January 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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Re: [math-fun] How we "do mathematics"
by Eugene Salamin 01 Feb '08

01 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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[math-fun] math in Word
by James Propp 31 Jan '08

31 Jan '08

Here's a math non-fun question (but I'm not sure whom to ask, so I'm asking y'all): Is there a way to do half-decent-looking integrals and summations in Word? I'd like to get things to look like / 1 \ f(x) dx / 0 and -- 100 \ / n(n+1) -- n=1 as opposed to / 1 \ f(x) dx / 0 and -- 100 \ / n(n+1) -- n=1 A quick Google search didn't turn up anything helpful. Thanks, Jim Propp

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[math-fun] Re: Hexgo (fwd)
by Richard Guy 31 Jan '08

31 Jan '08

This message was rejected by Math Fun, on account of the attachment. Without it, to make sense, you'll need to find: N E Clarke, S Finbow, S L Fitzpatrick, M E Messinger & R J Nowakowski, Seepage in directed acyclic graphs. It's evidently been submitted to Australasian Journal of Combinatorics ---------- Forwarded message ---------- Date: Wed, 30 Jan 2008 14:07:03 -0700 (MST) From: Richard Guy <rkg(a)cpsc.ucalgary.ca> Subject: Re: Hexago Dear all, I've looked at Adam Holer's research project, that Erich Friedman pointed me to: http://www.stetson.edu/mathcs/students/research/math/ms498/2006/aholer/prop… There are close relations between Adam's games and SEEPAGE [SEE PAGE?] Perhaps at least one of the 5 authors of the (no longer) attached should compare & contrast? Best, R. On Wed, 30 Jan 2008, rjn(a)mathstat.dal.ca wrote: > Richard et al, > > this last summer, a group of Maritimers looked at a graph game, > which we > called `Seepage', similar in nature to Hexago. The players are > Pollution and Green. The rules are: > > Let G be a directed, acyclic graph with root v. Vertex v is assumed to be > polluted at the start of the game but all other vertices are > clean and unprotected. On his turn Pollution is allowed to pollute exactly > one non-protected, clean vertex that is adjacent to some polluted > vertex. On her turn, Green is allowed to protect a non-protected, clean > vertex, which stays protected throughout the rest of the game. Pollution wins > if he can pollute a terminal vertex (we assume all terminal vertices to be > on the shores of a lake---if the lake gets polluted, it's "game over"); > Green wins if she can form a barrier between the root and the lake. > > The main three results were: > > 0) If Pollution can win, he can win by only polluting the > vertices of a path. > > 1) A characterization of trees that are Green-win. > > 2) It is too easy for Pollution to win in general, so we considered > letting Green protect more than one vertex. Let P_n={0,1,2,3,...,n} be a > path of length n. In the Cartesian product P_n x P_n x P_n rooted at > (0,0,0) consider the subgraph G_d consisting of all vertices at > distance at most d from (0,0,0). Now > > If d>=3 then, Green only needs to protect 2 vertices on each > turn to win on G_d. > > We didn't consider looking for game values though. > > Richard > > PS: Shameless Plug: I've attached a copy of the paper since at > least one of you expressed interest. > > ------------------------------------------------------------- > From Richard Guy > ------------------------------------------------------------- > I've just (re?)invented the game of Hexago: > > A Hexapod starts at the origin and moves > on a triangular grid, a distance equal to > one of the sixth roots of unity at each > move, and endeavors to reach the edge of > the board. Her opponent places Go stones > at grid points in an attempt to block her, > in the manner of the games of Chap 19 of > Winning Ways (Variants of Chessgo, Angel > & Square-eater, etc.) > > Problem: are there shapes & sizes of board > for which the game makes sense (allows a > complete analysis, makes for a reasonably > fair game, etc.)? E.g. > _____________________ ...... > / > / > / > / > * ??? > \ > \ > \ > \______________________...... > > Comments? R.

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Re: [math-fun] math in Word
by Henry Baker 31 Jan '08

31 Jan '08

My previous experience with Word's equation editor left much to be desired. Word supposedly handles "html" files. Does anyone know if it can handle html subscripts & superscripts? At 06:47 PM 1/30/2008, Kerry Mitchell wrote: >There's the equation editor embedded in Word. > >Kerry > >> Is there a way to do half-decent-looking integrals and summations in >> Word?

