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# 100 mathematicians, 100 rooms, and a sequence of real numbers

18 replies to this topic

### #1 Jrthedawg

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Posted 22 July 2013 - 12:49 AM

I am a collector of math and logic puzzles, and this must be the best I've ever seen.

100 rooms each contain countably many boxes labeled with the natural numbers. Inside of each box is a real number. For any natural number n, all 100 boxes labeled n (one in each room) contain the same real number. In other words, the 100 rooms are identical with respect to the boxes and real numbers.

Knowing the rooms are identical, 100 mathematicians play a game. After a time for discussing strategy, the mathematicians will simultaneously be sent to different rooms, not to communicate with one another again. While in the rooms, each mathematician may open up boxes (perhaps countably many) to see the real numbers contained within. Then each mathematician must guess the real number that is contained in a particular unopened box of his choosing. Notice this requires that each leaves at least one box unopened.

99 out of 100 mathematicians must correctly guess their real number for them to (collectively) win the game.

What is a winning strategy?

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### #2 TimeSpaceLightForce

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Posted 22 July 2013 - 10:18 AM

Nice puzzle ..

Can we label the rooms, boxes, real number - 00 to 99 ?

So that room 50 has a box with "50" written outside and "50" written inside..

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### #3 plasmid

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Posted 22 July 2013 - 02:48 PM

I get the feeling this will depend on how exactly you define communication.

Spoiler for If you just define communication as talking,

Since this is one of the best puzzles you've seen, your definition of communication would probably prohibit such a straightforward approach. But if you take an ultimately strict interpretation of communication, like "no mathematician can leave any evidence of their existence to any other mathematician after the game starts", then this should be indistinguishable from sending each mathematician into the game completely alone and there would clearly be no solution. So there must be some form of communication.

We'll need to know what's allowed and what isn't: whether mathematicians can rearrange boxes or see the boxes that other mathematicians have rearranged, whether the other mathematicians hear a guess that any other mathematician makes or whether or not it was correct, etc.
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### #4 Rainman

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Posted 22 July 2013 - 04:30 PM

No form of communication is necessary once they have entered their rooms. They can not see, hear, or otherwise perceive anything another mathematician does. We do need to allow some other wild stuff though.

Spoiler for Hints

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### #5 bonanova

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Posted 22 July 2013 - 07:25 PM

No form of communication is necessary once they have entered their rooms. They can not see, hear, or otherwise perceive anything another mathematician does. We do need to allow some other wild stuff though.

Spoiler for Hints

I think uncountably countably is all that's needed.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #6 Rainman

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Posted 22 July 2013 - 07:50 PM

No form of communication is necessary once they have entered their rooms. They can not see, hear, or otherwise perceive anything another mathematician does. We do need to allow some other wild stuff though.

Spoiler for Hints

I think uncountably countably is all that's needed.

The boxes themselves would only contain countable amounts of information, but I think the information the mathematicians need to bring into their rooms (i.e. their strategy) is uncountably infinite. Unless I'm missing a simpler solution.

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### #7 DeGe

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Posted 25 July 2013 - 03:22 PM

Do they hear the guesses of other mathematicians?

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### #8 bonanova

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Posted 26 July 2013 - 03:31 AM

Do they hear the guesses of other mathematicians?

DeGe,

No.

To answer plasmid's question about communicating, let's frame the question this way:

When a mathematician enters his room, the door locks behind him.

Consider the room to be an information black hole, except that each box communicates to a central computer.

Each box has a keypad capable of transmitting any real number (along with its box number) to the computer.

If and only if the computer receives correct (real number - box number) pairs from 99 or 100 rooms, the doors unlock themselves.

Otherwise, ... the world loses 100 mathematicians.

That is, it's not like the 100 prisoners with red or blue hats: they do hear each other.

Here, the strategy is so cool that they don't need to know each other's guesses.

Spoiler for A similar strategy will also solve this puzzle

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #9 plasmid

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Posted 26 July 2013 - 02:25 PM

This reminds me of the "Team of 15" problem, although it doesn't look like it can really be approached the same way.

Are you working under the premise that all boxes must contain a different number? It doesn't look like that's the case.
Spoiler for

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### #10 bonanova

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Posted 26 July 2013 - 09:48 PM

No shenanigans.

I don't think the real numbers have to all be different. OP just says each box contains a real number.
The probability of any two being the same is zero, but it could happen, and still solve the problem.

Edit:

When the OP says "countably" it has to mean "countably infinite."

There must be a box in each room for every counting number for the strategy to work.

Edited by bonanova, 27 July 2013 - 07:52 AM.
Clirify "countably" as used in OP has to mean "countably infinite"

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

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