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Although I'm not a professional mathematician by training, I felt I should have easily been able to answer straight away the following puzzle:

Three men go to a shop to buy a TV and the only one they can afford is £30 so they all chip in £10. Just as they are leaving, the manager comes back and tells the assisitant that the TV was only £25. The assistant thinks quickly and decides to make a quick profit, realising that he can give them all £1 back and keep £2.

So the question is this: If he gives them all £1 back which means that they all paid £9 each and he kept £2, wheres the missing £1?

3 x £9 = £27 + £2 = £29...??

Well, it took me over an hour of thinking before I finally knew what the correct answer to this puzzle was and, I'm embarrassed.

It reminds me of the embarrassement some professional mathematicians must have felt in not being able to give the correct answer to the famous Monty Hall problem answered by Marilyn Vos Savant:

http://www.marilynvossavant.com/articles/gameshow.html

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

Yes; you should switch.

It's also mentioned in the book: The Man Who Only loved Numbers, that Paul Erdos was not convinced the first time either when presented by his friend with the solution to the Monty Hall problem.

So what other simple puzzles are there which the general public can understand yet can fool professional mathematicians?

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    See [Grothendieck Prime](http://en.wikipedia.org/wiki/57_(number))2011-05-03
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    While this is nitpickery, I should note that the Monty Hall problem was posed and answered long before vos Savant came along, and the confusion and logical debates it engenders have lingered well after her 'answering' it - I'm not sure she did anything for the problem except making it marginally more well-known.2011-05-03
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    It should also be noted that the Monty Hall problem is notorious in part because slight variations change the correct answer, and vos Savant's original description of the problem was ambiguous enough to possibly include several variants with different possible solutions (e.g., not specifying clearly the host *always* opens a door he knows does not contain a car, *always* offers the choice to switch). And, how did Erdos get dragged into this? I was not aware he was part of that little imbroglio.2011-05-03
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    @Art: Not sure how true this is: http://archive.vector.org.uk/art100116402011-05-03
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    @Moron: So, not part of the vos Savant little imbroglio; of course, we run into the same problem with that description: were the conditions clearly stipulated to Erdos? If the host doesn't know where the car is and just happens to open a door with a goat, switching doesn't help; if the host doesn't always offer a switch, then it may be beneficial to stay, or it may amount to the same thing, depending on how the host decides whether to offer a switch or not, etc. If the problem is not *very carefully* specified, there is enough ambiguity for alternative answers.2011-05-03
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    @Arturo: I am not claiming anything. Just was answering the possible Erdos relation. I am as surprised as you are.2011-05-03
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    @Moron: Didn't meant to imply you were claiming something. But I think invoking Erdos above in contrast with vos Savant is somewhat misleading if this is the source of the connection. It *is* true that vos Savant received letters, some fairly nasty, from professional mathematicians giving the wrong answer to the *intended* problem. Mentioning Erdos in that context seemed to me to insinuate he was one of those calling vos Savant on the alleged error...2011-05-03
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    @Art: I agree, there is no need to mention names without any solid evidence.2011-05-03
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    @Arturo The Erdős story is true. Vazsonyi describes it in detail on pp. 130-131 in [Erdős on graphs: his legacy of unsolved problems.](http://books.google.com/books?id=Doc_ErUTDcAC&pg=PA131) Vazsonyi failed to convince Erdos using a decision tree. He had to resort to a Monte Carlo simulation in Excel. A few days later Ron Graham convincely explained to Erdős *why* to switch.2011-05-04
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    @Bill: The point is that the way that Erdos is mentioned seemed to imply that Erdos was involved in the *public* dispute *with vos Savant* (remember, vos Savant later was quite smug about the fact that many mathematicians wrote to her claiming her solutions was incorrect; I interpreted the juxtaposition of Erdos with "Monty Hall problem answered by vos Savant" as a claim that Erdos was involved in *that* particular altercation.2011-05-05
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    @Arturo Yes, I agree. My point was merely to give said reference. Perhaps it's also worth emphasizing that vos Savant is known to stretch the truth. For example, in [Boston & Granville's scathing Monthly review](http://www.dms.umontreal.ca/~andrew/PDF/VS.pdf) of her error littered FLT book they mention that she "disingenuously used four eminent mathematicians for the publicity of her book". After her prior very confused [Parade Sunday paper article](http://groups.google.com/groups?selm=WGD.93Nov29055015%40martigny.ai.mit.edu) I don't take her (or her ghost writers) mathematically seriously.2011-05-05
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    I heard about the Erdös story from Ken Binmore, who was present when Erdös heard about the puzzle (thy were playing cards or something). Erdös did indeed not get the problem right away.2012-04-17

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