Friday, August 14, 2009

NPR Sunday Puzzle (Aug 9): Would you like cream and sugar with that?

NPR Sunday Puzzle (Aug 9): Would you like cream and sugar with that?:
Q: A waitress walks up to a breakfast table with five logicians and asks, 'Does everyone here want coffee?'
The first logician says, 'I don't know.'
The second logician says, 'I don't know.'
The third logician says, 'I don't know.'
The fourth logician says, 'I don't know.'
And the fifth logician says, 'No.'
To whom did the waitress bring coffee — and why?
No doubt the waitress has to be a logician too. This reminds me of a related joke:

René Descartes walks into a diner and orders a cup of coffee. The waitress asks if he wants cream and sugar to which he replies, "I think not". He promptly disappears.

Edit: The answer isn't too difficult if you just think of each person's reply choices...
A: The waitress asks, "Does EVERYONE here want coffee?"

If the first logician didn't want coffee, he knows that not EVERYONE wants coffee so he could have replied "No." But he doesn't yet know what the others are thinking so he answers "I don't know." This tells us he wants coffee.

Similarly, the second logician has determined that the first logician wants coffee. Now we are in the same situation as the first logician. If the 2nd logician didn't want coffee, he could say, "No." But because he replies "I don't know." it means he also wants coffee.

The same logic applies to the 3rd and 4th logicians.

The fifth logican has determined that all the rest want coffee. If he also wanted coffee, he would reply "Yes" because he can accurately determine that everyone wants coffee. However, he doesnt want coffee and therefore answers "No", because he is the lone logician that doesn't want coffee and therefore NOT everyone wants coffee.

Summary:
The waitress (being the consumate logician herself) brings coffee to logicians 1 through 4. Logician 5 is not served coffee.

55 comments:

  1. Here's my standard reminder... don't post the answer or any outright spoilers before the deadline of Thursday at 3pm ET. If you know the answer, click the link and submit it to NPR, but don't give it away here. Thank you.

    ReplyDelete
  2. Once I went into a restaurant and ordered a coffee without cream. After a long delay the waitress came back and apologized that they were out of cream. She asked me if I would like my coffee without milk!

    ReplyDelete
  3. Be sure to look at Natasha's post at the end of last weeks' comments. It's a perfect comment for this week. And, knowing the abilities of the members of this group, the answer to her question is surely "yes."

    ReplyDelete
  4. I think the answers would be more like the puzzle's though.

    ReplyDelete
  5. I wonder if Mr. S will give style points in selecting the correct answer this week. Part of the challenge should be to explain the answer concisely and elegantly.

    ReplyDelete
  6. X^5 - 4*X^4 + 6*X^3 - 4*X^2 + X=0

    Not worth the trouble. Good jokes,though.

    ReplyDelete
  7. DaveJ, Re your comment posted at the end of last week's puzzle: Does a logician "assume"?

    ReplyDelete
  8. If I owned that restaurant I'd fire that waitress or send her back to school.

    ReplyDelete
  9. Answer submitted. I hope I not still in the jury room at the designated call time, however.

    ReplyDelete
  10. Since this week's logic puzzle assumes a logical waitress, here's one that is a favorite of mine and doesn't require table service. Perhaps a bit more challenging.

    # # #

    Three students are gathered in a room. The lights are turned off and a bin is passed around containing six berets, identical except that three of them are red and three are blue. Every student puts on a beret, the bin is closed, and the lights turned on. All three can now see the colors of the other two hats but not their own.

    "Who can see at least one red hat?", asks the professor. All three students raise their hands.

    "Who knows the color of their own hat?", asks the professor. About ten seconds pass, at which point one student then raises her hand.

    What color is her beret and how does she know?

    -- Other Ben

    P.S. He’s guilty, Mike.

    ReplyDelete
  11. Ben, funny how everyone says that. It's too late to get removed from the jury, so I just hope deliberations are not too contentious.

    ReplyDelete
  12. Other Ben, I think I know the answer. Definitely trickier than the NPR challenge.

    ReplyDelete
  13. is it just me or does this puzzle seem really simple?... i think i may be underthinking it, because with my logic i got a quite simple answer

    ReplyDelete
  14. Other Ben, the key to the solution to the beret puzzle is what can be deduced from the 10 seconds of silence.

    ReplyDelete
  15. Looks like at least a few others here share my feeling that this week's NPR puzzle is absurdly easy.

    - Original Ben

    ReplyDelete
  16. It is absurdly easy? I am second-guessing myself thinking that there is something I am missing, but my wife & I both have degrees in Philosophy, and can only deduce one answer/operator.

