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[math-fun] The outwardly meek continued fraction
by Bill Gosper 06 May '17

06 May '17

1²/(1 + 2²/(2 + 3²/(3 + · · · ~ .480175529546832 apparently does not simplify beyond 2 Hypergeometric2F1[2, 2, (5 - 1/√5)/2, 1/2 - 1/(2 √5)]/ ((-1 + 3 √5) Hypergeometric2F1[1, 1, (3 - 1/√5)/2, 1/2 - 1/(2 √5)]) Mathematica has been startling me with valuations of K[quartic,quadratic,...], but not this time. Does anyone know the simple CF 1² + 1/(2² + 1/(3² + · · · = 1.24328847839972? It's probably just a quotient of confluent hypergeometrics. Somewhere in my notebooks may lie a tool Julian built for doing it, but where? --rwg

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[math-fun] Brain teaser
by Henry Baker 06 May '17

06 May '17

Paul is looking at Linda; Linda is looking at John. Paul is married; John is not married. Is someone who is married looking at someone who is not married? A. Yes B. No C. Can't tell

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Re: [math-fun] Brain teaser
by Dan Asimov 06 May '17

06 May '17

The twins' last name is Winklevoss (not Winklevos). —Dan ----- Indeed. Or 'Winklevoses', since it derives etymologically not from the Greek 'Winklevos', but is instead a back-formation from the English word 'Winklevology': https://mathoverflow.net/a/60915/39521 -- APG. > Sent: Friday, May 05, 2017 at 6:25 PM > From: "Gareth McCaughan" <gareth.mccaughan(a)pobox.com> > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] Brain teaser > > On 05/05/2017 12:20, Veit Elser wrote: > > > One of my favorite moments in “The Social Network” was when the Zuckerberg character refers to the Harvard twins as “the Winklevii”. > > It ought to be "Winklevoi", though. -----

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[math-fun] Grandi valuations
by James Propp 05 May '17

05 May '17

If S is an eventually-periodic subset of {0,1,2,...}, then |S|_q, defined as the sum of q^n as n ranges over S, is a rational function. The map from S to |S|_q is a finitely-additive measure ("valuation" is probably a better word to use) taking its values in the ring of rational functions in q. If we order that ring "at 1", we get a way to compare sizes of sets of natural numbers. In fact, it is a total ordering (though not a well-ordering) of the set of eventually-periodic subsets of {0,1,2,...}. This may remind some of you of the ordered fields that you get by adjoining to R a formal infinitesimal or a formal infinity. In both cases, you've got the field R(x), and all that differs is the ordering you put on it. My proposal is similar, except that I'm ordering the field at 1 rather than 0 or infinity. Writing |S|_q as A / (1-q) + B + lower order terms, we can focus on just A and B. A measures the density of S, and B captures finer information. The first thing to notice about B is that it's sensitive to perturbations that don't affect the density; if you add an element to S, you increase B by 1, and if you remove an element from S, you decrease B by 1 (even though A doesn't change). But B conveys subtler information too: For instance, compare {0,2,4,...} with {1,3,5,...}; the former has B = 1/4 while the latter has B = -1/4. Informally, we say that sliding the set of even nonnegative integers over the right by 1 reduces the "size" of the set by 1/2. And that makes sense, because if you slide it over to the right again, "losing" half an element again, you get {2,4,...}, which is missing precisely 1/2 + 1/2 = 1 of the elements of {0,2,4,...}. (Yeah, this way of measuring sets is not translation-invariant. But you gain something nice in return: a proper subset of a set is always strictly smaller than the set.) I recently learned that Ilya Chernykh has been exploring the same circle of ideas (see http://mathoverflow.net/questions/215762/non-standard-numbers-and-exponenti…) but I don't know of anyone else who's fished in these waters. (You could argue that analytic number theorists know all this already and don't think it's interesting.) Anyway, the map that sends S to (A,B) should I believe be called a "Grandi valuation", since it gives a rigorous framework for making sense of Grandi's formula 1 - 1 + 1 - 1 + ... = 1/2. (Indeed, Bernoulli's formula 1 + 0 - 1 + 1 + 0 - 1 + ... = 2/3 can be understood in the same way: {0,3,6,...} exceeds {2,5,8,...} by 2/3 of an element, if we measure size using this valuation.) More precisely one might call the map the "Grandi-Abel valuation", since it uses Abel regularization of the summation of the indicator function of S. Anyway, other ways of taming divergent series exist, so there will be other Grandi valuations. And maybe the term should also be applied to the non-truncated valuation (the one that takes values in the ordered field R(q)), which gives the sets {0,3,4,7,8,11,...} and {1,2,5,6,9,10,..} different "sizes" (the former is infinitesimally larger). In the non-truncated valuation, we get a translation-covariance property that might console us for the loss of translation-invariance: shifting a set corresponds to multiplication by q. One can try to extend the Grandi valuation to sets that aren't eventually periodic, as I do in my MathOverflow post, and to some extent this works. However, one can construct sets that are incomparable, so if you're a fan of number systems that satisfy trichotomy, it's important not to overreach. I haven't given a definition of what a Grandi valuation is in general, but a hallmark is that a Grandi valuation should assign to any eventually periodic set S a value in some module from which the quantity I called "B" can be derived. Anyway, I have a bit of a crush on this notion, and I'd like to put it to some sort of use. My current target is a general uniqueness theorem in the context of one-dimensional packing. Jim P.S. It has occurred to me that when we measure infinite sets in this way, a cute name for this kind of generalized cardinality would be "grandiosity". But I think it's prudent to avoid introducing this term until I have a better idea for how it should be defined!

