taeto

It sure is. I am mildly disappointed though, if it really follows from the grammar of the english language that Jack's first comment actually already implies that the initial digit is 9 and the last one is 0. Something which I seemed able to deduce logically from the ensuing pieces of information. Surely you agree that as a riddle, it would be better constructed if essential pieces of the puzzle are not revealed at the start, rather than at a point later when they can be deduced logically.

I took this to mean that Jack does not know the values of all the digits. Possibly none, a single one, or any number up to three, but not all four. Can you explain any case, except when the range is 9, when Jack can truthfully make this statement? Because in this case he will know that \(a\) is 9 and \(d\) is 0. Anyway, we agree that it follows from the entire conversation that \(a\) is 9 and \(d\) is 0. I certainly do not agree that it follows from ordinary english usage that Jack is in any way indicating that he knows that the range is 9.

Great. So we can take \(S=2\) as an example sum. The possibilities are obviously just 1100 and 2000. At this point Jack knows that the range is 1 or 2. If the range is 1, then Jack will have to admit that 2111 may be a solution, in which case James may know the true answer for certain since the product is \(p=2.\) Jack has no basis for stating that James cannot know the answer. If the range is 2, it is similar, since 3111 may be the solution. In either case, knowing that the range is 1, or knowing the the range is 2, does not provide Jack with enough information to ensure that James cannot figure out the answer just from knowing the product \(p\).

James may actually know. I suspect you are missing this point. John has to work with the information available to him. James knows how much information is available to John (except for the precise value of the sum). So James may well be entitled to make statements about whether John can or cannot know the answer, based on having the sum available.

I assume you mean a function \(f : [0,\infty) \to \mathbb{R} \) with \(f(0)=90\) and \(\lim_{x\to \infty} f(x) = 300.\) The quality of the answer depends on your personal taste in functions. You can have something absolutely natural, but with an intricate look. Or with a very simple design, but properties that are ugly when you look into it. The function \(f \) given by \(f(x)= 90 + \frac{210x}{x+1}\) is an obvious first choice. It looks pretty enough in a graph. Analytically it is still a little questionable. I do not know how you feel about a function involving a hyperbolic arctangent as an alternative.

I am not sure where you are going. Certainly neither 0000 nor 1000 can be solutions, based on the fact that John would immediately realize this, based only on the value of the sum. If any of the guys are lying, there is no way the problem makes sense, so I will assume they are all truthful. Conversely, do you see how any of the numbers 9990, 9980, 9100, 9000 would not qualify? I see a problem exactly where you identify it as well, namely that neither of these purported solutions would allow John to deduce that Jack will also know it by then. Each of them looks the same from Jack's point of view, does it not?

Okay, I thought that this is how the range of solutions would be interpretated anyway. The sum 0 being the solution means that \( (0,0,0,0) \) is the solution. But James says that John will not be able to figure out the solution. However, John knows that the sum is 0. So James is lying when he says that.

I am not sure how you mean that. The sum 0 cannot be in the range of solutions, because James says (truthfully, I hope) that John cannot yet know the solution. And when (0,0,0,0) is still in the range of possible solutions, so far as James can ascertain, it would be a lie if James says that John cannot possibly know the solution yet at that point. At least James does not have enough information to make that statement.

Certainly it is suggested to plot \(y = \log(K) \) versus \( x = \log( 100M_s/L_T^3) = \log(100M_s) - 3\cdot \log(L_T). \) The reason being that the latter expression only has linear dependencies, meaning something like \(y = ax+b, \) as opposed to ratios and exponents that appear in your original. You will obtain a graphical representation which is pleasant to look at, and if you need to do a regreession analysis, these are the points you would want to do it on.

Yeah. How long have you been stuck with programming? But seriously, I actually love programming stuff, did a lot of research in pure maths by writing programs, letting them run for a bit, and check out what kind of garbage they would deliver as output. It really works quite well to get papers in pure maths and theoretical computer science. Your intuition is not unreasonable. I also thought that the answer would not be something linear. Though on reflection, it is the case that as you work through your load, it gets gradually lighter, though not by a whole lot.

