## Three Box Puzzle

Science fun for all. Science fairs, homemade experiments, amateur microscopy, comics, puzzles, videos, or anything else you'd like to share.

## Read the puzzle description. You are given a choice. Do you:

Keep the closed Box you originally chose.. because the odds are 50/50
4
31%
Choose the other closed Box.. because your odds are better with the other Box
9
69%

Total votes : 13

### Three Box Puzzle

Hi Folks

Ran across this little puzzle and thought you might enjoy it.

The Three Box Puzzle:

You have seen the Game Show where you get to pick one of three identical looking Boxes.

Two Boxes will be empty and one has a prize inside.

The Boxes are closed.. you don't know which one contains the prize.

So You pick a Box.

Of the two remaining Boxes, one has to be empty.
The Host knows which of the remaining Boxes is empty.
The Host selects that Box and shows you that it's empty.

That leaves the Box you originally picked and the remaining closed Box.

The Host now offers you the opportunity to swap your original Box choice with the remaining Box.

Do you..

A: Keep the closed Box you originally chose.. because the odds are 50/50.

B: Choose the other closed Box.. because your odds are better with the other Box.

Pick "A" or "B" and post your answer only.. Please.. no explanations.
(and no peeking at what others have picked)

I'll be back in a few days with the correct answer.

Best of luck,
Dave :^)

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### Re: Three Box Puzzle

WARNING: SPOILERS

It's called the "<censored - we'll tell you the name when you solve it :D ~ XCT>"; and it's generally regarded as a veridical paradox.

Here's why I don't think it deserves that status: it carries the unspoken assumptions that

1. The prize is initially equally likely to be inside each box
2. The host must open an empty box
3. The host must uniformly choose which box to open if both are empty
4. The host must make the offer to switch

Once these premises are made clear, the conclusions don't seem half so absurd. So, rather than being paradoxical, the problem is rather a lack of clarity.

(Sorry if I'm being a stick in the mud. I just thought it needed saying.)

Lomax

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### Re: Three Box Puzzle

Thanks Lomax

I liked what you said.. would have preferred the puzzle not be identified by name, else some may cheat and look it up. As a Moderator.. can you edit out the name part?

Best regards..
Dave :^)

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### Re: Three Box Puzzle

B is statistically better.
Terry
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### Re: Three Box Puzzle

I liked what you said.. would have preferred the puzzle not be identified by name, else some may cheat and look it up. As a Moderator.. can you edit out the name part?

Best regards..
Dave :^)

Haha, ok. Done.

___

EDIT: Oh, wait. There's no edit box for that post for some reason. This is SciameriKen's forum so I guess it's down to him.

Lomax

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### Re: Three Box Puzzle

Thanks Lomax,

This is a fun problem. I saw it on another site, long ago, before the solution was published. It was debated by many individuals at that time.

One side argued that, at the time of the final decision, it boiled down to 2 choices, regardless of what had transpired before that point. So it had to be 50/50.

The other side argued that what had transpired prior to the point of making the final decision, did affect the final odds, and it wasn't 50/50.

Don't feel shy about jumping in, even if you pick the wrong argument. Just don't present your argument, so we can see how divided the results might become. If you have seen this problem before or looked up the answer, please don't post a response here and bias the results. Let's see how it tallies up from readers fresh to this problem. For the record, I voted for the 50/50 side, way back then.

Next Friday, April 8th, I will post a link to the answer and present the answer too.. have fun.

Best wishes to all..
Dave :^)

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### Re: Three Box Puzzle

You might be interested to know that the basic principle behind this puzzle can also be used in science for deciding between competing hypotheses. Here is a post I did earlier this week which illustrates the idea: viewtopic.php?f=67&t=18175&p=176701#p176390

(although I misread the problem posed by Lincoln a little, who is the physicist on this forum who asked me to do this analysis)

Anyway, here's another puzzle in parametric decision theory, the branch of mathematics that governs this stuff.

I give you two envelopes to choose from, and you know one of the envelopes has 2X as much money as the other. You have to pick one.

After you pick one, you open it up, and look inside, and see it contains \$100.

