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# Which Chair you will choose ?

### #11

Posted 26 June 2012 - 12:33 AM

Someone find the right word for me

### #12

Posted 26 June 2012 - 05:27 AM

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

- Bertrand Russell

### #13

Posted 26 June 2012 - 10:25 PM

Got to say, it's __________ that five of us have completely different solutions, and no working in common.

Someone find the right word for me

"interesting". This means the puzzle is so good that it got each one of us to think differently. Thank you for the brain teaser, ujjagrawal.

Although after going through the posts, I am now convinced that bonanova has nailed the answer.

### #14

Posted 26 June 2012 - 11:00 PM

"interesting". This means the puzzle is so good that it got each one of us to think differently. Thank you for the brain teaser, ujjagrawal.

Although after going through the posts, I am now convinced that bonanova has nailed the answer.

Personally I find it bizarre. I've noted that the "signal to noise ratio" (if signal means correct/true/good reasoning/answer/process, noise means bad/incorrect/false) on brainden is worse than most ad hoc forums or messages, even for the trivially easy puzzles. One of these forms is where people give answers that don't mean a thing to anyone else; the words and logic is practically gibberish in English. How does this happen? Even for most people to acknowledge a correct logical answer seems to be impossible here. If this place lacks common sense, you have to wonder what "communal" value there is given the objectives of contributing here.

In this case, the fact that we have completely different solutions means most of us are completely wrong, and yet believe we are all on the right track.

I have no problem accepting I could be wrong, but I'm the only one who has given a proof, and I have implicitly disproved all other answers. But in my experience, proofs and disproofs mean nothing to most people.

### #15

Posted 27 June 2012 - 03:42 PM

The probability of four same color (thus first chair winning):

8/8 * 3/7 * 2/6 * 1/5 = 1/35. Same answer results from using combinations.

There are 70 (ordered) permutations of the 8 stamps, each with equal chance.

There are 16 (ordered) permutations of 4 of the 8 stamps, not with equal chance.

If this doesn't occur, then first chair cannot know the answer.

Second chair now knows that first chair does not see: YYXXXX

Specifically it leaves 68 permutations with equal chance.

Then, if he sees XXYYXX he wins. As in the first case, there are two permutations that satisfy this.

So he wins this way with 1/34 chance.

I believe he learns nothing else.

Third chair now knows that second chair does not see: XXYYXX

This eliminates 2 possibilities, leaving 66 with equal chance.

If he sees XXXXYY he wins, 2/66 chance.

Also if he sees XXYY?? then he wins knowing he has XY (since the others haven't won yet).

This could happen as BBWW**** or WWBB**** where **** is permutations of BBWW. 2 * 6 = 12 ways

However XXYY?? intersects with XX??XX at XXYYXX so there are 8 ways left.

So if he didn't win by XXXXYY he wins this with 8/64.

Equivalent overall: 2/66 + 64/66 (8/64) === 10/66 = 5/33

I believe he learns nothing else.

First chair:

He's survived this far, so there are 56 possibilities left. YYXX?? is not possible.

He applies same thing as third chair, he wins if he sees ??XXYY where he must have XY. Still 12 possibilities, minus 4 intersections.

Wins with chance 8/54 = 4/27.

I can't see him learning anything else.

Second chair:

48 possibilities left, YYXX?? and ??XXYY are excluded.

If he sees XX??YY he must have XY and he wins. 8 ways => 8/48 = 1/6 chance of winning like this.

If he doesn't win like that, he knows the XX, YY, XY pairs aren't there (no one sees them). XXYYXY and XXYYXX and XXYYYY aren't there, it must mean XX YY isn't there. What about XX XY? (XX XY YY, XX XY XX can't be). One mixed pair doesn't exist, this means there are two or three mixed pairs.

Thus if he sees XX??MM or MM??XX where MM = mixed pair, then he knows he has XY.

XXMMMM has 8 forms: 2x for swapping X with Y, 4x for two MM pairs as XY or YX. No more multipliers because the rest must be YY.

MMMMXX has 8 ways also.

He wins with 16/40 chance now = 2/5

Overall: 8/48 + 40/48 (16/40) = 24/48 = 1/2 chance.

Third chair: 24 left.

So I think there are 2 or 3 mixed pairs. Second chair would have won if one of 1st or 3rd pair was not mixed, therefore they must both be mixed.

I believe possibilities left are MMMMMM or MMXXMM. This would mewan third must have XY no matter what.

Can check this by seeing how many possibilities MMMMMM and MMXXMM form:

MMXXMM: 2x for XY or YX on third pair, 2x for XY/YX on 1st pair, 2x for identify of X, 1x for the rest must be YY. 8 ways.

MMMMMM: 8x for three pairs XY/YX. Remaining is also a pair, 2x for that. 16 ways.

Totals 24 so correct.