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Re: [math-fun] Brain teaser
by Henry Baker 05 May '17

05 May '17

I think you already know the answer, because you probably passed a large number of these types of tests (SAT's, etc.). There are standard "test-type" questions with "test-type" answers, and people who look for holes tend to be labelled by the testing process as "smart-alecks", and weeded out. Regardless of our rhetoric, we really can't allow people who "think different" into our hallowed halls. Thus, we *already* have a "bias favouring candidates neurologically similar to the examiners" -- something that blacks & minorities have complained about for years. On the other hand, when considering the state of the world, remember that the great majority of those college graduates who -- * develop our tax codes * develop our environmental regulations * oversee our health care system * overlook our financial institutions * sit on the Supreme Court * send our children into battle * negotiate our trade deals * diagnose our diseases * manage our pension plans * defend us in court * etc. -- can't pass this simple logic test. Perhaps tests like this could be used prior to spending large amounts of money "teaching everyone to code (badly)". At 05:00 AM 5/5/2017, Fred Lunnon wrote: ><< This particular question has been posed to large >numbers of people (college students, mostly, because >they are readily available and free), and only 10-20% >get it right. >> > >So what IS "the" answer, and why exactly is it "right" ? > > [ Interesting to speculate how a battery of such innocent-looking questions >at a job interview could result in a --- possibly unintentional and undetectable >--- bias favouring candidates neurologically similar to the examiners! ]

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Re: [math-fun] Brain teaser
by Henry Baker 05 May '17

05 May '17

Bingo! This particular question has been posed to large numbers of people (college students, mostly, because they are readily available and free), and only 10-20% get it right. *Training* in math may not be as important as *aptitude* in math. BTW, it should be frightening to those on this list that the overwhelming majority of our next generation of business, legal and political elites don't get this right. At 02:59 AM 5/5/2017, Gareth McCaughan wrote: >On 05/05/2017 09:36, Victor Miller wrote: > >>Is there some trick? If Linda is married, then the Linda-John pair is the >>married/not married pair. If Linda isn't married then the Paul/lLinda pair >>is the one. > >The only trick is that a surprising number of people, even >rather clever people, get only as far as figuring out that >they can't tell whether L&J are married nor whether P&L are >married, and conclude that the answer to the question is no. > >I think it likely that training in mathematics correlates >strongly with answering this correctly. I am not sure how >the explanation for this, if I'm right, should be divided >between the effect of learning about logic and the effect >of being the sort of person who is willing to spend time >learning about logic.

