February 2007 - mathfun@mailman.xmission.com

[math-fun] Inequality puzzle
by Daniel Asimov 09 Feb '07

09 Feb '07

Given fixed real numbers c_1 < c_2 < . . . < c_n, define f(sigma) = Sum_{1 <= j <= n} c_j c_sigma(j) for any permutation sigma of {1,...,n}. Prove that sigma = id_{1,...n} is the unique permutation that maximizes f(sigma). --Dan

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[math-fun] # of non-isomorphic polyomino graphs
by Thane Plambeck 07 Feb '07

07 Feb '07

I'm not finding the following sequence in the OEIS, probably because I'm blind to a mistake I've made in counting them. Consider the set of graphs G(n) constructed in the following way. Take the set of polyominos with n cells. For each polyomino P in turn, put a vertex at the center of each cell, and draw edges between two vertices if they share an edge in the polyomino, obtaining a plain old unidirected graph G with n vertices. I'm interested in the # of nonisomorphic graphs that are obtained in this way, for each n. Starting at n=1, I get the sequence 1,1,1,3,4,10 and hits in the OEIS that don't look like they fit for larger n -- Thane Plambeck tplambeck(a)gmail.com http://www.plambeck.org/ehome.htm

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[math-fun] Groups with exactly two conjugacy classes
by Eugene Salamin 06 Feb '07

06 Feb '07

Can some give an example of a group having exactly two conjugacy classes, other than the group with two elements? Necessarily, the non-identity class consists of elements of infinite order. Gene ____________________________________________________________________________________ Want to start your own business? Learn how on Yahoo! Small Business. http://smallbusiness.yahoo.com/r-index

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[math-fun] predict Pochhammer untuplication?
by R. William Gosper 03 Feb '07

03 Feb '07

Define oo /===\ i | | 1 - a q (a; q) := | | ------------, x | | x + i i = 0 1 - a q a product of |x| binomials when x is an integer. By Mathematica, 2 3 3 3 (a; q) = (a q ; q ) (a; q ) (a q; q ) 3 n n - 1/3 n n + 1/3 3 5 4 5 5 (a; q) = (a q ; q ) (a q ; q ) (a; q ) 5 n n - 2/5 n - 1/5 n 5 2 5 (a q; q ) (a q ; q ) n + 1/5 n + 2/5 4 5 2 5 5 3 5 = (a q ; q ) (a q ; q ) (a; q ) (a q ; q ) n - 2/5 n - 1/5 n n + 1/5 5 (a q; q ) n + 2/5 4 5 3 5 5 2 5 = (a q ; q ) (a q ; q ) (a; q ) (a q ; q ) n - 3/5 n - 1/5 n n + 1/5 5 (a q; q ) n + 3/5 What's likely to be the case for 7? 2, 4, and 6? --rwg BRAINSTEM TRIBESMAN b jdb(a)math.arizona.edu aka(a)pu.ac.in hic(a)iname.com aadler(a)sbcglobal.net misha(a)nasledov.com aaronl(a)vitelus.com srinivasamu rthyb(a)yahoo.com rjpetti(a)rcn.com jeffrey.golden(a)verizon.net fateman(a)cs.berkeley.edu djeffrey(a)uwo.ca Rob.Corless(a)uwo.ca ch ongo(a)mellis.com mikeh(a)maths.unsw.edu.au AnneStrass(a)aol.com cshapiro(a)panix.com whitfield.diffie(a)Sun.COM gregaleg909@yahoo .com ono(a)math.wisc.edu tjhuber(a)math.uiuc.edu ruiming_zhang(a)hotmail.com b elh(a)osots.com rudy(a)rudyrucker.com silver(a)pilgrim.com sk(a)pilgrim.com peter7387(a)aol.com rwellessprint(a)earthlink.net b danny(a)appliedminds.com meryt(a)worldnet.att.net djeffrey(a)uwo.ca Rob.Corless(a)uwo.ca smj(a)cirr.com thomasmrodgers(a)yahoo.com mtrott(a)wolfram.com oleg(a)wolfram.com pavlyk(a)wolfram.com countolaf148(a)hotmail.com

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Re: [math-fun] Quintic equation
by Eugene Salamin 02 Feb '07

02 Feb '07

----- Original Message ---- From: Christian Boyer <cboyer(a)club-internet.fr> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Friday, February 2, 2007 3:23:00 AM Subject: RE: [math-fun] Quintic equation Sorry, some mistakes in my previous hypothesis. With the 5th-degree equation: x(x² + a)(x² + b) = c x, a, b, c real numbers a, b > 0 c > (1+a)(1+b) x > 1 what is the fastest method to get an approximated value of x? (very few steps) Christian. For x > 0, the LHS is monotonically increasing from 0. If x = c^(1/5), LHS > RHS. Therefore there is exactly one root in [0, c^(1/5)]. It can be found by successively dividing the interval in half. Gene ____________________________________________________________________________________ Finding fabulous fares is fun. Let Yahoo! FareChase search your favorite travel sites to find flight and hotel bargains. http://farechase.yahoo.com/promo-generic-14795097

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[math-fun] Alex Doonesbury at MIT
by Daniel Asimov 02 Feb '07

02 Feb '07

I've been enjoying the Doonesbury strip when it features the 17-year-old Alex(andra) Doonesbury in her freshman year at MIT, as it does this week, starting with Monday. (See < www.ucomics.com/doonesbury >. Best to read strips in sequence; to do this click on "Yesterday" as soon as it appears and enter your "birth year" and "gender" when asked, then step back to Monday.) This week Alex and a classmate enter a Best Homemade Robot contest. --Dan

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[math-fun] another type of chess problem
by Erich Friedman 01 Feb '07

01 Feb '07

funsters, over the last month, i've become interested in the following type of chess problem: given two types of chess pieces (possibly equal, no pawns) A and B, and two positive integers m and n, is it possible to find a placement of A's and B's so that each A attacks exactly n B's (and no A's), and each B attacks exactly m A's (and no B's)? if so, what is the smallest rectangular board (in terms of area) on which this is possible? i have found many such arrangements, which can be found here: http://www.stetson.edu/~efriedma/mathmagic/0207.html does anyone know if this problem has been considered before? i am convinced that other cases are possible, and that many of my solutions can be improved. can anyone solve any of the unsolved cases, or prove they are impossible? can anyone find smaller solutions for some of these cases? other variations on this theme of pieces attacking only one other type of piece are also explored on the above page. my favorite, and one of the hardest, is: given three types of chess pieces A, B, and C, is it possible to find a placement of A's, B's and C's so that each A attacks 2 B's, each B attacks 2 C's, and each C attacks 2 A's (with no other attacks)? if so, what is the smallest solution? the solutions with A=bishop, B=king, C=knight and A=queen, B=king, C=knight were especially fun to find. erich friedman

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