April 2006 - mathfun@mailman.xmission.com

Re: [math-fun] placebos
by dasimov＠earthlink.net 11 Apr '06

11 Apr '06

Jim wries: << I'll be teaching about the placebo effect in my quantitative reasoning class later today, and I just remembered having read a year or two ago about a study that claimed to actually disprove the existence of the effect. Do any of you remember this? What is the current scientific consensus about the study? >> What I seem to recall is only that a study appeared to show the placebo effect is less than it was thought to be. Seem likely to me that its effect will depend highly on the population under study, and perhaps above all on the kind of ailment that is being treated. If the ailment has a high psychosomatic component, there would probably be a higher placebo effect. --Dan

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RE: [math-fun] high school mathematics
by Cordwell, William R 10 Apr '06

10 Apr '06

I have seen various college texts that define natural numbers to be the integers > 0, and others the integers >= 0. I'm fairly certain, that, in middle school, we were taught that the natural numbers were positive. BC -----Original Message----- From: math-fun-bounces+cordwell=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+cordwell=sandia.gov@mailman.xmission.com]On Behalf Of Henry Baker Sent: Monday, April 10, 2006 12:47 PM To: James Propp Cc: math-fun(a)mailman.xmission.com Subject: Re: [math-fun] high school mathematics Unfortunately, I haven't found mathematicians in college to necessarily be any more punctilious than those in high school when it comes to universal agreement on terminology.

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[math-fun] Re ... Chess Record at 330 moves
by Guy Haworth 10 Apr '06

10 Apr '06

I don't understand Richard's concerns - and I think there must be some chessic considerations missing. For example, with DTC = Depth to Conversion, i.e. #moves by each side to mate and/or change of force. maxDTC KQK = 10, maxDTC KQQK = 3, maxDTC KQQQK = 3, maxDTC KQQQQK =< 3 etc. Basically, White can sacrifice a Q from KQQK onwards in order to force a capture, when the bK has no alternative. Thus, the number of pieces, number of positions, move-fan-out, move-fan-in (all increasing) result in a maxDTC which is non-increasing. Not sure how this squares with the graph-metrics Richard cites but: White (assumed winning here) reduces DTC by 1, 0 or -w Black (assumed losing here) reduces DTC by 0 to b g

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[math-fun] high school mathematics
by James Propp 10 Apr '06

10 Apr '06

Has anyone written a good essay on the "two cultures" of high school mathematics and university mathematics? The former is not just a subset of the latter. For one thing, I think high school teachers tend to be punctilious about some matters that university mathematicians are sloppy about. A case in point is the meaning of "imaginary part". Is the imaginary part of a+bi equal to b, or to bi? I don't think that I myself am consistent about this. I got the impression that Barry Mazur, in his book "Imagining Numbers", used a convention at odds with what's being taught in high schools these days, because one irate amazon reviewer of the book opined that Mazur was not qualified to write a book on advanced math because his misuse of the term "imaginary part" showed that he didn't even know basic high school math! I also think that high school math culture enshrines some 19th century usages that have fallen into disfavor. For instance, the word "cone" in three-dimensional analytic geometry refers to a complete locus, not just the half that lies on one particular side of the vertex (and some high school teachers are rather pedantic about that); but from calculus onward, "cone" in university math means what high school teachers would call a half-cone. Part of what's going on is a time-lag between developments in research and the training received by high school math teachers, and the nature of the courses that future high school teachers take. To give just one example: I've seen more than one course in "modern geometry for future high school teachers" in which the discussion of non-Euclidean geometry is extremely axiomatic and historical, and hyperbolic geometry is illustrated by way of a single picture (the pseudosphere), maybe supplemented by pictures of the Klein or Poincare models, but with no upper half-plane (and no Escher pictures!). Who would write such a book? Someone who had read such a book at a formative age, and who had a very static view of mathematics. It would probably be easy to write a one-sided essay that simply contended "high school mathematics is bad mathematics" (cf. the essay on applied mathematics by Halmos). But I think there's more to it than that. Comments? Jim Propp

