February 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.

2 2

[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.

1 2

[math-fun] functions behaving badly
by James Propp 19 Mar '08

19 Mar '08

Are there any theories of integration out there that would make the function {-(2 Cos[1/x^2])/x + 2 x Sin[1/x^2] for x not equal to 0, f(x) = { { 0 for x = 0 integrable on the interval [-1,1]? Note that the function f(x), although discontinuous at x=0, is equal to the derivative of the function {x^2 Sin[1/x^2] for x not equal to 0, F(x) = { { 0 for x = 0 for ALL x (including x = 0). So, if we were to blindly apply the Fundamental Theorem of the Calculus without attending to the satisfaction of its hypotheses, we would conclude that the integral of f from -1 to 1 equals F(1) - F(-1). Proceeding less blindly, it seems safe to say that any theory of integration that assigned a value to the integral would have to assign it the value F(1) - F(-1). But the unbounded behavior of f(x) in the vicinity of x=0 makes things difficult. Indeed, the function f(x) is neither Riemann integrable nor Lebesgue integrable on [-1,1]. See http://jamespropp.org/142/FunctionsBehavingWorse.nb for a sketch of the graph. (Can nonstandard analysis help here? For some reason I have a feeling that the Stone-Cech compactification might be of use.) Another question arising from this example is: Can things be even worse? Say a function is locally unbounded at a if it is unbounded on every neighborhood of a. If a function F is differentiable everywhere, what can be said about the points at which the derivative is locally unbounded? Can it be dense? Can it be all of R? Jim

3 3

[math-fun] State Diagram of Hanoi
by Steve Witham 06 Mar '08

06 Mar '08

>From: "Guy Haworth" <g.haworth(a)reading.ac.uk> >[asked about numbering states] >From: Dan Asimov <dasimov(a)earthlink.net> >what is an algorithm for using the fewest moves [between two states]? >From: "Andy Latto" <andy.latto(a)pobox.com> >[proof that all states are connected, by induction adding >the smallest disk to a solution for the others] >From: "Fred lunnon" <fred.lunnon(a)gmail.com> >[obvious base-3 numbering, optimized base-2 numbering based on the standard solution] Call the biggest disk 1, the next smallest 2, etc. The state diagram for one disk is a triangle: 1.. .1. ..1 The state diagram for n+1 disks replaces each vertex in the diagram for n disks with a triangle. For instance, for two disks: 21 . . 1 2 . 1 . 2 | \ . 2 1 . 1 2 . . 21 2.1 -- 2 1 . . 21 . Notice that the position of disk 2 is rotated in each of the three sub-triangles. This is like a Gray code, which you can construct from binary by exoring each bit with all the previous bits. So, each state can be assigned an (x,y) by going from the biggest to smallest disk (maybe a right rather than an equilateral triangle?). Also pretty easy to find the three (or two) states reachable from here and their (x,y) positions. Here's an algorithm for finding one of the shortest paths from where you are to target state T: while you're not at T: move to (one of) the reachable state(s) closest (in space) to T Pretty easy to encode the x and y numbers within a single base-three number if you like. You can also just skip the (x,y), let each digit stand for one of three directions, and choose the best move based on the most-significant different digit between the reachable positions. Giving the states (x,y) positions shows ties sometimes, and breaks what are really ties other times, You might say Gray code is a simplified Towers of Hanoi problem. --Steve -- My apples and oranges are beyond compare!

5 6

Re: [math-fun] functions behaving badly
by James Propp 04 Mar '08

04 Mar '08

Here's another "dirty secrets of the calculus" question (arising from my preparing to teach integration by parts): If f and g are differentiable, must f'(x) g(x) have an antiderivative? (Note that the product rule implies that if f'(x) g(x) has an antiderivative, then so does f(x) g'(x), and vice versa But it's conceivable to me that there could be cases where neither of the two functions has an antiderivative.) Jim Propp

2 1

[math-fun] Re help with definite integral
by N. J. A. Sloane 04 Mar '08

04 Mar '08

The other day I asked for help in evaluating a definite integral Let h = 1/2, to save space. Int_{x = 0..h} Int_{r=-h-x..h-x} Int_{y=0..h} Int_ {s=-h-y..h-y} sqrt(r^2+s^2) dx dr dy ds = ? Thanks to David Cantrell, Eric Weisstein, Steve Finch, Emeric Deutsch and others who replied. To summarize the results: Integrals like this can be found in the books Finch, Encyclop. of Math. Constants (CUP); Santalo, Integral Geometry (Addison-Wesley) There is a web page in Spanish: <http://groups.google.com/group/es.ciencia.matematicas/browse_frm/thread/efe… d222ada58ab7> Mathematica can do this integral, but you have to put the variables in the right order, and you need a machine with the right operating system (it worked on a Sun Linux 64-bit machine, but - with the same version of Mma - failed on an Intel-based Linux 64-bit machine) Best regards Neil