    ReplyDelete
  17. Since each logician can only either want coffee or not, the answer seems quite logically obvious.
    I'm guessing more than 300 entries, but less than two thousand (how's that for a range ?)

    ReplyDelete
  18. The next question is how many dineros did the waitress get for a tip for the coffee.
    The beret puzzle seems obvious to me. Am I missing something?

    ReplyDelete
  19. I think I went for a wordplay answer instead of a logician answer. What if all 5 logicians were siblings and children of Mr. and Mrs. John Noe?

    ReplyDelete
  20. Natasha, I wouldn't call the beret puzzle "obvious". It is definitely more challenging than this week's NPR puzzle. My guess is that you drew an unwarranted conclusion from the students' initial answers. As I hinted,there is something about the 10-second interval during which none of the students responded that is key to the solution.

    ReplyDelete
  21. Lorenzo, ok ..will reconsider the interval. Maybe they traded hats!

    ReplyDelete
  22. I misremembered this having only heard it this morning. When I saw what the waitress actually said in writing the answer was pretty clear. I was thinking she said "Can I bring you coffee?". In that case Pee Wee Herman kept coming to mind -- "I don't know, can ya?". Or maybe they really said, "Aye. Don't. No.", and nobody had any.

    ReplyDelete
  23. re lorenzo’s statement of "part of the challenge should be to explain the answer concisely and elegantly."

    Is true. Is true. Is true.
    True is the challenge of the concise and elegant explanation. Very.

    Since the server knows to bring either zero coffees or at least one coffee to that specific table's five people, its persons' wants having nothing at all to do with anyone else's of the shop's customers, then she, indeed, is as was stated ... a logician, too.

    ReplyDelete
  24. This comment has been removed by the author.

    ReplyDelete
  25. DaveJ, each student sees at least one red beret. That means they see either one or two. Think about the girl who answered. What are the implications of each option for her considering that each of the others sees at least one red beret?

    ReplyDelete
  26. The ten second interval is very important.

    ReplyDelete
  27. I confess I had to google the answer to the red beret (not so for the NPR puzzle) and the method of solving is quite elegant ! I considered this a kind of 2nd order puzzle - the information presented is not enough to solve immediately, but the lack of identified solution + time = a solidly logical deduction - excellent !

    ReplyDelete
  28. Actually, there is a kink in WS's puzzle which makes it possible for the waitress to respond accurately. She would have been in doubt about what to do in some but not all cases if something had changed.

    ReplyDelete
  29. Yes, the joy of the beret puzzle is that it requires deductive instead of inductive reasoning. Or the other way around, I forget which.

    I'll post the solution tomorrow, unless anyone complains.

    And there is another kink in WS's puzzle, which I wasn't going to mention lest someone think I'm an a**hole.

    But since Andy Brown (obviously an alias) already mentioned wordplay variations, I should note that WS started the puzzle with "A waitress walks up to a breakfast table with five logicians and asks, 'Does everyone here want coffee?'"

    So perhaps the waitress was being tailed by the five logicians (who were obviously all apprentice waiters, or perhaps busboys who couldn't find any professional work as logicians) and then they approached a table of ten people all dressed as Juan Valdez?

    -- Other Ben

    ReplyDelete
  30. I believe the beret puzzle does require an unstated assumption to solve. You must assume the reason the first two students can't determine the color of their own hat is because they don't have enough information (therefore the other solution must apply). Since a Professor is mentioned, this implies college students. However, if they are kindergarten students, the response might not be so formative...

    ReplyDelete
  31. I don't agree with that assumption. The 10 second interval before the deduced answer implies a reasoning process. It's possible that one or both other students had sufficient information but did not reason the answer.

    ReplyDelete
  32. First, I agree with what was said about this week's NPR "challenge": I had it figured out by the time WS finished stating the question. Sorry if I sound conceited, but this was so easy as to be prohibitive--I have memories from back in sixth grade when we used to do these kinds of puzzles, at which time they really *were* challenging.

    I also agree there are two kinks that determine the solution to this puzzle, one on the question side and one on the answer side; the conversation had to go exactly the way it did and only the way it did in order for the solution to be absolutely certain.

    Liane appropriately commented on WS's puzzle, "Sounds like the beginning of a bad joke." Which takes me to Blaine's "related joke": Unlike WS's puzzle, which only requires logic, this one requires logic plus one piece of knowledge... Again, not to brag, but I do like this Descartes joke! 8-D

    Finally, about the beret puzzle: you do *not* need any "assumptions" on top of the premises provided. In fact, there is a logically compelling answer for each of the two other students as to what color his or her beret is.

