November 2007 - mathfun@mailman.xmission.com

[math-fun] Smallest magic squares of powers
by Christian Boyer 01 Sep '08

01 Sep '08

Oct 7th, 2007: the smallest known magic square of 4th powers is now a 44x44 square constructed by Jaroslaw Wroblewski (Poland) and Hugo Pfoertner (Germany). Jaroslaw built the semi-magic square in June, then Hugo needed a very impressive computing job of 3 months, on 57 processors of a SGI computer http://www.sgi.com/products/remarketed/origin3000/, in order to find two magic diagonals in this square! A big progress, because the previous smallest known was a 243x243 square, coming from the 243x243 tetramagic square of Pan Fengchu (China). The best known results, and their first authors, are now: Magic square Smallest known ------------ --------------- Of squares 4x4 (Euler, 1770) Of cubes 9x9 (Boyer, 2006) Of 4th powers 44x44 (Wroblewski-Pfoertner, 2007) Of 5th powers 729x729 (from the pentamagic square of Li Wen, 2003) ON ANY POWER, WE DO NOT KNOW WHAT THE SMALLEST POSSIBLE MAGIC SQUARE IS! For example, as you know, we still do not know if a 3x3 magic square of squares is possible... 4x4, 5x5, 6x6, 7x7, and 8x8 magic squares of cubes are unknown... And the problem of the diagonals is extremely difficult, as proved by the impressive computing search of Hugo. It remains a big gap between the smallest known magic and semi-magic squares (=with or without magic diagonals) of 4th powers: 44x44 versus 4x4... Here is the 4x4 semi-magic square of 4th powers (Morgenstern-Boyer, 2006): 14101^4 938^4 931^4 37762^4 35866^4 20881^4 21038^4 13393^4 24806^4 30191^4 30418^4 9263^4 413^4 32026^4 31787^4 1106^4 Christian.

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Re: [math-fun] How we "do mathematics"
by Eugene Salamin 02 Feb '08

02 Feb '08

----- Original Message ---- From: David Gale <gale(a)math.berkeley.edu> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Thursday, August 9, 2007 3:17:48 AM Subject: RE: [math-fun] How we "do mathematics" ... >A (very) old one I like : > >Two Death Valley marathonians friends complete their run >together and seek shadow in their tent nearby the finish >line. > >The first one had prepared a fresh 5-liter tank of water >in a cooling box and his friend a similar 3-liter one. > >They were about to start drinking when a journalist enters >the tent desperately asking for water. > >OK, the water is divided into three equal parts -- and drunk. > >The thankful journalist insists to pay his share and leaves >8 dollars to be split between the two friends. > >How? > >---- Why is this problem so fascinating? The two friends are paid $8. The guy who contributed 5 l gets $5, and the guy who contributed 3 l gets $3. Or maybe the $8 goes towards paying for their campsite. I must be missing something. Gene ____________________________________________________________________________________ Pinpoint customers who are looking for what you sell. http://searchmarketing.yahoo.com/

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Re: [math-fun] Compute a Function From its Fourier Series?
by Bill Gosper 17 Jan '08

17 Jan '08

GFee>See the following paper. >Authors: Gottlieb and Shu Title: On the Gibb's phenomenon and its resolution Journal: SIAM Review Issue: Vol.39, No. 4, pages 644-668, December 1997 Also see http://www.tweedledum.com/rwg/gibbs.htm --rwg --------------------------------- Tonight's top picks. What will you watch tonight? Preview the hottest shows on Yahoo! TV.

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[math-fun] extend exp(x) for quickly convergent Fourier series
by Greg Fee 04 Dec '07

04 Dec '07

Define a function f(x) on the interval -1 <= x < 0, to create a function g(x), such that: 1. g(x)=f(x) for -1 <= x < 0 2. g(x)=exp(x) for 0 <= x <= 1 3. The period 2 Fourier series of g(x) on the interval -1 <= x <= 1, of the form: a[0] + sum(a[i]*cos(i*Pi*x),i=1..infinity) + sum(b[i]*sin(i*Pi*x),i=1..infinity) converges to exp(x) on the interval 0 <= x <= 1, faster than choosing f(x)=exp(x) Can you find an f(x) such that g(x) has the quickest converging Fourier series to exp(x) on [0,1] ?

