February 2016 - mathfun@mailman.xmission.com

[math-fun] Math revolution in the U.S. (The Atlantic)
by Henry Baker 10 Feb '16

10 Feb '16

FYI -- http://science.slashdot.org/story/16/02/09/149224/an-advanced-math-educatio… The Atlantic has an extended article on the recent surge in advanced math education at the primary and secondary levels in the U.S., arguing that last year's victory for the U.S. in the Math Olympiad was not a random anomaly. Participation in math camps, after-school or weekend math "academies," and math competitions has surged in recent years, with many programs having long wait lists. Inessa Rifkin, co-founder of one of these math academies, argues that the problems with math education begin in the 2nd and 3rd grades: ""The youngest ones, very naturally, their minds see math differently.... It is common that they can ask simple questions and then, in the next minute, a very complicated one. But if the teacher doesn't know enough mathematics, she will answer the simple question and shut down the other, more difficult one." These alternative math programs put a greater focus on problem-solving: "Unlike most math classes, where teachers struggle to impart knowledge to studentsÂwho must passively absorb it and then regurgitate it on a testÂproblem-solving classes demand that the pupils execute the cognitive bench press: investigating, conjecturing, predicting, analyzing, and finally verifying their own mathematical strategy. The point is not to accurately execute algorithms, although there is, of course, a right answer... Truly thinking the problem throughÂcreatively applying what you know about math and puzzling out possible solutionsÂis more important." http://www.theatlantic.com/magazine/archive/2016/03/the-math-revolution/426…

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[math-fun] How difficult are math problems? The "Putnam."
by Warren D Smith 09 Feb '16

09 Feb '16

It is hard to measure how "difficult" it is to solve mathematical problems. However, it occurs to me that Putnam exam questions are a somewhat standardized unit of mathematical difficulty. https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition Specifically, each year, the Putnam has 12 questions, each worth 10 points, to be solved in a total of 6 hours, i.e. 30 minutes per question. The questions are cooked up by experts with the intentional goal of making them so difficult that the median score on the exam (which is taken by good math undergraduate students Canada+USA-wide) is zero; but in such a way that the best 1% students in the country will score above about 42 points, and the best student in the country will score near 120. And this standard presumably is time-varying. Undergrads know different things now, than they knew in 1938. They as far as I know are not allowed to use electronic assistance on the Putnam, but perhaps one day that would/should be permitted, which if so would change things considerably. Pierre de Fermat presumably was considerably smarter than I, but he'd probably have no chance against me in a competitive math exam because he was so ignorant because math was just not very developed at that time. And some fans have been compiling online databases of all past Putnam problems. So consider 1 Putnam exam problem, as one "unit of difficulty." What, then, is the difficulty level of the hardest math problems that humanity has been able to solve? That 1 person has been able to solve? That you ever solved? Etc? It seems to me, if a problem requires solving about 5 sub-parts, each 1 Putnam difficulty, that's probably about the best I am capable of. It sort of grows on you exponentially. I mean, if I were given a Putnam problem, there is perhaps 50% chance I could solve it. (Depends how motivated I was, if I had a lot of motivation maybe 90%?) So anyhow, assuming the 50% figure, then if I had to solve 5 in a row to solve the uber-problem, presumably there would be about 1/32 chance I could. And then of course you'd have to figure out the 5 sub-parts. So in this sense, I'm not very smart. I mean, if an undergrad can solve something in 20 minutes, it wasn't all that hard. And if the best I can hope to do is equivalent to about 5 of that, not very impressive. Now we have the advantage of standing on the shoulders of giants, electronic assistants, and database searches to locate the previous works by the giants. Plus, there's the collaborative effect where, if the problem requires solving 10 Putnam-level subproblems, and some of them have already been solved by somebody else, then even though you could not have managed all 10 by yourself, you still might be able to do it. So then what is reachable? The "classification of finite simple groups" for example, is one of the hardest math problems that humanity has (allegedly) managed to solve, and certainly beyond the ability of any one human. How many Putnams of difficulty were involved? Hard for me to say. I would guess this is less than a KiloPutnam, but more than 50 Putnams? How soon do you think a computer AI will be able to compete on the putnam exam? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Textbook sleaze
by Bill Gosper 09 Feb '16