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[math-fun] high school mathematics
by emcnicho＠email.arizona.edu 10 Apr '06

10 Apr '06

I went to a very interesting talk by Karen King on this topic. At the time she was affiliated with Michigan State University, but I believe she is now with NYU. She would be a good resource if you are looking for articles on the subject. Sincerely, Erin :!:!:!:!:!:!:!:!:!:!:!:!:!:!:!:!:!:!:!:!:!:!:!:!:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Erin McNicholas Department of Mathematics University of Arizona (520) 621-6896 "Things are not all so comprehensible and utterable as people would mostly have us believe; most events are unutterable, consummating themselves in a sphere where word has never trod..." -Rainer Maria Rilke !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !

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[math-fun] ways a number is a sum of 2 distinct primes
by Emeric Deutsch 10 Apr '06

10 Apr '06

Dear Seqfans, The least number expressible as a sum of two distinct primes in exactly n ways is given by sequence A087747, starting with 5, 16, 24, 36, 48, 60, 78, 84, 90, 114, 144, 120, 168, 180, 234, for n=1,2,3,... [5=2+3; 16=3+13=5+11; 24=5+19=7+17=11+13] Numbers which can be expressed as the sum of two distinct primes in exactly two ways are given by A077914: 16, 18, 20, 22, 26, 28, 32, 62, 68 Numbers which can be expressed as the sum of two distinct primes in exactly three ways are given by A077969: 24, 30, 34, 40, 44, 46, 52, 56, 58, 98, 122, 128 Numbers which can be expressed as the sum of two distinct primes in exactly four ways are given by A078299: 36, 42, 50, 74, 80, 82, 86, 88, 92, 94, 152, 158 and so on 5 ways, A080854: 48, 54, 64, 70, 76, 104, 106, 118, 124, 134, 136, 146, 148, 164, 166, 188; 6 ways, A080862: 60, 66, 72, 100, 110, 116, 172, 178, 182, 194, 206, 212, 218, 226, 248, 278, 326, 332, 398 7 ways, not in OEIS: 78, 96, 112, 130, 140, 142, 176, 208, 214, 224, 232, 272, 362 8 ways, not in OEIS: 84, 102, 108, 138, 154, 160, 184, 190, 200, 202, 242, 254, 256, 262, 266, 284, 292, 296, 302, 308, 314, 346, 368, 458 These sequences have no other terms up to 7000. Are they finite? Is there any literature on this? So far it SEEMS that the largest number expressible as a sum of two distinct primes in exactly n =2,3,4,... ways is given by 68,128,158,188,398,362,458,... (last terms in the finite (?) sequences given above). I'd appreciate any input on this. Thanks, Emeric P.S. I have used the following g.f. for the number of partitions of n into 2 distinct primes: sum(sum(x^(p(i)+p(j)), i=1..j-1), j=1..infinity), where p(k) is the k-th prime. See A117929.

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[math-fun] sports trivium
by R. William Gosper 10 Apr '06

10 Apr '06

The width of a regulation NFL field is half the average of its diagonal and its 100 yd length. --rwg PS, in reply to your privately sent anagrams, Si', Argentine tangerine is delicious lousicide.