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Re: [math-fun] Puzzle: Where was the white king?
by Dan Asimov 03 May '17

03 May '17

Apologies if my post was confusingly worded. I meant I detect two solutions to RWG's chess problem (at bottom): r3k3/p1p5/Q3K3/8/8/8/8/8 w . The Smullyan retrograde problem definitely has only one solution! —Dan >From: Bob Hearn <bob(a)hearn.to> >That is my very favorite retrograde problem. And I’m pretty sure there’s only one solution. > >It’s the cover of Smullyan’s Chess Mysteries of the Arabian Knights: https://images-na.ssl-images-amazon.com/images/I/91B-xKpMIFL.jpg > >”The White king, Haroun al Rashid, has made himself invisible. Where is he?” > >(Incidentally Smullyan autographed my copy “I wish you the best, but I refuse to sign this book! Raymond Smullyan”.) >> On May 2, 2017, at 12:37 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >> >> Cute! >> >> (I detect two solutions; another piece could be added to prevent a second option.) >> >> I'm still very fond of a retrograde chess problem from a Martin Gardner column, attributed to Raymond Smullyan: >> >> >> . . . . . . . . >> >> . . . . . . . . >> >> . . . . . . . . >> >> . . . . . . . . >> >> . r . b . . . . >> >> B . . . . . . . >> >> . . . . . . . . >> >> . . . . . . . . >> >> . . . k . . . . >> >> After a legal move has been completed in a legal chess game, >> the white king was accidentally knocked off the board. >> >> Puzzle: Where was the white king? >> >> (No fair if you've already seen this.) >> >> —Dan >> >> >>> From: Bill Gosper <billgosper(a)gmail.com> >> >>> r3k3/p1p5/Q3K3/8/8/8/8/8 w --rwg

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[math-fun] Puzzle: Where was the white king?
by Dan Asimov 02 May '17

02 May '17

Cute! (I detect two solutions; another piece could be added to prevent a second option.) I'm still very fond of a retrograde chess problem from a Martin Gardner column, attributed to Raymond Smullyan: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r . b . . . . B . . . . . . . . . . . . . . . . . . . . . . . . . . k . . . . After a legal move has been completed in a legal chess game, the white king was accidentally knocked off the board. Puzzle: Where was the white king? (No fair if you've already seen this.) —Dan >From: Bill Gosper <billgosper(a)gmail.com> >r3k3/p1p5/Q3K3/8/8/8/8/8 w --rwg

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[math-fun] tricky mate in 2
by Bill Gosper 02 May '17

02 May '17

r3k3/p1p5/Q3K3/8/8/8/8/8 w --rwg

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[math-fun] mathematical understanding and creativity via neural nets (?)
by Thane Plambeck 01 May '17

01 May '17

the recent amazing successes of "model free" neural net algorithms (alpha go, outstanding language translation, ultra-realistic sounding voice synthesis, etc) makes me wonder whether somehow these techniques eventually could be, or perhaps already have been, applied to the creation of new and interesting mathematics. if things were to proceed along the lines of the progress to date on other problems, someone might possibly start modestly from a large training set of data (say 10,000 first year calculus exams, together with their grades), and somehow train a neural network to recognize the difference between incorrect and correct calculus question-answering. (i am just making this up). or, perhaps take textbooks and their answer keys. or polya's problems and someone's answers. it seems to still be far away from something likely to succeed, somehow, in generating a proof of a hard theorem or even creating say a simple paper i might want to read, but i had a similar reaction to what I was reading about alpha go in its early days. at the very least, a useful virtual mathematical interlocutor seems possibly feasible from a suitably large corpus of training data of "actual" mathematical back and forth interactions between teachers and students, or between researchers, maybe?

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