February 2005 - mathfun@mailman.xmission.com

[math-fun] Beyond base 10
by David Wilson 22 Feb '05

22 Feb '05

I observe that the adoption of numerical symbols distinct from the writing symbols is a major cultural and mathematical advance. The main benefit is the ability to freely intermix words and numbers with little chance of misinterpretation. Clearly, a separate numerical system is superior to a system in which numbers must be represented by words, or in which letters double as numbers. I have always thought that the use of letters for digits in higher-base numbers (i.e, A = 10, B = 11, C = 12, etc.) is rather a step backwards. The letter values are unintuitive, words and numbers become confusable, a fairly universal digit set is extended with culturally proprietary symbols, numbers now have case issues, and for all that we run out of symbols again at base 37. An extensible set of digits would be useful. For example, the digit 10 might be represented by a glyph incorporating the number 10 (a circled, boxed, barred, or otherwise joined digits of 10). Such a glyph system would be extensible to any reasonable base, and digit values would be unmistakable wherever Arabic numerals are recognized. I was wondering, prior to the computer age, were higher bases used widely? Was there an accepted representation for higher-base numerals? Did A = 10, B = 11, etc, arise before the computer age, or did it arise out of hexadecimal computer applications? I always assumed it was just a pragmatic adoption of the most suitable ASCII characters as additional digits. -- No virus found in this outgoing message. Checked by AVG Anti-Virus. Version: 7.0.300 / Virus Database: 266.2.0 - Release Date: 2/21/2005

3 2

[math-fun] Arnold's "strange infinite sequence of degrees"
by N. J. A. Sloane 22 Feb '05

22 Feb '05

To Math-Fun, Seq-Fans: There is an article by Arnold which has appeared in several places. There are two versions in English: MR2080045 Arnold, V. I. From Hilbert's superposition problem to dynamical systems [ MR1733564 (2001h:01031)]. Amer. Math. Monthly 111 (2004), no. 7, 608--624. 01A65 (01A60 37-03 54H20) MR1733564 (2001h:01031) Arnold, V. I. From Hilbert's superposition problem to dynamical systems. The Arnoldfest (Toronto, ON, 1997), 1--18, Fields Inst. Commun., 24, Amer. Math. Soc., Providence, RI, 1999. (Reviewer: Stanis\l aw Janeczko) 01A65 (01A60 37-03 54H20) Footnote 2 says: "Starting from degree 9, one can kill one more coefficient. The known possibilities to kill more coefficients occur along a rather strange infinite sequence of degrees." He is refering to the problem of putting a polynomial equation f(X)=0 into canonical form. E.g. any 5th degree equation can be reduced to x^5 + aX + 1 = 0. My question is, what is this "strange infinite sequence of degrees"? Thanks Neil Sloane

1 0

Re: [math-fun] Ulam number picture
by David Wilson 21 Feb '05

21 Feb '05

OK, let's start again. To all: I have a picture of the Ulam numbers that illustrates their quasiperdiocity. I created this picture a long time ago and sent a copy to Eric Weisstein thinking that he might use it in Mathworld, but apparently no interest. To all, (esp. Hugo Pfoertner, Dan Asimov): I am planning on getting rid of the image. Before doing so, I will send a copy to whomever requests. To NJAS: I will send you a copy, which you may place on your web site if you wish, in association with A002858. The image is actually fairly simplistic, and easily reproducible, and I do not care about attribution. To Daniel Asimov: A PNG is a Portable Network Graphic file. PNG images are fairly common on the Internet and viewable using all major browsers and PC utilities. I prefer PNGs because they are (1) public domain (unlike most other image formats) (2) small (unlike Microsoft BMP or Compuserve GIF images) (3) faithful (unlike JPG images, which sacrifice image quality for small size).

2 2

[math-fun] the generous grader problem
by James Propp 21 Feb '05

21 Feb '05

Martin Hock, a computer science grad student in UW Madison who was working for me as a grader in Spring 2004 did some work on a problem we called the generous grader problem, and we're wondering if it has been described in the literature before. The problem arose from an unforeseen collision between my announced grading policy ("I will drop your two lowest homework scores") and the fact that not all of the problem sets had the same weight (each problem was worth 10 points, so that problem sets with more problems were weighted more heavily). It turns out that dropping the two lowest scores is not always the best thing a grader can do for the purpose of boosting a student's score; e.g., it might be better to drop the third and fourth lowest scores if those two problem sets carried a lot of weight. This made us wonder about the general problem lurking here: How do you find the subset N of {1,2,...,n} of fixed cardinality m that maximizes sum_{i in N} s_i ---------------- sum_{i in N} h_i (where s_i is a student's actual score on the ith assignment and h_i is the maximum possible score for that assignment)? Hock devised an algorithm that always finds the optimal choice of N, and the algorithm appears to be quite fast in practice (although Martin hasn't proved a polynomial bound on the running time yet). Jim Propp

1 0

[math-fun] Ulam number picture
by David Wilson 20 Feb '05

20 Feb '05

I have a PNG showing the Ulam numbers. Numbering the pixels lexicographically, the Ulams are black, the other numbers white. It clearly shows the regular periodic clumping of the Ulams.

