October 2005 - mathfun@mailman.xmission.com

[math-fun] 6x6 multiplicative magic squares
by Christian Boyer 14 Oct '05

14 Oct '05

After 3x3, 4x4, 5x5, what about 6x6? My algorithm needs to be better optimized to attack the question. I will work on it in the next days. The smallest published 6x6 seems to be an example constructed by Borkovitz & Hwang, with magic product = 2,000,376,000. My question: can somebody produce an example with a smaller product? Michael Kleber? Rich? Unfortunately the latin squares method, used in previous messages for 4x4 and 5x5, does not work for 6x6: the famous "36-officers problem" of Euler has no solution! Christian.

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[math-fun] Atlas of Natural Numbers
by David Wilson 12 Oct '05

12 Oct '05

Another update on the Atlas. A contributor mailing list is at http://groups.yahoo.com/group/atlasofnaturalnumbers/ to submit and discuss facts about numbers. A developer mailing list is at http://groups.yahoo.com/group/aonndev/ to discuss Atlas developer issues (i.e, help me out). The Atlas home page is at http://www.freewebs.com/davidwwilson/. You photographically inclined may also be interested in helping me out with my mailbox project. -------------------------------- - David Wilson

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RE: [math-fun] Non-commutative magic squares
by dasimov＠earthlink.net 12 Oct '05

12 Oct '05

Rich wrote: << . . . trying out non-commutative squares. This would let us return to the idea of using all the members of a well-defined set. The two most interesting cases to try seem to be using one of the non-commutative groups with 16 elements to make a 4x4, or using 2x2 matrices with elements 0,1 or maybe +-1, also to make a 4x4 square. The 0,1 matrix case would need a magic constant of [00,00]. You need some rules on the order of combining things, and whether the up-going diagonal reads forward or backward. >> Am trying to think of a nice non-commutative group of order 16. Here's one: G = { az + b | a in {1,i,-1,-i}, b in Z[i]/2Z[i] }. Here b is uniquely = one of 0,1,i,1+i mod 2Z[i]. (Imagine a square torus T tiled by 4 congruent subsquares. Then G is the group of oriented rigid motions of T that preserve subsquares.) Then define g o h via (az+b)o(cz+d) = (ac)z + (ad+b) where ad+b is represented by its remainder mod 2Z[i] = one of 0,1,i,1+i. Thus 1/(az+b) is given by (1/a)z + -b/a. Questions: 1. Does this G admit any 4x4 magic squares? 2.If so, is the magic constant uniquely determined? [A priori we can just assume group multiplication of a row is left to right, and of a column is top to bottom.] (I'd guess the most promising magic constant is the identity. Also, if it's the identity then any cyclic permutation of a given row or column will multiply to the same thing: rstu = 1 <=> stur = 1 avoiding the sticky problem of how to define the magic product.) Hmm, another nice non-commutative group of order 16 is the isometries of a regular octagon (= D_8). This seems less interesting than G, but the same questions apply.) --Dan ----------------------------------------------------- P.S. (I say: diagonals, schmiagonals. Requiring the two diagonals to have the same magic constant just strikes me as a random rule with little redeeming recreational math value.)

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[math-fun] Mathematicians solve evolution debate (fwd)
by Richard Guy 11 Oct '05

11 Oct '05

Glad that's settled! R. ---------- Forwarded message ---------- Date: Tue, 11 Oct 2005 08:21:49 -0600 From: Gary Margrave <margrave(a)ucalgary.ca> To: GGDept <glgp-l(a)majordomo.ucalgary.ca>, dept(a)math.ucalgary.ca Subject: Mathematicians solve evolution debate Breaking News: Mathematicians solve evolution debate In the ongoing confrontation between science and religion over evolution there has been little progress until today, said leading mathematicians. At the crux of the debate is whether, as the scientists put it, the earth is 4.5 billion years old and life today evolved from primitive forms; or, as the Christian conservatives say, the earth and all beings on it were created six thousand years ago by God. In recent years the struggle has focused on the teachings of our origins in public schools and religious conservatives have put forward the "intelligent design" hypothesis as an alternative to evolution, demanding that it be taught along side Darwin's theory. Now, in a brilliant synthesis, hailed as a breakthrough on both sides, mathematicians have stepped into the fray with a unifying hypothesis. They first note that both sides agree that the earth and its inhabitants are very complex. Then, drawing upon the well-established theory of complex-valued functions, they propose an earth model that brings both viewpoints into a complete theory. Since complex-valued functions have both real and imaginary parts, mathematicians suggest that it is the real part of the earth that is 4.5 billion years old while the imaginary part was created just six thousand years ago. In other news ...