09 Feb '16

gosper.org/textbooksleaze.png Prob 13 clearly wants you to add the two rectangles, which, in light of the slanted southern border, is clearly inferior to simply neglecting the panhandle. But it isn't if you use their numbers and compare with http://www.city-data.com/states/Oklahoma-Location-size-and-extent.html ! "The total area of Oklahoma is 69,956 sq mi (181,186 sq km), of which land takes up 68,655 sq mi (177,817 sq km) and inland water 1,301 sq mi (3,369 sq km). Oklahoma extends 464 mi (747 km) E-W including the panhandle in the NW, which is about 165 mi (266 km) long. The maximum N-S extension is 230 mi (370 km)." This is because the textbook shaves 2 mi off the actual width and 8 mi off the actual height of the non-panhandle! This reduces its area by half the area of the panhandle. But since the total area is easier to research than the actual dimensions, poor Johnny will probably get shot down for eyeballing the truth. A big boost for his self doubt. --rwg

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[math-fun] Can a nonconstant continuous function on the reals
by Bill Gosper 08 Feb '16

08 Feb '16

assume only rational values? Related puzzle: Find an x in closed form s.t. CantorStaircase[x] is irrational. --rwg Spoiler: "2/)]2/1 ,0 ,3[atehTcitpillE + 1-( == ]]3/1 ,0 ,3[atehTcitpillE + 1-[esacriatSrotnaC"

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[math-fun] Re; Can a nonconstant continuous function on the reals (Bill Gosper)
by Warren D Smith 08 Feb '16

08 Feb '16

answer is no, reason is https://en.wikipedia.org/wiki/Intermediate_value_theorem A more interesting question is to find a continuous function (the smoother the better) which maps rationals to rationals, but is not a rational function.

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[math-fun] Fwd: Fourier synthesis of (cos(4 sin t), sin(4 sin t))
by Bill Gosper 08 Feb '16

08 Feb '16

Here's (cos(9 sin t), sin(9 sin t)): gosper.org/Besser9.gif Note that the animation does not reverse after the pauses marking rotor axis alignment. The boxes formed by the rotors have convex sides just before the end of the clockwise phase, but just after the end of the anticlockwise phase. Meanwhile, the rotors just keep going. In two hours Mathematica was *barely* able to disgorge this, swelling past 26GB and getting only 6% useful cpu. Mathematica runs it on the fly at an almost watchable speed. It would probably roar if it had the magic lisp machine triangle primitive. That relatively dinky machine could run this live to 20 radians of modulation. But how to use the primitive would be a bear to document. E.g., if you paint a polygon with an side AB, and another with side AC sharing part of AB, i.e. with C lying on AB, and finally add a third polygon with a side CB, then you should also draw the degenerate triangle ABC to magically heal the pixel mismatches caused by the slopes of AB, BC, and CA being slightly unequal due to quantization. --Bill On Sat, Feb 6, 2016 at 8:13 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > (overmodulated FM): > gosper.org/Besser.gif > The harmonics are Bessel coefficients. --rwg >

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[math-fun] Fwd: Fourier synthesis of (cos(4 sin t), sin(4 sin t))
by Bill Gosper 07 Feb '16

07 Feb '16

(overmodulated FM): gosper.org/Besser.gif The harmonics are Bessel coefficients. --rwg Someone complained of seeing only an inanimate blue rectangle. Let me know if you too see a stiff gif.

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[math-fun] plane-filling fractal
by David Wilson 06 Feb '16

06 Feb '16

Is there a fractal set M a la the Mandelbrot set where both M and its complement are dense on the plane?