1 0

[math-fun] Wolfbang Andanoyus
by R. William Gosper 18 Feb '05

18 Feb '05

Those of you who use the Mozart effect against tough math and programming problems may want to read a short flame: www.ippi.com/rwg/shredwood.txt . --rwg

1 0

[math-fun] Wandering clock
by Eric Angelini 18 Feb '05

18 Feb '05

Hello math-fun ans seqfan [crossposted to rec.puzzles] This should look like an ordinary clock (without hands) : 12 11 1 10 2 9 3 8 4 7 5 6 Consider the 15 digits above and imagine them suddenly moving clockwise to another place on the clock: --> the "1" digits move 1 step clockwise; --> the "2" digits move 2 steps clockwise; --> the "3" digit moves 3 steps clockwise; etc. A « jump » sees the 14 digits moving simultaneously, accordingly to their nature (0, the 15th digit, never moves, of course). So after jump 1 we would have this configuration (digits on the same place assemble to form the smallest integer): 116 1 1 50 127 . . 4 28 . . 39 Jump 0 to jump 5 configurations are represented here: Jump0 1 2 3 4 5 6 7 8 9 10 11 12 Jump1 1 127 . 28 . 39 . 4 . 50 1 116 Jump2 11 1 159 2 . 26 . . 37 0 . 148 Jump3 1 11 1 147 . 2 . 258 . 0 . 369 Jump4 5 1 113 18 1 6 . 24 9 20 7 . Jump5 . . 1 11 1 13579 . . . 20 . 2468 ... Question: at what jump will the « jump 0 » configuration appear again? Best, É. ---- ObSeqFan : We could associate an integer to every jump : it's "horizontal" sum (81 for jump 0; 366 for jump 1; etc.) Find the sequence linking jump 0 to jump 0' (same config. again)

1 0

RE: [math-fun] 100 Digrams -- 3059
by Mike Stay 16 Feb '05

16 Feb '05

Rich wrote: > I'm puzzled about this. > What are we supposed to do about including 00,01,...? The digrams occur in the concatenation of the terms in the sequence, so 00 occurs in 100 and 01 in 100122. > And why can't the sequence below be improved by replacing > 188 with 18 and 88? Then 1888 has two copies of 88 [1(88)8 and 18(88)], which is not allowed. > > Rich > > -----Original Message----- > From: > math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com > [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmissi > on.com] On Behalf Of Eric Angelini Sent: Wednesday, > February 16, 2005 11:52 AM To: math-fun; > seqfan(a)ext.jussieu.fr Subject: [math-fun] 100 Digrams -- > 3059 > > > Still better, by Geoff Bailey (original post is on > rec.puzzles). Best, > É. > ------------------------------- > > 8 19 > 20 23 24 25 26 27 28 29 > 30 34 35 36 37 38 39 > 40 45 46 47 48 49 > 50 56 57 58 59 > 60 67 68 69 > 70 78 79 > 80 89 > 90 91 > 100 122 133 144 155 166 177 188 > > Total 3059 > > Cheers, > Geoff. > ---------------------------------------------------------- > ---------- Geoff Bailey (Fred the Wonder Worm) | > Programmer by trade -- ftww(a)maths.usyd.edu.au > | Gameplayer by vocation. > ---------------------------------------------------------- > ---------- > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

1 0

RE: [math-fun] 100 Digrams -- 3059
by Schroeppel, Richard 16 Feb '05

16 Feb '05

I'm puzzled about this. What are we supposed to do about including 00,01,...? And why can't the sequence below be improved by replacing 188 with 18 and 88? Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com] On Behalf Of Eric Angelini Sent: Wednesday, February 16, 2005 11:52 AM To: math-fun; seqfan(a)ext.jussieu.fr Subject: [math-fun] 100 Digrams -- 3059 Still better, by Geoff Bailey (original post is on rec.puzzles). Best, É. ------------------------------- 8 19 20 23 24 25 26 27 28 29 30 34 35 36 37 38 39 40 45 46 47 48 49 50 56 57 58 59 60 67 68 69 70 78 79 80 89 90 91 100 122 133 144 155 166 177 188 Total 3059 Cheers, Geoff. -------------------------------------------------------------------- Geoff Bailey (Fred the Wonder Worm) | Programmer by trade -- ftww(a)maths.usyd.edu.au | Gameplayer by vocation. --------------------------------------------------------------------

1 0

[math-fun] 100 Digrams -- 3059
by Eric Angelini 16 Feb '05

16 Feb '05

Still better, by Geoff Bailey (original post is on rec.puzzles). Best, É. ------------------------------- 8 19 20 23 24 25 26 27 28 29 30 34 35 36 37 38 39 40 45 46 47 48 49 50 56 57 58 59 60 67 68 69 70 78 79 80 89 90 91 100 122 133 144 155 166 177 188 Total 3059 Cheers, Geoff. -------------------------------------------------------------------- Geoff Bailey (Fred the Wonder Worm) | Programmer by trade -- ftww(a)maths.usyd.edu.au | Gameplayer by vocation. --------------------------------------------------------------------

1 0