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Re: [math-fun] high school mathematics
by dasimov＠earthlink.net 09 Apr '06

09 Apr '06

Jim wrote: << The former is not just a subset of the latter. For one thing, I think high school teachers tend to be punctilious about some matters that university mathematicians are sloppy about. A case in point is the meaning of "imaginary part". Is the imaginary part of a+bi equal to b, or to bi? I don't think that I myself am consistent about this. I got the impression that Barry Mazur, in his book "Imagining Numbers", used a convention at odds with what's being taught in high schools these days, because one irate amazon reviewer of the book opined that Mazur was not qualified to write a book on advanced math because his misuse of the term "imaginary part" showed that he didn't even know basic high school math . . . . . . >> My own experience is quite the opposite. In high school I was never clear about what the imaginary part of a complex number a+bi is (ib or b ?). In college and subsequently I've learned that there is a universal, unequivocal convention: Im(a+bi) = b. Likewise, I recall in H.S. thinking the phrase "square root", or the square-root sign, meant *both* square roots of a complex number, and especially of a nonnegative real. In college & ever since, it's been made very clear that the square-root sign -- when applied to a nonneg. real -- is always the square root >= 0. Mazur glosses over this point and others, since there is no such thing as "the" square root of -15: it has two. (But it's no problem, since his intended audience is not at all the technically minded.) Based on the Amazon reviews, he seems to have succeeded well. In the same vein, all inverse functions that have multiple branches were described only vaguely in H.S. (Above all, in going from cartesian to polar coordinates, the inverse tangent was bandied about with great insouciance.) But in college I learned that, at least for the six inverse trig functions on the reals, there is a standard domain always assumed. (But it's entirely possible that, more recently, H.S. teachers have become more punctilious about such questions.) Finally, cones: the cone is one of many words in math that, unfortunately, are used differently in different contexts. In some contexts (e.g., real quadratic forms) modern researchers still discuss quadric surfaces and use cone to mean both nappes, as in H.S. In probably most math contexts a cone is generalized to mean any subset of a linear space L that is the union of all rays from the origin through any point in some subset S of L -- i.e., C(L) = {tv : v is in S and t >= 0}. In topology there is even a third, slightly different meaning of cone. --Dan

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[math-fun] Chess Endgame Depth Record Broken Again
by Guy Haworth 08 Apr '06

08 Apr '06

On 10th March 2006, Marc Bourzutschky announced that Yakov Konoval's program had computed the DTC/Z endgame table (EGT) for KQBNKQB and found: maxDTZ/C = 330 moves: maxDTZ position = 1k6/1b5q/N7/8/8/1Q6/8/B1K5+b DTC = Depth to Conversion, i.e., change of force and/or mate DTZ = Depth to (move-count) Zeroing (move), i.e. P-push and/or Conversion This depth leaves all chess records of this kind well behind: 193m = the longest decisive game in classical chess: Stepak-Mashian (1980) 199m = longest (rapid) decisive chess game: Petrosian-Milanovic (2005) 262m = the greatest known EGT maxDTM(ate), in KRNKNN 269m = longest (drawn) chess game: Nikolic-Arsovic (1989), 290m = the deepest known Chess Problem - by O.T.Blathy 290m = the previous maxDTC/Z record - in KRRNKRR Generation and verification of the EGT required 3.5 weeks of a 3.6GHz processor. The first two attempts to generate this EGT were frustrated by interrupted power supplies. guy

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Re: [math-fun] Sum of reciprocals of 8877690 integers with distinct digits
by Schroeppel, Richard 08 Apr '06

08 Apr '06

[This is from Eric Weisstein, sent a week ago.] ------ On Thu, 30 Mar 2006, Robert Baillie wrote: > Problem E2533 in the American Mathematical Monthly (1975 or 1976) says there > are 8877690 integers that have distinct digits. The problem is to estimate > the sum of their reciprocals. > > I thought I read someplace that someone had computed the exact sum. Does > anyone know how big the numerator and denominator are? And, by the way, what > is the approximate sum? Assuming I have not made a silly computation error, the exact sum is 6745140117...4801548051 (14816583 digits) ----------------------- 7553392231...8000000000 (14816582 digits) given to 100 digits by 8.929948174755443424171179403981525048667286767255408480936913993018768120557239657437064181411273952 which will appear shortly on http://mathworld.wolfram.com/Digit.html http://www.research.att.com/~njas/sequences/?q=A117914 Cheers, -Eric

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