All this pompous commentary and nothing of substance advanced!Don't you people have better ways to waste your life? Just for once, pretend you asked this question of yourself in the privacy of your own mind. Have a life, bro.

<t>Now you prefer confusion under the guise of "It can be fun getting lost!"?How dare you say "Especially with this problem, which wasn't really that difficult if you want a classical high school answer..."?There are many major factors that can cast this problem into a whirlwind of interminable frus...

<t>QuoteOriginally posted by: Pauldunrewpp, you are far too easily upset!! Neither of the forum members posting here has shown any troll-like behaviour that I am aware of, they are genuinely trying to answer the question. Did you not say "Feel free to discuss at will" in the original post? Ok, size ...

<t>QuoteOriginally posted by: LandscapeMy intention was not to change the conditions of the problem but to illustrate that all conditions you mentioned matters and will produce different results on earth and moon. For me this became clearer when I considered these extreme cases.I agree with katastro...

<t>You are changing the givens of the problem. Sand is just it - sand - not fluid. It is not just a single stone either. Each grain of sand has a diameter smaller than the 'bottleneck' portion of the hourglass. Sand is not feather either - it is beach sand. And the size of the hour glass is just 12 ...

<t>If it takes a 12-inch hourglass, consisting of two identical 6-inch spheres, to empty in one hour on Earth, how long will it take for the hourglass to empty on the Moon?Now there are many parameters of interest that are not mentioned in this question. For example, no mention is made of the amount...

All this bloviating, lecturizing, directionless math talk! Just (re)state the problem accurately and precisely, and get on with a solution. What a self-satisfying waste of time... phew!

<t>Your discussion expands my original problem. Of course the answer to a probability question depends on how much you know about a given situation. A particular perspective gives a corresponding probability answer for the same event. Different people will arrive at different probabilities for the s...

<t>Suppose you fall into a very long and deep slumber and next time you wake up, you do so on some random day (of the year). When you are alert enough to understand a word, you are told that a rare day is awaiting you with 100% certainty. You are not told what that rare day is; only that when it doe...

<r>Your counterexample cetainly does not counter what my original post says. Recall x>0.As I have said previously, for x rational, the case is too trivial and so I have tacitly assumed x is irrational.It's more confusing to change (a,b) to (m,n). Just leave the line as y=mx, with m irrational. And r...

<r>In my proof I've sort of tacitly assumed x is irrational, since the rational case is too trivial. If N=0 for some n, then x(n+1), being always positive, would have to be bigger than x(n) -- a contradiction since the sequence {x(n)} is strictly decreasing in the case of x being irrational.You said...

<t>What could the phrase "The precise formulation + proof" possibly mean? There is no such thing as "The precise formulation" as if my formulation (as posted) is not precise or my proof is not 'really' good. For any valid claim there are possibly an infinite number of proofs and equivalent formulati...

<r>The wiki entry does not make the same assertion as my post. Wiki says:for any real number a and any positive integer N, there exist integers p and q such that 1 <=q <=N and |qa-p|<1/(N+1).Notice that p and q can be any integers, including zero. So, for a=0.00000001 and N=1, q has to be 1 and p ha...

<r>I agree with paulptli.QuoteOriginally posted by: paulptliyongge: what's wrong with p=1?QuoteOriginally posted by: yonggeThe wiki entry is not correct.I copied the wiki entry below:for any real number α and any positive integer N, there exists integers p and q such that 1 ≤ q ≤ N and counter examp...

<t>Thanks for your novel approach. But there are problems with your proof, some trivial, some maybe serious. First the trivial problems: There is a typo in one line, quoted here:"Let m_0=M, we have S_1=."I think "m_0" is incorrect and you meant to say: "Let m_1=M". Am I right?Now, on to some not so ...