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[math-fun] Re: math in Word
by James Propp 31 Jan '08

31 Jan '08

Kerry writes: >There's the equation editor embedded in Word. I didn't know about that; thanks! As it happens, I'm using the Mac version of Microsoft word (Word:mac), which apparently doesn't include the equation editor. (I tried Insert -> Object, but "Microsoft Equation 3.0" wasn't one of the options that showed up in the resulting menu.) According to the on-line help, >To insert an equation by using the Equation Editor, you must first install >the Equation Editor from the Microsoft Office Value Pack. For more information >on installing Value Pack components, refer to the Guide to the Value Pack on >your Microsoft Office CD. So now I have to track down the Microsoft Office CD that came with my Mac (which I suppose I must have used to install the Office suite of programs on my PowerBook G4 when I purchased it three or four years ago...). One thing that concerns me is that even if I track down the CD and install the Equation Editor, other folks who have Word:mac won't be able to view the equations. So my follow-up questions are: (1) Can equations made in Equation Editor be viewed with versions of Word that don't have Equation Editor? (2) What's the cheapest version of Word for the Mac that includes the equation editor (in case I'm not able to find the Microsoft Office CD that came with my Mac)? Thanks! (And sorry to bring up something so un-fun...) Jim

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[math-fun] Hexgo
by Richard Guy 30 Jan '08

30 Jan '08

I've just (re?)invented the game of Hexgo: A Hexapod starts at the origin and moves on a triangular grid, a distance equal to one of the sixth roots of unity at each move, and endeavors to reach the edge of the board. Her opponent places Go stones at grid points in an attempt to block her, in the manner of the games of Chap 19 of Winning Ways (Variants of Chessgo, Angel & Square-eater, etc.) Problem: are there shapes & sizes of board for which the game makes sense (allows a complete analysis, makes for a reasonably fair game, etc.)? E.g. _____________________ ...... / / / / * ??? \ \ \ \______________________...... Comments? R.

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Re: [math-fun] Hexgo
by Dan Asimov 30 Jan '08

30 Jan '08

Richard describes the asymmetrical 2-player finite perfect-information game of Hexgo, saying: << I've just (re?)invented the game of Hexgo: . . . Problem: are there shapes & sizes of board for which the game makes sense (allows a complete analysis, makes for a reasonably fair game, etc.)? >> I have no answers, other than to mention that "Hexago" seems to trip off the tongue more easily than "Hexgo". But in the mid-1980s I (re?)invented the likewise asymmetrical 2-player finite perfect-information game of Can as a way to play 3D Hex: The board is topologically a solid cylinder D^2 x D^1, and the cells upon which plays occur are the semi-regular truncated octahedra that tile space (as the 3D analogue of hexagons in 2D). (Or if you prefer, equivalently, play occurs on the nodes of a portion of a the dual graph: a bcc lattice with each node connected to its 8+6=14 near neighbors.) in Can*, play proceeds just as in Hex, with one player's goal to span the opposite ends of the Can with a path, and the other players goal to span the separate the opposite ends of the Can with a surface. I mention this now mainly because of its corresponding problem of determing how to set up an asymmetrical game so it will be fair. --Dan _______________________ * named in part as homage to an early name for Hex: John. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] How small can it be?
by Dan Asimov 29 Jan '08

29 Jan '08

Let w denote a primitive nth root of unity. The ring Z[w] is dense in the complex plane for n = 5 and n >= 7, so we assume n lies in the set {5} u {7,8,9,...}. Every element of Z[w] is of the form k_1*w + . . . + k_(n-1)*w^(n-1) for integers k_j, 1 <= j <= n-1. For convenience let K denote (k_1,...,k_(n-1)), and call this element of Z[w] by the name w[K]. Given real M > 0, what can be said about sm(M,n) := min {|w[K]| : ||K|| <= M, and w[K] is not 0} in terms of M (where ||K|| is the Euclidean norm)* ? Is there a reasonably sharp asymptotic upper bound for sm(M,n) as M -> oo ? --Dan _______________ * or else for the taxicab norm ||K|| = |k_1|+...+|k_(n-1)| _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Reflections on Orientation
by Dan Asimov 29 Jan '08

29 Jan '08

Fred wrote: << . . . It's now evident that a major source of confusion was that [coming from a computational geometry direction --- orientation?] I'm concerned rather with orientation of a subspace within the whole space. If we consider more general geometries --- Moebius / conformal / Poincar\'e, Lie sphere / physicist's conformal, and so on --- the Lie group corresponding to Euclidean isometries has 4 connected components rather than 2. On the other hand, orientation of subspaces does not appear to be any more difficult than before. So the idea occurs to me that perhaps the solution to my problem is to consider instead a 4-valued "sub-orientation"? Why should this be justified? >> If a concrete math problem or situation were stated, then I might understand what is being discussed here. Like: * What is it about the "orientation of a subspace within the whole space" that is of interest to you? What kinds of spaces? * With the "4 connected components" thing -- what is one concrete example of what you mean here? * What is it about "orientation of subspaces" that may or may not be "difficult" ? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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