    Does everyone on this blog have the answer to the beret puzzle by now? -- I don't know. >;@>

    ReplyDelete
  33. The Beret Puzzle, Answered:

    Three students are gathered in a room. The lights are turned off and a bin is passed around containing six berets, identical except that three of them are red and three are blue. Every student puts on a beret, the bin is closed, and the lights turned on. All three can now see the colors of the other two hats but not their own.

    "Who can see at least one red hat?", asks the professor. All three students raise their hands.

    "Who knows the color of their own hat?", asks the professor. About ten seconds pass, at which point one student then raises her hand.

    What color is her beret and how does she know?

    ANSWER:

    Her beret is red. In fact, all three are red. The answer lies in the ten second pause before she raises her hand.

    We'll call her student C. She sees student A with a red beret and student B also with a red beret. All three students, of course, raise their hand noting that they see a red beret.

    The professor asks "Who knows the color of their own hat?" and nobody raises a hand.

    Student C realizes that if she wore a blue beret, then students A and B would know the color of their hats to be red. Why? Because C sees A and B wearing red.

    C imagines life from A’s perspective. If A had seen B wearing red and C wearing blue, then A had noted B raising her hand that B saw a red hat, then A could have guessed that A’s hat was indeed the red hat that B was seeing when she raised her hand. So if A is logical, then C’s hat cannot be blue.

    Likewise, if B had seen A wearing red and C wearing blue, then noted A raising her hand that A was seeing a red hat, then B could have guessed that HER hat was the red hat A was seeing.

    Since nobody raised their hand in those ten seconds, C knew that her hat, like the other two she can see, is red.

    -- Other Ben

    ReplyDelete
  34. I think you do need assumptions in the beret puzzle. The assumptions being that the students all "know" if it is immediately "knowable" or not and if they know they make it known that they know.

    ReplyDelete
  35. This comment has been removed by the author.

    ReplyDelete
  36. Shorter explanation:
    Only one of two scenarios support each student seeing at least one red hat: either 2Red/1Blue or 3Red.
    If anyone saw a blue hat, they could immediately deduce that their own hat was not blue ('cos then someone would have seen 2 Blues and that is not viable )
    Thus after 10 seconds, no-one deduces that they don't have a blue hat, therefore the only possibility is 3 reds (anyone of three students could deduce this by the way).

    ReplyDelete
  37. Other Ben,

    Thanks for the beret puzzle, and indeed, given the premises, the answer has to be "red." About the other berets, though, I think one of them still needs to be blue, at least if you want the puzzle to be purely logical. The reasoning behind your answer that all berets are red goes beyond the premises, as it invokes assumptions made by student C about the nature of that ten-second period of silence.

    As you pointed out yourself, the only way to establish logically, for certain, that your own beret is red is if you see one of the other students wearing a beret that is blue. Ten seconds is a pretty good time for anyone to figure that out, don't you think? Another ten seconds, and the other student who was also wearing a red beret might have come up with that result for herself.

    I see where you are going with student C inferring from what *appears to be* uncertainty on the part of the other students that all berets are red. But again, if you wanted the solution to this puzzle to be determined by logic and by logic only, then you would need for one of the three berets be blue. That way, one more student could infer their beret is red, and the third student could answer that a logical conclusion is impossible based on the visual information available to them.

    I have a logic-related issue of a different kind with this puzzle: Per your explanation, all of the three students are females, since at some point you refer to each as "she" or "her." The professor, however, is quoted as saying, "Who knows the color of their own hat?" If these are the professor's *exact* words, then there must be at least one male among the three students; otherwise, it wouldn't make sense for the professor to use the gender-neutral "their." So... which one is it, A or B? Or perhaps it is both--since they were both lagging behind the smart one? ;-)

    ReplyDelete
  38. TWO BERET PROBLEM:

    Natasha and Dave are wearing berets. The berets are either red or blue. They can see each other’s beret, but not their own which they must guess. If either person gets the color right, then they both receive $1000. They cannot communicate with each other in any way after the hats have been placed on their heads and they must both say their guess at exactly the same time. However, they can meet in advance to decide on a strategy. Find a strategy that guarantees a payoff.

    ReplyDelete
  39. Wow, I feel like a celebrity!
    Dave we need to discuss the strategy.

    ReplyDelete
  40. Blaine, very nice puzzle with an elegantly simple solution (which I won't reveal until others have a chance to try).

    Dave and Natasha: I have a deal for you. I'll give you the strategy if you each agree to give me half of your winnings.

    ReplyDelete
  41. Hmm, $1,000 in Blaineworld dollars - wonder what the exchange rate is on Second Life...

    Unless standing feet apart for red/together for blue doesn't count as "communication" I'm kind of stuck for ideas...