Patterns:

1 XXXX??, 2 ways

2 XX??XX, 2 ways

3 ??XXXX, 2 ways

3 XXYY??, 8 new ways

1 ??XXYY, 8 new ways

2 XX??YY, 8 new ways

2 XXMMMM or MMMMXX, 16 ways

3 MMXXMM or MMMMMM, 24 ways

By intuition or otherwise, the order of events is independent of the ways of winning (must be better way to explain). Anyhow, you can just add up the number of ways.

First chair: 10 ways

Second chair: 26 ways

Third chair: 34 ways

So the logical choice is third chair, with 34/70 chance of winning, assuming the other two students are not (color)blind and are equally logical.

nice work.... your approach seems most correct to me... third chair is my choice too...

This is not a pure probability problem... it is a combination of probability and logic... the person sitting on a chair will not make any guesses based upon probability but he will try to deduce the answer by logic... incase he is not able to... he passes on... and in worst case senario... correct answer can be logically deducable until 6th turn i.e by person sitting on 3rd chair... that also hold maximum probability(chances) for a person to logically deduce the answer. Anyone with a different views/answer with explanation is most welcome...

### #16

Posted 27 June 2012 - 07:40 PM

Personally I find it bizarre. I've noted that the "signal to noise ratio" (if signal means correct/true/good reasoning/answer/process, noise means bad/incorrect/false) on brainden is worse than most ad hoc forums or messages, even for the trivially easy puzzles. One of these forms is where people give answers that don't mean a thing to anyone else; the words and logic is practically gibberish in English. How does this happen? Even for most people to acknowledge a correct logical answer seems to be impossible here. If this place lacks common sense, you have to wonder what "communal" value there is given the objectives of contributing here.

In this case, the fact that we have completely different solutions means most of us are completely wrong, and yet believe we are all on the right track.

I have no problem accepting I could be wrong, but I'm the only one who has given a proof, and I have implicitly disproved all other answers. But in my experience, proofs and disproofs mean nothing to most people.

I've been following this thread kind of haphazardly, but I have to come in and say that what you're saying, whether you're trying to or not, is discouraging people from posting with their ideas/thoughts on a puzzle, which I think is a key aspect of BD being a 'community' rather than just a website.

Even if people are wrong, they may say something that inspires thoughts in others, and that in itself is valuable. I am a scientist, and highly value collaboration. Pretty much all great discoveries, although often credited to one person or the other, were made by a collaboration or at least one person building the past works of others.

Also as a scientist, I've noticed that in science, economics, game theory, and even math, there are still problems 'experts' disagree on the correct approach to.

And personally, as a puzzle maker who likes to make somewhat complex puzzles, I highly encourage people to work together and share their working thoughts. It usually 'gets the thread going' and builds a kind of happy, fun atmosphere as well.

To be honest, I have not given this particular puzzle enough analysis to form an opinion on who is 'correct', my beginning line was with probabilities such as your first line, but I never took the time to finish. I recognize that you've put a lot of work and thought into your answer, and you gave a really good and thorough

*explanation*...but I have to point out, it does not constitute a

*proof*...your own 'comments' in red show the parts that need to be completed for it to be a proof.

Welcome to the den ;P.

**Women are definitely stronger. We are [Fe]males, after all...***Some of what makes me me is real, some of what makes me me is imaginary...I guess I'm just complex. ;P*

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When life hands me lemons, I make invisible ink.

### #17

Posted 27 June 2012 - 07:44 PM

Personally I find it bizarre. I've noted that the "signal to noise ratio" (if signal means correct/true/good reasoning/answer/process, noise means bad/incorrect/false) on brainden is worse than most ad hoc forums or messages, even for the trivially easy puzzles. One of these forms is where people give answers that don't mean a thing to anyone else; the words and logic is practically gibberish in English. How does this happen? Even for most people to acknowledge a correct logical answer seems to be impossible here. If this place lacks common sense, you have to wonder what "communal" value there is given the objectives of contributing here.

In this case, the fact that we have completely different solutions means most of us are completely wrong, and yet believe we are all on the right track.

I have no problem accepting I could be wrong, but I'm the only one who has given a proof, and I have implicitly disproved all other answers. But in my experience, proofs and disproofs mean nothing to most people.

I understand your frustration of receiving no feedback after a well-reasoned and meticulous post of the solution. It is obvious from your post that you are quite smart, and I think you would fit right in this community. Personally, I generally find puzzles here to be excellent (this topic is an perfect example), and the community here includes some very fastidious, clever, and brilliant logicians. I have found myself more than once awed and humbled by the creativity and the sheer elegance of some puzzles and solutions that the Denizens come up with. I hope you will give Brainden some more time to change your opinion of its worth.

