June 2003 - mathfun@mailman.xmission.com

[math-fun] addition-subtraction chains
by Richard Schroeppel 23 Jun '03

23 Jun '03

I noticed the sequence A003064: 2,3,5,7,11,19,29,47,71,127,191,379,607,1087,1903,3583,6271, 11231,18287,34303,65131,110591,196591,357887 Name: Smallest number with addition chain of length n. 1903 = 11*173 is the first composite, but after that it's about 50:50. It looks like the limiting ratio might gradually approach 2. I get 2 3 5 7 11 19 29 53 87? for the corresponding addition-subtraction chain. (I'm working by hand, errors all too likely.) Rich rcs(a)cs.arizona.edu

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[math-fun] Calculus I problem
by R. William Gosper 22 Jun '03

22 Jun '03

with a nice shortcut and an interesting answer: Q: A cylindrical water tank, radius 1, rests on its side, skewered by a coaxial bar of radius r. Choose r to maximize capacity if the water level may not exceed the top of the bar. A: .retal etaitnereffid od gniog tsuj er'uoy ecnis )2^x-1(trqs largetni rotces eht od ot deen t'nod uoy taht si tuctrohs ehT .)2^ip+1(trqS/1

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Re: [math-fun] Emacs Tetris & statistics (fwd)
by Erik Demaine 20 Jun '03

20 Jun '03

On Fri, 20 Jun 2003, Hugh Everett wrote: > Dean Hickerson suggested the game: > Player 1 chooses a tetromino, and player 2 then places it according to > the rules of > tetris (any sequence of rotate, left, right, down). With optimum > strategy, can > Player 1 force the board to fill? This is the usual game of Tetris, except that in normal Tetris, player 1 is "nature" in the sense that it plays stochastically. [See Christos Papadimitriou's paper "Games Against Nature".] It is known that nature can foil player 2, with some simple distributions on pieces. This result implies what you want, if you allow randomized strategies for player 1. There are also deterministic strategies, e.g., alternating between Z and S eventually dooms player 2 for boards of width 2n for odd n. Here are the references: John Brzustowski. Can you win at Tetris? Master's thesis, University of British Columbia, 1992. Heidi Burgiel. How to lose at Tetris. Mathematical Gazette, July 1997. See the bottom of page 3 of this paper: http://theory.lcs.mit.edu/~edemaine/papers/Tetris_COCOON2003/ Erik -- Erik Demaine | edemaine(a)mit.edu | http://theory.lcs.mit.edu/~edemaine/

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Re: [math-fun] Lots of Triangle Centers
by Henry Baker 20 Jun '03

20 Jun '03

This page is dying for an animation... You can take multiple gif's and compose them into an animated gif. There may even be free software online somewhere to do this. At 02:45 PM 6/20/03 -0700, Ed Pegg Jr wrote: > I was curious about how 800 different triangle centers moved around as the >underlying triangle changed, so I generated lots of pictures. You can see them >at http://www.mathpuzzle.com/TriCent/TriangleCenter.html . On my front page, >I've added links to Kimberling's page, Weisstein's page, and a list of 2000 >centers compiled by Edward Brisse. > >--Ed Pegg Jr, http://www.mathpuzzle.com

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[math-fun] Emacs Tetris & statistics
by Henry Baker 20 Jun '03

20 Jun '03

I have become intrigued by Tetris, not as a game, but as an online packing problem. I'm not interested in the optimum solution, per se, but in the statistics of the problem. Normally, Tetris has 10 columns and 7 tetraminos -- 2x2 "block", 2 "L"'s (L, L', mirror images), 2 "Z"'s (Z, Z', mirror images), 1 "T", and a 4x1 "line". Each tetramino has an area of 4 units. These tetraminos show up randomly, but with a uniform distribution -- the 7 different types are equally distributed. I'm curious about the interaction between the packing strategies/heuristics and the probability distribution of the pieces. Everyone who has played the game has noticed how the 4x1 pieces appear critical to achieving a high score -- if they show up too rarely, you quickly run out of space as things stack up. Tetris appears to be very close to "criticality", in the sense that with the right heuristics and statistics, one might achieve an unbounded score. One way to understand this phenomenon is to embed enough quality-playing heuristics, and then play (automatically) lots of games with slightly varying statistics of the pieces to see how the average score changes. Clearly, on a Tetris board with 14 columns, one could devote every pair of columns to a single piece type. Since each piece type can efficiently pack with itself in a 2-wide column, and since each type has the same probability and the same area, each of the column pairs will fill at the same rate, on average. So only large variances will cause the game to terminate. When the game is squeezed to only 10 columns, things get much more interesting, as there does not appear to be any similar strict "ghetto" strategy. I have taken the standard Lisp tetris code distributed with Emacs and made some radical changes to allow the parameters of the game to be modified. For example, you can now change the width (# columns) of the game and choose the probability distribution (I use Walker's algorithm, see Knuth, v.II). I intend to implement backing up (so you can practise better), additional levels of lookahead, automatic placement according to certain heuristics, etc. If anyone else on this list is interested, I'll be happy to post the code on my web site. I'll go with the usual Gnu GPL, and my changes are public domain if anyone cares. I make no guarantees about quality or reliability of the code, and it will get modified sporadically with no warning. Henry Baker