    ReplyDelete
  42. DaveJ, feet placement is communication, I believe. Lorenzo I wish I knew the answer you have in mind.

    ReplyDelete
  43. How about if Natasha agrees to say the opposite of Dave's color and Dave agrees to say the same color as Natasha's hat ?
    Since their hats are either going to be the same color or opposite, I think one of them will be right - yeah ? (or as Mark Knopfler said "two men say they're Jesus, one of them must be wrong...)

    ReplyDelete
  44. Blaine,

    Is there any relationship between the two berets?

    Or are we starting with a sufficiently large supply of berets so any combination is possible?

    -- Other Ben

    ReplyDelete
  45. The berets are either red or blue, so Natasha and Dave either have the same color or opposite color hats. If one bets the same and the other bets opposite - seems like they would have both cases covered...

    ReplyDelete
  46. Lorenzo and DaveJ are correct. The key is *either* of them can be right. If they cover both cases (same or different), one of them will be right and they'll win the $1000 prize.

    ReplyDelete
  47. Blaine, when can I expect to receive my share?

    ReplyDelete
  48. Great puzzle by the way Blaine - please keep 'em coming ! I donate my share of the prize to the retired logistics waitress' fund...

    ReplyDelete
  49. Natasha and Dave, here are your 1000 (virtual) dollars. You have to decide how to split the prize.

    Incidentally, I can't take credit for this puzzle. It came from a post on a blog entitled Division By Zero which in turn came from a talk at MathFest.

    If you liked this puzzle, try out the *infinite* hat problems in the same post. Infinity was too much for my little brain to handle so I stopped at two hats. :-)

    ReplyDelete
  50. Re: Beret puzzle

    I understand the answer and logic, but an assumption is necessary. Specifically, that one student is more logical than the other two. Or at least a faster thinker.

    When the puzzle states that one person raises his hand after ten second it implies that this one person was able to come to his conclusion as to what color beret he's wearing uniquely and that he had information unavailable to the others that allowed for his deduction. However, all three had enough information to confidently raise their hands once they saw no one raise their hands after a few seconds. But only one did?

    The wording is, at least, misleading.

    ReplyDelete
  51. My thoughts exactly, Patdugg. Glad to see your post.

    ReplyDelete
  52. If that waitress had asked around the table in the reverse order, beginning with the 5th logician, they all would have answered, "no".

    She would have brought NO coffee to their table. Four of the logicians would have been thirsty. They would have complained to the manager, and the waitress would have been fired! What kind of "consummate logician" is that? As I have commented, this puzzle was defective!

    ReplyDelete
  53. Well ... I have BEEN that consummate server: talented and skillful at it – and, more importantly to you the customer, to you my manager ... ... I loved doing it ---- Louise Sawyer – style in one of the busiest, long – distance hauling, interstate truckstops within the very midst of middle America. ... And, further, because I was so good at it 40+ – of – my – Woodstock years ago, very well could, if job situations came about again in this economic era such that I needed to, I could very well step right in and perform for a solvent living food - servicing serving ... again ... today.

    When a truckstop server IF I had, at the first, asked logician #5 with her no answer, THEN what I wanna know from folks here in BlaineNation is: how is it that the minds can be read of logicians’ #4, #3, #2 and #1 – to know ... now ... what then after #5’s no answer, their subsequent responses are to have become?

    Someone states "they all would have answered no." This is known ... now ... how exactly? Further stated is, “yada, yada, yada ... she would have been fired” cuz “what kind of consummate logician yada, yada?” “defective puzzle” and so forth.

    Again, I ask: How is it that minds can be read re logicians’ #4, #3, #2 and #1?

    Mr Shortz’s Sunday, 09 August 2009 puzzle challenge is not defective.

    ReplyDelete
  54. Very simple. Number 5 does not want coffee, so she knows that the answer to the waitress's question is "no". All the rest of the logicians, although thirsty for coffee, also know that the only logical answer is "no", that they don't ALL want coffee!

    So the puzzle is defective. A waitress-logician would never have fallen into such a trap by asking the wrong question! If she had asked it, and gotten a sequence of 5 "no" answers, she would have been smart enough to follow it up with a roll-call vote to identify any thirsty individual.

    That is why I consider the puzzle defective.

    ReplyDelete
  55. By the way, the waitress' name was not Suze, but Ivy (geddit ! - lighten up people..)

    ReplyDelete

For NPR puzzle posts, don't post the answer or any hints that could lead to the answer before the deadline (usually Thursday at 3pm ET). If you know the answer, submit it to NPR, but don't give it away here.

You may provide indirect hints to the answer to show you know it, but make sure they don't assist with solving. You can openly discuss your hints and the answer after the deadline. Thank you.