But now all quadruples \( (9,9,9,0), (9,9,8,0), (9,1,0,0), (9,0,0,0) \) seem possible?! They are the possible \( (9,b,c,0) \) for which the sum \(9+b+c+0 \) with \(9 \geq b \geq c \geq 0 \) determines \(b\) and \(c\) uniquely. I must be doing something wrong 

If "range" is simply \(r:=a-d,\) then we have: Lemma. \(r \in \{7,8,9\}.\) Proof. Jack knows that the product \(p:=a\cdot b\cdot c \cdot d\) is not enough information to determine \(abcd\). But if the range is one of the numbers \(1,2,4,6\), then the value \( (a,b,c,d) = (r+1,1,1,1)\) would be the unique quadruple with \(p = r+1,\) since \(r+1\) is a prime, which means that the value of \(p\) determines the solution uniquely. Also \(r=0\) is excluded, because of \( (9,9,9,9) \). For \(r=3\) the quadruple \( (5,5,5,2) \), and for \(r=5\) the quadruple \( (7,7,7,2) \), are determined by \(p\). That leaves the three possible values stated. Does that make sense? Edit: \(r=7\) is excluded, due to \( (9,9,9,2 ) \) as well. Edit: after a further think, \(r=8\) is also excluded, because of \( (8,0,0,0) \). Because now James and John both deduce \(r \geq 8 \) from the information that Jack is giving. John's sum \(s = a+b+c+d \geq a + 0 + 0 + 0 \geq r \geq 8 \) can be equal to 8 only in this particular case. Which means that James cannot deduce that John will not be able to get the correct answer. The upshot is that we have \(r=9\) and \(p = 0\), which implies \(a=9\) and \(d=0.\)

The condition \(a\geq b\geq c\geq d\) eventually got made explicit by the OP, hence it should be safe to assume. Which assumption do you work with for the use of "range" though? Maybe the question is on a higher level, where you have to figure out the unique use of "range" which produces a single possible answer. That would be nasty.

Yes please. I am confused about the use of the term "range". Conventionally it would mean the set of numbers \(\{a,b,c,d\},\) but it looks like it is used for the difference \(a-d\)? It could also mean the interval \([d;a]\) of numbers from smallest to largest?!

I would think that \(i\) is just a shorter, and commonly used, way of writing the complex number \(0 + 1\cdot i\)?

It is quite common in enumerative combinatorics to apply this kind of series expansion in situations where you want to count the number of different possible objects of some kind. In your example, the series \(A(x) = \sum_{n=0}^\infty x^n = 1 + 1\cdot x + 1\cdot x^2 + \cdots = \frac{1}{1-x}\) would give the answer to the "problem" ''How many natural numbers are there of size \(n\)?'' You get the answer by looking at the coefficient of \(x^n.\) In this application it makes no sense to "evaluate" a series at some value of \(x.\) The radius of convergence of the series does not play any role. In fact you may consider a series which counts the number of permutations \(n!\) of the numbers \(\{1,2,\ldots,n\}\): \[ P(x) = 0! + 1!\cdot x + 2!\cdot x^2 + \cdots = \sum_{n=0}^\infty n!x^n.\] Which has no convenient shorthand similar to \(A(x) = 1/(1-x)\) for the series \(A(x).\) In fact \(P(x)\) has zero radius of convergence; there is no value \(x \neq 0\) for which \(P(x)\) converges. And yet the series makes complete sense combinatorially.

If your dog eats \(x\) of the articles before you begin, and you start from only \(90-x\) articles instead of \(90,\) yes. Then \(Y(x) = \frac{90-x}{1-0.7}\) would be the total number of articles you would have to read. Remember that \(Y\) is the name of the function, and \(x\) is the name of its argument. It is \(Y\) that is the function, not \(Y(x),\) which is the value of the function \(Y\) at \(x.\) If for every 2 articles you have read, you have to add .7 more articles, then the 1 would become a 2, etc.

It comes to 300 articles because \(\frac{90}{1 - 0.7} = 300,\) no?

It looks like you confused \((-i)^2\) with \(-i^2\). The former is equal to \(-1,\) the latter to \(+1.\) The complex numbers \(\mathbb{C}\) form a field with addition and multiplication as defined for complex numbers. In particular \( (\mathbb{C},+) \) is a group. In an additive group, the inverse of an element \(x\) is written \(-x.\) Minus something denotes the additive inverse to that thing.

A 5 year old who sincerely asks for explanation of the meaning of infinite things in mathematics should preferably be given at least some bits of actual information. It seems that the arithmetic is basic and a fine place to start. I worry about how the philosophical stuff sounds circular and worthless. Some people have to look up things on wikipedia or the like, especially if they are not familiar with the basics, and even so, if they look for some further idea to spice up an explanation. But materials should at least be currently relevant, not only having had possible value in ancient times. If the 5 year old asks about the source of combustion, I would feel it is misguided to explain the phlogiston theory as my answer.