I'll have to admit that sometimes it is hard to get some feedback on this board. In this case, however, you could probably initiate some discussions about the correctness of the solution by examining other solutions (bonanova's in post #14, for instance) to see whether it is incorrect or whether you missed anything. I have examined both of your solutions, and I believe both have some errors. I may be wrong, and I often am, but I'll include those possible errors here

bonanova's solution

Spoiler for Analysis of the 70 distributions of stampsSpoiler for 'possible error

voider's solution (the spoiler tags are added by me)

Spoiler for solution'Spoiler for 'possible error

Having said that, I'll contribute a solution of my own. This solution will give a winning chance of 42/70 or 60%.

**Edited by bushindo, 27 June 2012 - 07:53 PM.**

### #18

Posted 27 June 2012 - 10:31 PM

I've been following this thread kind of haphazardly, but I have to come in and say that what you're saying, whether you're trying to or not, is discouraging people from posting with their ideas/thoughts on a puzzle, which I think is a key aspect of BD being a 'community' rather than just a website.

Even if people are wrong, they may say something that inspires thoughts in others, and that in itself is valuable. I am a scientist, and highly value collaboration. Pretty much all great discoveries, although often credited to one person or the other, were made by a collaboration or at least one person building the past works of others.

Also as a scientist, I've noticed that in science, economics, game theory, and even math, there are still problems 'experts' disagree on the correct approach to.

And personally, as a puzzle maker who likes to make somewhat complex puzzles, I highly encourage people to work together and share their working thoughts. It usually 'gets the thread going' and builds a kind of happy, fun atmosphere as well.

To be honest, I have not given this particular puzzle enough analysis to form an opinion on who is 'correct', my beginning line was with probabilities such as your first line, but I never took the time to finish. I recognize that you've put a lot of work and thought into your answer, and you gave a really good and thoroughexplanation...but I have to point out, it does not constitute aproof...your own 'comments' in red show the parts that need to be completed for it to be a proof.

Welcome to the den ;P.

Referring to the website, not this particular thread: (Maybe in the wrong section of the forum, but it is a reply.)

Firstly I agree on the value of collaboration and openness to creativity. But I believe in the context of logic puzzles there is usually a distinction between "something I haven't thought about" and "something that makes no sense, and is based on assumptions that are difficult to intelligently comprehend". Obviously I must have some pride in what I believe to be logical or at least reasonable thinking (even if it is inaccurate) because it affects me when others can't see this distinction, or get it the wrong way round. In fact, this is the first thing I discovered on brainden, other than that it had puzzles and problems on it. Of 200+ comments on the Epimenides Paradox, I dare say 20% or less showed accurate comprehension of its meaning and solution. I discussed this with rookie and proposed a way to improve people's

**thinking**, not just define an answer, a rare opportunity. It didn't come together. I know that generally you can't change people nor the way they think, and that's why my prior post is just an expression of my observations. If it is assumed that nothing can be done about it, and no one has tried, then what does it say about an appropriate environment on brainden? Is it appropriate that most people don't "get" one of the most famous paradoxes, doesn't it virtually encourage the notion that "it doesn't matter what's true, as long as others agree with you", or even "it doesn't matter what's true, as long as you agree with yourself [, and have fun in the process]" (deliberate hint of Hedonism, it's hard to initiate a discussion on that level) ? I would imagine a problem-solving context to do a bit better than that.

Can you express whatever you think about a logic puzzle on brainden? Yes

Should you be able to express it without second thought, and expect it to be accepted for what it is? I don't know.

Do all expressions have value to the "system"? I don't believe so. Spam, redundancy, etc.

Although not directly relevant, I will point out that the ideal logic problem solver would probably read problems, solve alone (never give up) and never come to a forum. But if I'm communicating any sense to you, you'll see that I think the worth of the community with respect to its social and objective values (if any) is a conundrum. You can't have everything, and I'm observing what brainden doesn't have.

**Edited by voider, 27 June 2012 - 10:34 PM.**

### #19

Posted 28 June 2012 - 04:13 AM

Spoiler for possible error

Oops, upon second thoughts, I'll revise the claim I made here

### #20

Posted 28 June 2012 - 06:33 AM

I have examined both of your solutions, and I believe both have some errors.

I may be wrong, and I often am, but I'll include those possible errors here

bonanova's solution

The spoiler did not get into the quote, but you suggest thatmmbgives the win to chair 2.Spoiler for here is my analysis of mmb case

Having said that, I'll contribute a solution of my own. This solution will give a winning chance of 42/70 or 60%.Spoiler for approach

This is a nice analysis to maximize probability that your answer is correct. I had understood the puzzle to require that when you answer you must be certain that your answer is correct. Then, each chair gives you a particular probability of being the first to be able to answer with certainty of your color.If the student cannot logically deduce the colors, she will move on to the second chair, then the third chair. If that does not decide the issue, she will continue around the circle of chairs until one of you gives the correct response, with correct reasoning, based on the stamps that are visible and the other students' answers.

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

- Bertrand Russell

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