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Re: To Greg Re: [math-fun] extend exp(x) for quickly convergent Fourier series
by Greg Fee 03 Dec '07

03 Dec '07

Dan Asimov wrote: > Hi, Greg. > > Interesting type of question! > > Do you have a specific way of measuring speed of convergence? > > Regards, > > Dan Asimov > We could choose the function, such that we need the fewest number of terms in the truncation of the Fourier series, so that the uniform norm of the difference between the Fourier "polynomial" and exp(x) is less than a given value epsilon, say epsilon=1/2^11 for "computer graphics" accuracy.

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[math-fun] The 3 terms after "3" cannot be 4, 5 or 6.
by Eric Angelini 29 Nov '07

29 Nov '07

Hello Math-fun, (thanks for the private answers to my earlier posts on other topics) --- Start the sequence S with a(1)=1. The next term is the smallest one not yet present in S with the rule: the a(n) terms immediately after a(n) must be < a(n) or > 2*a(n). Example: The 7 terms immediately after 7 cannot be 8, 9, 10, ... 14. The 15 termrs immediately after 15 cannot be 16, 17, 18, ..., 30. ... The term immediately after 1 cannot be 2. S starts like this (I think): s = 1, 3, 2, 7, 15, 4, 31, 63, 127, 255, 5, 8, ... Could please someone interested compute a hundred or so terms and find a formula giving the n-th term? I hope this is not old hat, Best, É.

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Re: [math-fun] Re: math-fun Digest, Vol 57, Issue 17
by Mike Stay 29 Nov '07

29 Nov '07

On Nov 28, 2007 11:10 AM, Steve Witham <sw(a)tiac.net> wrote: > >From: "Fred lunnon" <fred.lunnon(a)gmail.com> > > > > > http://math.ucr.edu/home/baez/square.root.html > > > >I've read this webpage three times, and I still don't understand it. > > I didn't make much sense of it... I feel compelled to say that this is, perhaps, the least illuminating of all Dr. Baez' writings. He's actually well-known for the clarity of his presentations, and works very hard to explain things so well that what once was "mysterious" looks almost trivial in retrospect. -- Mike Stay metaweta(a)gmail.com http://math.ucr.edu/~mike

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RE: [math-fun] The Axiom of Choice for roots of z^2 + 1
by Dan Asimov 28 Nov '07

28 Nov '07

<< If i and -i were treated as indistinguishable, wouldn't this destroy the map of the complexes onto the XY plane? >> << I think . . . Rather than addition, start with . . . i=-i . . . >> -------------------------------------------------- My point about the complex roots of X^2 + 1 being indistinguishable was not to say that in any sense they are equal*. Just that there is no way of specifying one rather than the other. --Dan __________________________________________________ * This might be a feasible approach, but it would, I agree, sure make a mess of the complex plane.

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[math-fun] smallest "non-Web" positive integer?
by James Propp 28 Nov '07

28 Nov '07

A Google search indicates that the number 184362536 (chosen more or less at random, after a half dozen failed attempts) does not occur anywhere on the Web. What is the smallest positive integer with this property as of right now? Jim Propp

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Re: [math-fun] Re: mirrors (was The Axiom of Choice for roots of z^2 + 1)
by Fred lunnon 28 Nov '07

28 Nov '07

On 11/26/07, Steve Witham <sw(a)tiac.net> wrote: > ... > Mirrored pairs are called "right-handed" and "left-handed"-- > is that the "badly-formulated linguistic trope" you mean, Fred? Yes indeed. That's what I had in mind when I queried to association of left-handed writing with left-handed coordinates. > Or is it the whole treatment of "swap" as a verb, or...? This could be queried as well, I suppose --- there's an implicit "modulo proper isometries", as several people pointed out. Of course, with a little surgery, the mirror could be implanted along the observer's plan of (approximate) bilateral symmetry, when it really does "swap left with right" ... WFL > >From: Bill Thurston <wpt4(a)cornell.edu> > > > > Ultimately, I mellowed. I realized that people were unconvinced by > >each other's explanations because they weren't actually confused > >about what a mirror does. There's not an actual question here without > >words. Sure --- but understanding takes place at several levels. If you are caught out by the question --- as most people are, unless they've made a specific study of the Euclidean group --- then I claim you have something to gain by grasping exactly why. WFL

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