bonanova

We've come from what seems at first to be an intractable problem and found some very nice results. Since the table edges seem like a separate issue, let's separate the rectangular table case from the infinite plane case. One has edge effects, the other does not. It seems we're an inch away from two nice general proofs. So ...To generalize the finite table problem, let's amend the OP to say that N plates were used initially. This precludes inspection of specific table dimensions. For the infinite plane problem, HH can finish his hex packing analysis to obtain a very elegant result.

11 has been given, not quite the purpose here, but I might have said, think of the first three letters. With the O in the middle, what prefix might suggest the intended meaning. That would suggest CON or COM [from latin ***: with or together] CONfer - work together COMfort - with strength or aid CONfluence - flow together. That approach might help with the others, or not. ------------------ Note to helpers, read the sticky - try to help with the process. That being said, this particular problem does not lend itself to process.

Because I almost play the piano, I'll grab this one:

1 + 1 = 1 + i is formally valid. It is also decidable. Only because of that can we say that it's incorrect, or logically it's false. Your "Cow" statement is tantamount to something like 1 + = x 4 - x 3 - / i 13 / = 45 / -. That is formally invalid and cannot be evaluated or discussed as to truth, meaning, decidability. You can't say it's false, because you can't even compose an assertion from it. That is the point. This discussion is about pinning down the meaning of valid. You are being a bit slippery about it because on the one hand you say if a statement is false or if it's undecidable, then it's invalid. Then you say if a statement is invalid, that means you can't even discuss it. [i.e. carry on a discussion of OP] Because calling it invalid is to have discussed it, you can't have both meanings. ------------ Consider your ability to say that someone had just told you a lie. Well, you couldn't. Because by your definition, it would be an invalid statement. Being invalid, you would be unable to determine whether it was true. If you say no, first you determined that is was not true, then called it invalid, you're no better off: You still have an invalid statement on your hand and you'd have to apologize to yourself for having determined it to be false. Because, of course, you can't analyze invalid statements. Net: I agree that you cannot analyze an invalid statement. A statement has to have formal validity for it to be analyzed as to meaning. I do not agree - it seems it only needs to be stated - that a false [or undecidable] - statement is not therefore invalid. False statements can [must] be formally valid.

xucam, See if one of these statements helps: If you work with only a subset of OP-allowed cases, you have not ensured coverage. To ensure coverage for all possible configurations, you have to cover the worst case. If OP does not disallow a particular case, you can't rule it out.

What invalid means to you is still unclear. Two comments: Your simile seems shaky: "This statement is false." is formally valid, both logically and syntactically; equally valid in a formal sense as "This statement is true." is. Given the formal validity of the statement, it becomes possible to discuss its meaning, its truth value, its decidability, etc... That the statement can be discussed cannot be even questioned: It is being discussed in this forum. To disqualify it as being not discussable ignores that fact. To disqualify it by saying it's invalid, is to beg the question: invalid is defined by its usage, producing justification by assumption: viz, a circular argument. (a) It's disqualified because it's invalid. (b) It's invalid, because to say so lets me disqualify it. By stark contrast, "Cow touch throw at mail." is not formally valid in either sense; and cannot be discussed. You use the term invalid to disqualify a formally valid statement, rather than discuss it. But neither your simile nor your disqualification of the statement sheds light on why disqualification is permissible. If your meaning of invalid were out in the open, one could take the statement to that definition and justify [or not] disqualification. To that end, A term may be defined in [at least] three ways.By giving examples - as in "Cow touch throw at mail.", but that likeness is in dispute. By explaining its usage - but how to use invalid, once understood, is probably not in dispute. By substituting another word [already understood] or phrase - with sufficient precision to justify the intended use of the term.

Consider an acute triangle ABC with angles α, β, γ as shown. The figure is not necessarily to scale. Let the bisector of angle BAC [α] intersect BC at P. Let the bisector of angle ABC [β] intersect AC at Q. Let AB + BP = AQ + QB. Let α = 60o. Find β and γ.

Can anyone construct a counter example? If not, this stands as the lowest [not disproved or retracted] answer to date.

It's clearly undecidable. When you say [not] valid do you mean [not] decidable? Validity usually has a formal sense, in which statements of the form "This A is B." are permitted, independent of A and B. By your definition, is "An invalid statement is neither true nor false." valid? [Not] provable or [not] decidable are more useful terms to use here I think. You must be permitted to make a statement; then decide on [a] it's truth or provability.

Read the spoiler if you want to know the answer, or PM me the answer YOU think it is, or just PM me if you think I've done it wrong. As others have noted, this is different from the Liar paradox "I am lying." or "This statement is false." The assertion "This statement cannot be proved." is a close approximation to a proof of Gödel's first incompleteness theorem. Logicians up through Hilbert, but starting with the Greeks, envisioned a system of proof which, given a useful set of axioms, could compute proofs of all valid theorems, therefore creating a complete and self-consistent system of logic. Didn't happen. Gödel blew that dream up by showing that any system of inference, sufficiently rich to be useful, had a fatal flaw: It was either incomplete or it was inconsistent. For example, it is impossible to prove the statement in the OP. Being incapable of proof, it is true. But to be complete, the system must be able to prove it. Thus the system is incomplete. So this is an example of a Gödel statement. It's true by the fact the mere fact that it cannot be proved. And that fact - that the system contains a true statement that eludes proof - makes the system incomplete. It takes a moment or two of thought: it's almost as if Gödel is saying you simply have to say the system is incomplete for it to be so. Which doesn't seem intuitive. But it stands. P.S. Gödel was not liked when he did this. To show their disdain for Gödel, logicians have simply gone ahead and used the system to achieve useful results, anyway

My answer to this chestnut has always been: If a falling tree can be postulated without a corroborating observer, then its sound can equally be postulated without a corroborating listener. Corollary: If a man speaks in a forest. and no woman is present to listen, is he still wrong?

You all have it.

Excellent work all. Given that observation, does a proof suggest itself? Can the proof be made rigorous? We have a number that is sufficient to cover the table; is it also necessary? [i.e. optimal?]

OK, now having fixed the OP, get the total to 100.

We assume the crate has rectangular, perpendicular sides. That is, it's a right rectangular pyramid. And we want the outer crate to cost less than the inner crate.

Consider in two dimensions a rectangle can hold a longer rectangle [albeit a narrower one] at an angle. It's easy to prove the containing rectangle has a greater perimeter in that case. Can you extend that proof, or come up with another proof, for three dimensions?

Don't assume that. The new containing crate is to have a lower sum of dimensions but contain the first crate thus saving money.

Place as many "+" signs as you like between the seven non-0 digits 1234567, in that order, to get 99 100. Example: 12 + 34 + 567 = 613 [that's a little high, but you get the idea.] After you find it, do it again, another way.

Place as many "+" signs as you like between the nine non-0 digits 987654321, in that order, to get 99. Example: 98 + 76 + 543 + 21 = 738 [that's a little high, but you get the idea.] After you find it, do it again, another way.

The cost of shipping a crate is specified by the sum of its length, width and height. You'd like to save money by packaging your crate inside a cheaper one. Prove that this is [or is not] possible. Assume crates have perpendicular, rectangular, and negligibly thin sides.

99.99 ... uses only four nines. And some periods, which in fairness the OP does not permit: only 9's and +, -, x, / Consider that "...." is a mathematical notation for infinite repetition. Thus any number, including two, 9's suffice to stand for an infinite number of 9's. 99.9 .... = 99.99 .... = 99.999 .... = 99.9999 .... = 100.

Nope.