources for preparation like this one. All they get is a chance to discuss the situation. Second, here you merely have to align responses, while there you need to actually communicate information, and that just doesn't ring a bell.

Title: Re: A collaboration game for three
Post by towr on May 9^th, 2006, 4:47am

I really don't see how you get the result you seem to be getting..
I mean, with ample preparation time we could build radios, but no communication after the start of the game is allowed, so that won't help. ::)

Title: Re: A collaboration game for three
Post by Icarus on May 9^th, 2006, 2:50pm

I probably would not have gotten it either, if Jock hadn't been so kind as to bring up the necessary trick elsewhere the day before. (It was nothing new to me, but the reminder meant that it was fresh in my mind when he posted this.)

Title: Re: A collaboration game for three
Post by rmsgrey on May 10^th, 2006, 9:26am

Seems like it'd involve some sort of "spooky action at a distance"...

Title: Re: A collaboration game for three
Post by SMQ on May 10^th, 2006, 2:09pm

OK, so following the hints so far I've got me a bunch of [hide]quantum-entangled particles[/hide]. I'm presumably attempting to use these in such a way that when I receive a choice between +2 and -2 I can reliably choose the opposite sign as whichever of my friends received the other choice between +2 and -2, right? Or am I using them to discern whether we're facing a product-of ones or product-of-twos choice?

Either way, how, exactly, are we proposing that such a scheme would work? I'm not seeing a way to arrange it with only [hide]entangled pairs[/hide], and I'm not familiar enough with the properties of [hide]entangled triples[/hide] to see their use either. Don't the same restrictions which prevent [hide]superluminal communication[/hide] also prevent [hide]communicating any information using the qbits alone[/hide]?

--SMQ

Title: Re: A collaboration game for three
Post by Icarus on May 10^th, 2006, 6:17pm

(No more hiding)

I'm not sure about the light-speed limitation, but I am told that Bell's theorem proves that you cannot communicate by quantum entanglement (I've never looked that far into it myself). Thus my comment above about "ringing a bell"

The general feel of the set up reminded me of a quantum entanglement experiment, so I was sure this was Jock's intent. That was the cause of my original post. (The "cakewalk" was in reference to the thread Jock posted the day before, which linked to a paper explaining quantum entanglement using cakes in ovens as the entangled particles. Unfortunately, that paper has been removed only two days later, so you can no longer follow the link.) Jock's reply assures me that this really was his intent.

But now I find myself in the same shoes as SMQ. While I am sure that QE was the intended solution, I can find no way to set it up. As far as I can see, there seems to be no way of aligning everyone's response without at least one bit of information being passed between 2 of the players.

It turns out my "cakewalk" involves carrying some really hefty cakes.