I then give you another choice - you can keep your envelope with \$100, or switch.

Do you:
(A) Keep the \$100
(B) Go for the unknown envelope

Say your choice and while you're at it, an explanation would be appreciated!

Cheers,

~XCT

------------------

EDIT: @Dave_Oblad: I changed your post so now it's a poll. I also answered according to the way I was taught in probability classes.

xcthulhu
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### Re: Three Box Puzzle

Dave,

Is this the famous Marilyn vos Savant puzzle? Sure sounds like it..........
ronjanec
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### Re: Three Box Puzzle

Oh wow..
Thanks to XCT for adding the polling function. I was afraid no one would join because their name would be attached to their vote and be shy about playing.. Out Standing!!!!!!!!!! Bravo!

Highest regards,
Dave :^)

P.s That envelope problem.. I won't cheat but let me think about it a bit.. Instinct tells me not to switch but we all know how much my instinct is worth...lol.

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### Re: Three Box Puzzle

I sent XCT my answer to his Envelope Puzzle but won't reveal it here incase He decides to make another game of it.. Best wishes to all from Dave :^)

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### Re: Three Box Puzzle

Dave_Oblad wrote:I sent XCT my answer to his Envelope Puzzle but won't reveal it here incase He decides to make another game of it.. Best wishes to all from Dave :^)

Nah, you can reveal it if you want...

But maybe we should wait for Lomax's answer to see what he thinks of the envelope puzzle.

xcthulhu
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### Re: Three Box Puzzle

Lomax wrote:EDIT: Oh, wait. There's no edit box for that post for some reason. This is SciameriKen's forum so I guess it's down to him.

Being a PCF mod only lets ya edit stuff there. Likewise, I'm not a PCF mod, so I can't edit things on that side.
Natural ChemE
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### Re: Three Box Puzzle

xcthulhu wrote:Nah, you can reveal it if you want...

But maybe we should wait for Lomax's answer to see what he thinks of the envelope puzzle.

I've heard it I'm afraid, sorry. There's a 50% chance the other envelope will contain \$50 and a 50% chance it will contain \$200, so the idea is to switch (because the mean potential value of the other envelope is \$125).

ronjanec wrote:Is this the famous Marilyn vos Savant puzzle? Sure sounds like it..........

'Tis.

Lomax

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### Re: Three Box Puzzle

Lomax wrote:
xcthulhu wrote:Nah, you can reveal it if you want...

But maybe we should wait for Lomax's answer to see what he thinks of the envelope puzzle.

I've heard it I'm afraid, sorry. There's a 50% chance the other envelope will contain \$50 and a 50% chance it will contain \$200, so the idea is to switch (because the mean potential value of the other envelope is \$125).

ronjanec wrote:Is this the famous Marilyn vos Savant puzzle? Sure sounds like it..........

'Tis.

Thanks Lomax. I thought so.
ronjanec
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### Re: Three Box Puzzle

Lomax wrote:
xcthulhu wrote:Nah, you can reveal it if you want...

But maybe we should wait for Lomax's answer to see what he thinks of the envelope puzzle.

I've heard it I'm afraid, sorry. There's a 50% chance the other envelope will contain \$50 and a 50% chance it will contain \$200, so the idea is to switch (because the mean potential value of the other envelope is \$125).

Figured the chances were high that you had heard it before. I'll start another thread in the logic forum (don't want to spam this forum) about another, more esoteric puzzle, but I'm starting to have my doubts that I can pitch you one you haven't already seen before.

xcthulhu
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### Re: Three Box Puzzle

Nah, I'm sure there are a ship load I haven't heard before.

There's a math trick reminiscent of the envelope puzzle that I read once. I don't know to whom it owes, but I discovered it through the popularizer Julian Baggini:

Suppose you have two envelopes, A and B, and you know that one of them contains twice as much as the other. Pick up an envelope but don't open it.

The envelope (suppose it's envelope A) contains \$x.

This means that there's a 50% chance that envelope B contains \$2x and a 50% chance it contains \$0.5 x. So, the mean potential value in envelope B is \$1.25 x.