And yet I seem blind to it . Unless this idea presupposes that "infinity" is already available as the amount of time, or the number of steps, that something can take. I have not. But maybe physicists have their own idea of the concept, separate from philosophers? Is it to "use infinity" to integrate products involving wave functions over \(\mathbb{R}^3\) to calculate the expected value of an observable in QM? If so, then I have seen that a lot. Maybe I should explain why "something larger than any natural number" is a meaningless explanation in mathematics. I assume that natural numbers are ordered in their natural sequence. Also I assume that \(\{1\}\) is not a natural number (as e.g. in Von Neumann's classical construction of \(\mathbb{N}\) ). Then let \(\infty = \{1\}\), and extend the natural ordering \(<\) of \(\mathbb{N}\) to \(\mathbb{N}\cup \{\infty\}\) by letting \(n < \infty\) for every natural number \(n.\) Then it is clear that \(\infty = \{1\}\) satisfies the requirement for being infinity. You can replace \(\{1\}\) by anything that is not itself a natural number.

Doing fine, thanks a lot! It is not quite that I do not know stuff. Maybe I know quite a lot. But I have trouble to look up the right pieces of stuff and put them together into some appropriate picture which it seems everyone else can see. In the explaining infinity to a 5 year old situation, the focus turned to explaining how for every number you can add one to it and get a larger number. This is covered by integer arithmetic. There is no object of integer arithmetic that is larger than any integer or has no bound; nothing can be infinite. The "set of all natural numbers" belongs to set theory, not to arithmetic. You will have to add a concept of repeating the addition, and make a mental picture of "repeating forever". Is it correct to say that this already presupposes the concept of time being able to extend infinitely into the future? In which case the "infinity" stuff is something that you have to add extra into the mathematics and which is not already present from the outset? The "potential infinity" construction is well-liked by cranks especially. You save the day by adding "potential" in front of the despised and/or scary "infinity". How does this actually work semantically: is it a similar kind of construction like when you add "real" in front of "number", to get "real number", which is an object altogether not the same as a "number" (by default a natural number) that is "real"? If so, then introducing "potential infinity" in order to explain "infinity" is a red herring, just like you do not explain natural numbers by making reference to real numbers. And if the construction is to make a new concept "potential infinity" starting from the concept "infinity", thus meaning "the kind of infinity that is potential", then it also does not help to explain how to come up with "infinity" in the first place. Sorry for rambling. Maybe at least I get my confusion across.

Very well then. Everything in mathematics is philosophical. But the converse does not seem to apply without restriction. Is everything philosophical also a mathematical concept? What I have is from the Wikipedia page: "Infinity is a concept describing something without any bound, or something larger than any natural number." Since this is not a mathematical explanation, I take it to be an explanation of the philosophical concept. The sentence following it states that "Philosophers have speculated about the nature of the infinite..." Maybe "Infinity is a concept describing something without any bound, or something larger than any natural number" is not even a philosophically valid statement, in which case I admit to be barking up the wrong tree. I simply am not sufficiently familiar with philosophy to tell. Although then the sentence about "Philosophers have speculated..." would seem largely irrelevant if that is the case.

I am sorry, but I feel that I really need this explained. Prompted by a very nice account that I read on math stack exchange from a parent who tried to answer questions from a 5-year old child about "infinity". The reactions turned into suggestions about whole numbers and "numbers that are larger than all numbers that you can imagine", and similarly. Numbers? I looked up the wikipedia page on "infinity". Similar story. The first sentences establish that "infinity" is a philosophical concept concerning something without any bound. Then it is said that "modern mathematics uses the general concept of infinity..." I may be a terrible example of a parent to a 5-year old. But I do know that mathematics in no way uses the philosophical concept of "infinity" (or indeed any other purely philosophical concepts). The infinity symbol \(\infty\) is used in mathematics to abbreviate expressions involving limits, typically in summations and integrals. There is no "concept" involved, since you could freely choose to replace the symbol by the more tedious original limit expressions without any change of the meaning. The symbol is also used in situations, in topology, elliptic curves, and real and complex analysis, etc., when it is practical or necessary to add an additional element to an existing structure to obtain a larger structure with desired properties. The "extra" element is traditionally named using the symbol \(\infty\). There is no concept whatsoever involved in this choice, since you could use any other symbol desired. The statement "mathematics uses a concept of infinity" seems a basic misunderstanding, from my traditional mathematical point of view. Maybe someone can point to a usage of "infinity" in mathematics, which is actually related to the philosophical concept and not just as a choice of a symbolism?

If they really give away an actual paperback book, then I would be more happy to receive that, in lieu of reading everything here . Edit. I see now that the paperback version is still $9.85 at Amazon. I was hoping to save a bit more . Oh, it does say kindle, my mistake .