May 2008 - 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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[math-fun] Superfluous Multinomials
by wouter meeussen 04 Aug '08

04 Aug '08

dear funsters, when posting stuff like this, I presume it is generally known, or at least could be. Motivation: just fun and beauty. 1/. Intro consider the expansion of (a+b+c+d)^3 it is the sum over M(u,v,w,x) a^u b^v c^w d^x with positive or zero integers u+v+w+x=3 and M(u,v,w,x) is the multinomial u! v! w! x! /(u+v+w+x)! When we drop the multinomial coefficients, then we get the 'compositions' of 4 elements in groups of three. in Mma: Expand[(a+b+c+d)^3 /3! ]/. q_^n_ ->q^n n! gives a^3 + a^2*b + a*b^2 + b^3 + a^2*c + a*b*c + b^2*c + a*c^2 + b*c^2 + c^3 + a^2*d + a*b*d + b^2*d + a*c*d + b*c*d + c^2*d + a*d^2 + b*d^2 + c*d^2 + d^3 this is equivalent to Plus @@ Apply[Times,{a,b,c,d}^#& /@Compositions[3,4],{1}] 2/. Generalise replace the (a+b+c+d) above by Sum(j=1..n, t[ j ] ) , t stands for 'temporary', now replace t[j] with E^(2 Pi I j/n) and it's obvious that this sum equals zero. And that any power of zero is also zero. But now, dropping multinomials, we find: Table[ Plus@@ Apply[Times,(Exp[2 Pi I/n]^Range[n])^#& /@ Compositions[k,n],{1}],{n,5},{k,n}] and we find that it's also zero, except it's 1 when k equals n, that is, we take all n roots of (-1) and put it to the n-th (multinomial-less) power. 3/. an instructive typo : Exp[2 Pi I/k] instead of Exp[2 Pi I/n]: Table[ Plus@@ Apply[Times,(Exp[2 Pi I/k]^Range[n])^#& /@ Compositions[k,n],{1}],{n,5},{k,n}] equals Table[Ceiling[n/m],{n,9},{m,n}] alias A120885 That leads to this silly remark: Table[ Plus@@ (Exp[2 Pi I/k]^Range[n])^k ,{n,16},{k,n}] is integer, except for the following couples of {n,k} : {7, 5}, {8, 5}, {9, 7}, {10, 7}, {10, 8}, {11, 7}, {11, 8}, {11, 9}, {12, 5}, {12, 7}, {12, 8}, {12, 9}, {12, 10}, {13, 5}, {13, 8}, {13, 9}, {13, 10}, {13, 11}, {14, 8}, {14, 9}, {14, 10}, {14, 11}, {14, 12}, {15, 9}, {15, 10}, {15, 11}, {15, 12}, {15, 13}, {16, 7}, {16, 9}, {16, 10}, {16, 11}, {16, 12}, {16, 13}, {16, 14}, ... and, where's the system in *this* madness? Wouter.

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[math-fun] Morpion Solitaire
by Christian Boyer 06 Jun '08

06 Jun '08

An easy game: Only paper and pencil needed. A game for one player. And an open game: Who will improve the current world records??? Here is my new website dedicated to "Morpion Solitaire": www.morpionsolitaire.com Fully bilingual, English/French. Feel free to report any error, or any missing information. As you can read in the "Origin" page, any new information on the origin of this game will be welcomed. Some other questions. Is it possible to do best than the two grids given in "Other 5"? Or worst than my very bad grids at the end of the "Limits" page? Christian.

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Re: [math-fun] becoming a 21st century mathematician
by N. J. A. Sloane 04 Jun '08

04 Jun '08

For any work involving group theory, number theory, algebraic geometry, combinatorics, algebra, coding theory, ring theory, etc., etc., - mathematics generally - MAGMA is indispensable Neil

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[math-fun] In the display labeled (d183)
by rwg＠sdf.lonestar.org 30 May '08

30 May '08

near the end of www.tweedledum.com/rwg/idents.htm, suppose a,b>0. How can the 4th equand be negative, but not the 5th? --rwg

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[math-fun] becoming a 21st century mathematician
by James Propp 30 May '08

30 May '08

I'd be interested in hearing from people about software they've found to be especially useful for any aspect of the math research enterprise. I'm getting a couple of new laptops, and I'm trying to figure out what to install on them. I've got some access to funds from my university, so price is not necessarily an obstacle. Some programs that are fairly indispensable to me already are LaTeX (for typsetting) and Maple / Mathematica for computations (I prefer Maple but am training myself to use Mathematica as well). I'm hoping some of you will steer me towards programs that, a few years from now, will have become just as indispensable to me! (or, failing that, will at least be a lot of fun). One thing I could really use is a good system for creating illustrations. Fifteen years ago there was a simple little Mac drawing program called idraw that I loved, though everyone else I knew preferred xfig; nowadays there's something that calls itself idraw, but I don't think it's related to the idraw that I used back then. I admire folks who are proficient in PostScript, and maybe I'll be one of them someday; for now, I'm tooling along with pstricks, and spending far more time creating pictures than I ought to have to. (As an aside, I'll mention that I've managed to do some non-trivial math research in Excel. I would not ordinarily have thought to do this, but frequently I find myself with no computer more powerful than my Treo, and my Treo happens to have a version of Excel running on it, and one finds ways to get by with whatever one has.) While we're at it, I'm also open to suggestions for software that people have found useful for their teaching. I'm also open to suggestions for hardware. I already own a couple of flash drive sticks, and they've gotten me out of a couple of tight spots by permitting me to give a talk on someone else's laptop at the last minute. I have a hard time believing that flash drives aren't a product of extraterrestrial technology. Are there other things as cool as flash drives that I don't know about? I'm also open to suggestions for websites. Of course I already use OEIS. Speaking of which, three cheers for Neil for deservedly winning the latest Robbins Prize! Since back in the 1970s, the Handbook and Encyclopedia have made math research both more productive and more fun for me than it would otherwise have been. Jim Propp

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[math-fun] Math pun
by Steve Witham 29 May '08

29 May '08

Penrose = pentagon rose? (A name dictionary says, "All are named with Celtic pen 'head', 'top', 'end' + ros 'heath', 'moor'.") --Steve

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Re: [math-fun] becoming a 21st century mathematician
by N. J. A. Sloane 29 May '08

29 May '08

Miks Stay asked how to get MAGMA. The home page is http://magma.maths.usyd.edu.au/magma/ They are a non-profit group, at the University of Sydney, but they do charge for it (though not unreasonably). Neil

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Re: [math-fun] golden
by N. J. A. Sloane 28 May '08

28 May '08

there are millions of places where "phi" is used and where "tau" is used for (1+sqrt(5))/2 ! it is too late to try to unify the notation, sorry! Neil

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Re: [math-fun] phi, pho, phoo
by Steve Witham 28 May '08

28 May '08

>From: Steve Witham <sw(a)tiac.net> > >Wikipedia & Mathworld both say (1 - sqrt5)/2 is called the "golden >ratio conjugate" & Mathworld adds "silver ratio". Wikipedia rationalizes >(heh) that it's the absolute value of the actual conjugate. Meant (-1 + sqrt5)/2 of course. Sewing enough confusion already. > -+- / > (|) or (/) capital Phi > -+- / Aka "pho", the golden noodle, eye of the noodle, etc. ) (( ) --Steve

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