So, it seems rational to switch to envelope B. You don't know how much is in there, except that it's \$1.25. So now, there's a 50% chance that envelope A contains 2 (\$1.25 x) and a 50% chance it contains 0.5 (\$1.25 x), so the mean potential value of A is \$1.5625 x.

By this logic, we can indefinitely improve our cash winnings by just switching the envelopes back and forth, back and forth...

It's not a veridical paradox like the last one of course, but I thought it was interesting.

Lomax

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### Re: Three Box Puzzle

When I think of this problem I think of Deal or no deal :D

In that case you have 25 suitcases and you choose one, if you survive long enough and you eliminate all the other suitcases you'll get the choice of taking yours or the one that is left.

Now I think the logic says - you take the other case --
Thinking of it as - in the beginning you have a 1/25 chance of picking the right suitcase which means the right suitcase is a 24/25 chance to be left behind -- thus that final decision is between one suitcase that was 4% likely and the other which represents a 96% likelihood.

Assuming the million dollar case in the remaining 24 -- that means as you eliminate it you take a chance of finding it -- thus on the first there is a 1/23 chance of finding it,, then a 1/22, then a 1/21 ... and so on even down to 1/2 chance of it being there... now what is more likely - that the million dollar case successfully avoided all these pick attempts or that it was not in the pool of 24 in the first place?

The Three box puzzle is a scaled down version of this and in my opinion is truly a 50-50 :)

SciameriKen
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### Re: Three Box Puzzle

Hello SciameriKen,

Well this is why I think the three box puzzle (can we name it yet?) is deceptive; because it doesn't explicitly state that the jackpot prize cannot be eliminated, but that is nevertheless assumed in the question.

So, in Deal Or No Deal terms, it is as if Noel alone is choosing which suitcase to remove at the end of each round, and is not allowed to remove the million dollar suitcase. That's why it's better to switch.

Lomax

Lomax

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### Re: Three Box Puzzle

Good points lomax --

my thoughts more or less require active querying of the remaining boxes...

of course there is only one way to deal with this - and that is experimentation -- I think I know my next Scientific AmeriKen :)

SciameriKen
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### Re: Three Box Puzzle

Ok folks.. time to reveal the answer.

I chose to call it the 3 Box Puzzle to hide it's original name, which is: "The Monty Hall Problem". An explanation and analysis is on wikipedia at:

http://en.wikipedia.org/wiki/Monty_Hall_problem

When I explain this simply, I put it like this:

You have 3 boxes.. pick one.

Now you can keep that single box.. or.. you can have "BOTH" of the other boxes. Now which deal do you go for?

That's a 1 out of 3 chance to win.. OR.. a 2 out of 3 chance to win? Go ahead and think it over...

Of course you would take the 2/3 chance to win. You know one is empty for sure but you still effectively get both boxes for the price of one. The fact Monty shows you which of the other two boxes is empty doesn't change your 2/3 odds of winning by giving up your original 1/3 chance.

Lomax was correct.. It's deception is in the way the puzzle is worded.

Thanks all.. that was fun.

Best wishes,
Dave :^)

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### Re: Three Box Puzzle

Dave_Oblad wrote:That's a 1 out of 3 chance to win.. OR.. a 2 out of 3 chance to win? Go ahead and think it over...

Of course you would take the 2/3 chance to win. You know one is empty for sure but you still effectively get both boxes for the price of one. The fact Monty shows you which of the other two boxes is empty doesn't change your 2/3 odds of winning by giving up your original 1/3 chance.

Lomax was correct.. It's deception is in the way the puzzle is worded.

Thanks all.. that was fun.

Best wishes,
Dave :^)

Thanks for the good topic Dave-- What do you think of the variation to this game where say you pick one of the two remaining boxes -- if its in the one you picked the game ends.. if it isn't you then have to choose whether to keep yours or take other...

SciameriKen
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### Re: Three Box Puzzle

Greetings all -- Finally completed my experiment on this topic!!

http://www.scientificameriken.com/threebox.asp

Will be happy to discuss any thoughts and opinions!

-SciameriKen

SciameriKen
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