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[math-fun] NYTimes: Nobel came after years of battling the system
by hbaker1 11 Oct '05

11 Oct '05

More on the H. Pylori Nobel Prize in a NYTimes article, below. NYTimes blames drug companies. I personally think that peer review is more than adequate to kill new ideas without having to play the "big bad business" card. http://www.nytimes.com/2005/10/11/health/11docs.html --- On another note, Tony Tether's recent robot vehicle prize was an outstanding success. Let's hope it inspires more research prizes.

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[math-fun] Atlas Design Pages
by David Wilson 11 Oct '05

11 Oct '05

Relative the Atlas of Natural Numbers, I have added some design pages at http://www.freewebs.com/davidwwilson/ -------------------------------- - David Wilson

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Re: [math-fun] Atlas Replies + Help
by David Wilson 07 Oct '05

07 Oct '05

For the time being, I have created a Yahoo mailing list. Please visit http://groups.yahoo.com/group/atlasofnaturalnumbers/ This is not an optimal solution, however, I am sticking with it for the time being.

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[math-fun] Re: Ratio of two Bessel function values
by Richard Guy 06 Oct '05

06 Oct '05

Many thanks! R. On Tue, 4 Oct 2005, Eric W. Weisstein wrote: > On Tue, 4 Oct 2005, Richard Guy wrote: > >> I recently asked, on one of these networks, >> about the number whose continued fraction >> is {1,2,3,4,...} and someone was kind >> enough to provide an explicit answer, which >> I've stupidly deleted. Can it be repeated? >> >> Incidentally, is there a more general >> result, that a continued fraction whose >> partial quotients form an arithmetic >> progression (or a set of APs) can be >> expressed in terms of Bessel functions? >> >> There are a few examples which are >> rational functions of e. R. > > Yes; see > http://mathworld.wolfram.com/ContinuedFractionConstant.html > > Cheers, > -Eric >

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[math-fun] More Median Musings
by David Gale 06 Oct '05

06 Oct '05

I've noticed some nice number theoretic properties of the median which are probably well known, but I'd be interested in references A sequence of real numbers (a_1,a_2, . . ,a_n) has a PERFECT MEDIAN if there is an index m such that a_1+. . +a_m = a_m+ . . +a_n The sequence has a DOUBLE MEDIAN if there exists an index m such that a_1+ . . +a_m = a_(m+1)+ . +a_n Example: The constant sequence (c) has a perfect median if n is odd, double medians if n is even. For which n does the sequence (1,2, . ,n) have a perfect median? If n=1 then m=1 is a perfect median x If n=8 then m=6 is a perfect median x xx xxx xxxx xxxxx mmmmmm xxxxxxx xxxxxxxx The next case is n=49,m=35. For which n does the sequence (1,2, . ,n) have a double median? If n=3 then 2 and 3 are medians x xx ======== xxx If n=20 then 14 and 15 are medians x xx xxx xxxx xxxxx xxxxxx xxxxxxx xxxxxxxx xxxxxxxxx xxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxx xxxxxxxxxxxxx xxxxxxxxxxxxxx _======================== xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxx The next case is n=119 with medians 84 and 85. It turns out one can parameterize all solutions in both cases by solutions of a suitable Pell equation. Again, references please. David

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[math-fun] telephonic mystery
by Thane Plambeck 06 Oct '05

06 Oct '05

Why do so many businesses have the phone number 214-748-3647? [ Try googling it, for example ] http://www.google.com/search?hl=en&lr=&c2coff=1&q=214-748-3647&btnG=Search -- Thane Plambeck http://www.plambeck.org/ehome.htm

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