## 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:

RenĂ© 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.

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.

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!

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."

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

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.

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

Not worth the trouble. Good jokes,though.

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

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

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

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.

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.

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

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

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

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

- Original Ben

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.

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 ?)

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?

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?

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.

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

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.

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.

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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?

26. The ten second interval is very important.

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 !

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.

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

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...

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.

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. >;@>

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?

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

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.

35. This comment has been removed by the author.

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).

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? ;-)

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.

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

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.

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...

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

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...)

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

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...

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.

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

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

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. :-)

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.

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!

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.

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.

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