3 2

[math-fun] Help with polynomial calculations in Mathematica or Maple
by Dan Asimov 01 Mar '08

01 Mar '08

I find I quickly lose patience when the online "help" files in either of these packages doesn't seem to be helping no matter how I phrase my query. Can someone please tell me how to accomplish the following: I want to multiply out some polynomials (with symbolic coefficients) to get a big hairy polynomial (so far no problem) . . . and then reduce the result modulo setting certain (small) polynomials equal to 0. (It seems that in Mathematica's Simplify command, the Assumptions parameter is intended to do just this. But apparently I can't figure out the syntax, and the "help" files aren't telling.) (What I tried was Simpify[hairypolynomial, {smallpoly = 0}] but it didn't like the syntax for {smallpoly = 0}.) Thanks in advance for any assistance with this. --Dan P.S. In an alternative attempt to get Mathematica to enforce the simple condition that x^5 = 1, I tried just setting x = Exp[2 Pi i/5]. It certainly understood this, since when I typed x^5 it responded with: 2 i Pi Out[2]= E . But an attempt to Simplify that expression left it unchanged. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

4 3

Re: [math-fun] Mathematica Viewer
by Steve Witham 28 Feb '08

28 Feb '08

>From: James Propp <jpropp(a)cs.uml.edu> > I >should mention that there's now free software for viewing Mathematica >notebooks): See http://jamespropp.org/142/FunctionsBehavingWorse.pdf . You only have to buy a big enough computer! --Steve

1 0

[math-fun] horrors calculus
by James Propp 26 Feb '08

26 Feb '08

In addition to teaching honors calculus four times a week, I'm also currently working out on a treadmill in my home four times a week or so, and this week I'm watching the movie "Lemony Snicket's A Series of Unfortunate Events" on the video player that faces the treadmill. The two influences have gotten a bit mixed up in my mind, and are inspiring fantasies of a gothic honors calculus supplementary text that begins something like this: "The calculus of Newton and Leibniz is one of the most beautiful and useful creations of the human mind. It is founded on two wonderful operations called integration and differentiation that fit together like hand and glove, forming a tight closed circle that admits no intrusions of discord or doubt. If you start with any function you like (or rather any function a sensible, emotionally balanced person would like), perform integration on it, and then perform differentiation on the result, you will find yourself back where you started. You will not have gotten anywhere. But why would you want to go anywhere? Why not stay where you are, where all is bliss and harmony? "If this is the sort of story you want to hear, Reader, I advise you to shut this book immediately and go purchase any of the hundreds of light-hearted calculus books on the market, many of which have colorful pictures of happy people doing happy, calculus things. "If, however, you want to know about the pathological functions that Newton and Leibniz never knew about --- the monstrous constructs from which nineteenth century mathematicians recoiled in horror --- then read on, and meet the subtle, mind-numbing creatures that have been relegated to the Calculus Closet. Let us open the door together: slowly, slowly..." Jim

2 2

Re: [math-fun] functions behaving badly
by James Propp 26 Feb '08

26 Feb '08

Dan Asimov writes: >Using this function g and an enumeration >{q_n}, n = 1,2,3,... of the rationals, define > > h:R -> R >via > h(x) := Sum_{n=1,2,3,...} g(x-q_n)/2^n I like Dan's idea of translating the graph of x^2 sin 1/x^2 horizontally by a dense set of displacements and summing (with coefficients falling off quickly enough to encourage summability). But I'm worried that if an adversary picks the enumeration of the rationals, the sum won't converge everywhere (because x^2 sin 1/x^2 can be pretty big when x is large). Fortunately this issue can be ducked by using the bounded function x^2 sin 1/x^2 exp -x^2. Andy Latto asks: >Can you provide a hint of the proof that it is not Lebesgue integrable? I believe that the total signed area between the graph of f(x) and the x-axis is not absolutely summable. I.e., the integral of max(f(x),0) on [-1,1] is infinite. Fred Lunnon writes: >Only somebody with Nota Bene installed is going to be able to view this! Actually ".nb" here means "Mathematica notebook". I'll make everyone's life easier by putting a PDF on the web (though I should mention that there's now free software for viewing Mathematica notebooks): See http://jamespropp.org/142/FunctionsBehavingWorse.pdf . Jim

1 0