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[math-fun] computer go player kicks human butt
by Warren D Smith 06 Feb '16

06 Feb '16

https://gogameguru.com/tag/deepmind-alphago-lee-sedol/ says the Sedol match will be March 8-15 in Seoul; more details to be posted later. There also are some pro players commenting on AlphaGo's games vs Fan Hui, but their commentary is too advanced for me to follow. Go, unlike(?) chess, has the property that if you read a go book that is too advanced for you, it makes virtually no sense at all. (I've done that, and I could not believe how senseless it was. Then later when I was a stronger go player, I re-read one of those books, and, it now was making sense! Amazing!) -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Continued logarithm redux
by Bill Gosper 05 Feb '16

05 Feb '16

Let x>=1. Iterate x <- 1/(x/2^n -1), where 2^n is the largest power of 2 <=x . The sequence of n's quantifies x analogously to a continued fraction. http://www.tweedledum.com/rwg/cfup.htm incorrectly stated that this sequence is periodic for quadratic surd x. The first x^2 = integer counterexamples appear to be 13, 14, 15, 29, 31, 35, 39, 41, ..., but they might just have huge periods. E.g., In[435]:= NestList[Simplify[1/(#/2^Floor[Log[2, #]] - 1)] &, Sqrt[167], 69] Out[435]= {Sqrt[167], 8/(-8 + Sqrt[167]), 1/89 (39 + 8 Sqrt[167]), 1/46 (25 + 4 Sqrt[167]), 2/97 (21 + 4 Sqrt[167]), 1/79 (55 + 8 Sqrt[167]), 206 + 16 Sqrt[167], 128/(-25 + 8 Sqrt[167]), 1/(-1 + 128/(-25 + 8 Sqrt[167])), ( 2929 + 512 Sqrt[167])/5534, 5534/(-2605 + 512 Sqrt[167]), ( 8159 + 1024 Sqrt[167])/8119, (2 (8079 + 1024 Sqrt[167]))/13529, ( 2725 + 512 Sqrt[167])/5374, 5374/(-2649 + 512 Sqrt[167]), ( 8383 + 1024 Sqrt[167])/7663, (2 (6943 + 1024 Sqrt[167]))/16561, [...] 3363 + 1024 Sqrt[167])/13697, ( 5167 + 512 Sqrt[167])/2494, 9976/(-4809 + 512 Sqrt[167]), ( 6943 + 1024 Sqrt[167])/7663, (2 (8383 + 1024 Sqrt[167]))/13681, ( 2649 + 512 Sqrt[167])/5374, 5374/(-2725 + 512 Sqrt[167]), ( 8079 + 1024 Sqrt[167])/8119, (2 (8159 + 1024 Sqrt[167]))/13369, ( 2605 + 512 Sqrt[167])/5534, 5534/(-2929 + 512 Sqrt[167]), ( 6863 + 1024 Sqrt[167])/10063, 1/128 (25 + 8 Sqrt[167]), 128/(-103 + 8 Sqrt[167]), 2/97 (55 + 8 Sqrt[167]), 1/46 (21 + 4 Sqrt[167]), 2/89 (25 + 4 Sqrt[167]), 1/103 (39 + 8 Sqrt[167]), 1/8 (8 + Sqrt[167]), 16/(-8 + Sqrt[167]), 1/89 (39 + 8 Sqrt[167]), 1/46 (25 + 4 Sqrt[167])} has menacing coefficients, but period 66. And, strangely, √13 +3, 1, -1, and -2 all have period 4. With large x, observable periods become increasingly sparse. Negative n permit 0<x<1. Actually, NestList[Simplify[1/(1-#/2^Ceiling[Log[2, #]])] &, Sqrt[k], 69] is much richer in periodicity, the first apparent blowups being k = 19, 27, 31, 61, 71, 76, 85, 86, 87, ... . Near blowups make for longish periods, e.g. period(√65) = 127±, period(√69) = 395±, p(√70) = 399±, p(√75) = 310±. The patterns of rationals within the periods are intriguing. --rwg

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