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[math-fun] nome
by R. William Gosper 20 Jun '03

20 Jun '03

Someone privately requested the definition of "nome". Not to be confused with the garden pest ( http://alyssadolls.homestead.com/potterica.html http://www.angelfire.com/stars2/harrypotterland/degnome.html http://membres.lycos.fr/justpotter/studies4.html http://www.geocities.com/hogwarts_29/ww/2/2c.html http://meisetsu.topcities.com/enchantments/beasts2.html http://neocortex.by.ru/index/d1.html#Degnoming) a nome is merely a crude approximation to a pole, as in the Alaskan city whose famed beauty pageant annually crowns "Miss Nomer." It can also refer to the little dot on the bottom of a fruit opposite the stem, as in nome de plum, (in deference to Asian- Americans, no longer called "berry button.") Finally, of course, there is the dyslexic animated fish. Hmm. The guy with the funny cigarette is no longer upwind, so let's try again. For what might seem an unmotivated web of definitions, see http://mathworld.wolfram.com/Nome.html . In summary, "nome" is merely the name for the second argument of a theta function, and is usually denoted by q. This is the same q as the "base" in "basic-" or q-hypergeometric series, so they both could have been called "nome" or "base". In fact, q is just a parameter, i.e., it tends to remain fixed, undefined, or even implicit for a given application of theta functions. E.g., the tumbling of an asteroid can be described with quotients of theta functions wherein the first argument is proportional to time, while the nome remains fixed. See, e.g., www.emis.de/proceedings/7ICDGA/VI/puta.ps . No, that's the name of the author, not a piece of asteroid. So why didn't they call it "parameter" instead of "base" or "nome"? Because "parameter" was already usurped to mean two other things: modulus^2 in the case of theta, and (incompatibly) a constituent of a series coefficient in the hypergeometric case. Finally, do not be discouraged that there are four theta functions. They are just four translates of the same function, like sine vs cosine. But unlike sine and cosine, thetas almost never appear alone, but rather paired in quotients. It is as if you never needed sine, just tangent and cotangent. --rwg

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[math-fun] NYTimes: Madam, I'm Adam!
by Henry Baker 20 Jun '03

20 Jun '03

This is no joke! According to a front-page story in the NYTimes yesterday, the male human Y chromosome is (actually, contains) palindromic! By folding back upon itself, the redundancy can be used to repair bad DNA sequences. So this palindrome is now a pun!

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[math-fun] The Santayana Corollaries
by Thane Plambeck 20 Jun '03

20 Jun '03

"Those of us who forget the past are condemned to repeat it." (George Santayana) "Those of us who remember that it was Santayana who said, 'those of us who forget the past are condemned to repeat it,' are condemned to repeat it." (Plambeck) And: By the G.S. principle itself, even those of us who forget that it was George Santayana who said, "those of us who forget the past are condemned to repeat it," are condemned to repeat it... Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm

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RE: [math-fun] Geometry/topology question about planar graphs
by Andy Latto 17 Jun '03

17 Jun '03

> -----Original Message----- > From: John Conway [mailto:conway@Math.Princeton.EDU] e. > However, the > "Carpenter's Ruler" problem (are the plane embeddings of a > hinged ruler > interconvertible in this way, keeping the segment-lengths fixed) is > quite difficult - indeed I think it's still unsolved, This was finally solved, in the affirmative, by Erik Demaine, Robert Connelly, and Guenter Rote. http://theory.lcs.mit.edu/~edemaine/papers/Linkage/ > and the > corresponding assertion for arbitary graphs is false. Is there an easy-to-describe counterexample? Andy Latto andy.latto(a)pobox.com

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Re: [math-fun] Gardner's 6th book of games
by Jud McCranie 17 Jun '03

17 Jun '03

At 10:37 PM 6/16/2003, you wrote: > > Is Martin Gardner's "Sixth book of mathematical games/diversions" > > also known by a different name? Can someone tell me the name of the first chapter? Thanks.

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