bonanova

Nice. Gardner simply asked the question (which he helpfully worded: "What's the only word that ...) I gussied it up with the story.

The first step - making it the sum of two quantities - is correct. From there the minimum can be calculated directly.

Without resorting to differentiation (sorry, Y-San) find the minimum for (0< x< pi) of the function

Would it help to say that clues abound in this puzzle?

Not sure what you mean by flat. What am I missing?

Natural numbers are positive or non-negative

Much more satisfying, as well! Good one.

Hi shepster, Yeah, an oldie but goodie. You got it. Welcome to the Den.

I have four gold chains, each having three links: (trust me, they are interconnected.) OOO OOO OOO OOO How many links must I cut and rejoin to make a circular chain of twelve interconnected links? O O O O O O O O O O O O

Yes, it's true. Otto never loved Yolanda. His assistance in the cleaning of her parrot cage signaled the notion that he did, but with time it became apparent that his affection was for the parrot, not for Yolanda. Yolanda took it quite well, considering. But, for the purposes of this puzzle, none of this sad story really matters. Sorry. A bit of a red herring. What does matter is the fact that the nine-word sentence that describes Otto's cage cleanup admits ten different meanings, slight differences though they be, when a single, common English word is inserted at any of the possible locations in the sentence: in short, in any of the blanks shown below: ____ Otto ____ helped ____ Yolanda's ____ daughter ____ clean ____ Yolanda's ____ parrot's ____ cage ____ yesterday ____. What's the only word that makes sense in every blank, where the sense is different in each case? Credit: Martin Gardner.

@BMAD by figures with "identical areas" do you mean figures having "equal areas" or figures having "congruent shapes"?

Is this a fair paraphrase of the OP? If Proposition x were put to me, I would say it's true. If Proposition x were put to UTM, it would respond "false." Give an example of Proposition x.

@BMAD "Devise a true proposition ... " does its truth depend on UTM's response? If not, it seems impossible.

You're right. They only force a "false" response (unless the machine wants to out itself.)

Solved by k-man. for gavinksong

I know this isn't exactly what the OP is asking for, but it seems that's as close as I can get it. You're right, I think; the OP asks for a true statement.

Is this an accurate paraphrase of the OP? The machine takes a proposition and responds with one of {"true", "false", "undecidable"} Given its responses to these three particular inputs, ... devise a true statement to which the machine will respond "false".

All correct conclusions. k-man was first, and his post was thoughtful enough to signal the answer without spoiling the puzzle. I'll mark it solved after giving some time to come up with a words-only solution, which Erdős would very likely agree belongs in The Book, and earn the gold star. Specifically, neither the enumeration of cases nor an infinite sum, both of which give the correct answer, are needed. It will also show that dgreening's last comment does not quite apply to k-man's expected cost, which is exact.

Is this claim for any nxm table or only when n = m ? Does OP ask for every nxm, or a particular nxm?

Does this account for all 1000 patrons?