December 2004 - mathfun@mailman.xmission.com

[math-fun] (Fwd) [NOBNET 4625] Mathematical Games
by Bernie Cosell 09 May '05

09 May '05

I know that in addition to your likely being interested in this, I know that several of you have contributed to Mathematical Games over the years and so might want to get in touch with Ed... /Bernie\ ------- Forwarded message follows ------- Date: Mon, 8 Nov 2004 06:30:37 -0800 (PST) From: ed pegg <ed(a)mathpuzzle.com> Subject: [NOBNET 4625] Mathematical Games To: nobnet(a)iijnet.or.jp Martin Gardner just recently celebrated his 90th birthday. Also, maa.org is *finally* releasing the entire collection of Mathematical Games on CD rom next month. There were frightening amounts of copyright issues to work out, but they have been resolved. Catalog Code: TDG/FW04 ISBN 0-88385-545-3 List Price: $54.95 This is all 15 Mathematical Games books on a CD-Rom. I'd like to do a "behind the scenes" look at Mathematical Games for an article. Martin did lots, but he avidly used a lot of help from hundreds of people. I'd like to write an article about the background help. If you ever assisted in one of Martin's columns, I'd like to hear from you -- a note about what you did. I'll use these notes in my article. Ed Pegg Jr http://www.maa.org columnist http://www.mathpuzzle.com ------- End of forwarded message ------- -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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[math-fun] A missing sequence from group theory?
by N. J. A. Sloane 04 Feb '05

04 Feb '05

To MathFun and SeqFans: A possibly new sequence? Could someone calculate it? Let t(G) = no. of unitary factors of the abelian group G, and let T(n) = Sum t(G) taken over all abelian groups of order <= n. There are several papers giving estimates for T(n), e.g. T(x) = c_1 x (log x + 2 gamma - 1) + c_2 x + ... My question is, how does the sequence T(n) begin? References and definitions follow, from MSN. NJAS MR1850168 Zhai, Wen Guang On the average number of unitary factors of finite abelian groups. II [MR1813450 (2001j:11092)]. (Chinese) Acta Math. Sinica 44 (2001), no. 4, 667--672. 11N45 MR1813450 (2001j:11092) Zhai, Wenguang On the average number of unitary factors of finite abelian groups. II. Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 4, 549--554. (Reviewer: R. C. Baker) 11N45 (11L07 20K01) MR1641042 (99e:11124) Zhai, Wenguang; Cao, Xiaodong On the average number of unitary factors of finite abelian groups. Acta Arith. 85 (1998), no. 4, 293--300. (Reviewer: R. C. Baker) 11N45 MR1613290 (99f:11124) Wu, J. On the average number of unitary factors of finite abelian groups. Acta Arith. 84 (1998), no. 1, 17--29. (Reviewer: R. C. Baker) 11N45 (11L07 20K01) 1225427 (94k:11108) Schmidt, Peter Georg(D-MRBG) Zur Anzahl unitrer Faktoren abelscher Gruppen. (German) [On the number of unitary factors in abelian groups] Acta Arith. 64 (1993), no. 3, 237--248. Part of this review follows: Let $\scr A$ denote the set of all abelian groups. Under the operation of direct product, $\scr A$ is a semigroup with identity element $E$, the group with one element. $G_1$ and $G_2$ are relatively prime (German: teilerfremd) if the only common direct factor of $G_1$ and $G_2$ is $E$. We say that $G_1$ and $G_2$ are unitary factors of $G$ if $G=G_1\times G_2$ and $G_1,G_2$ are relatively prime. Let $t(G)$ denote the number of unitary factors of $G$, and let $T(x)=\sum_{G\in {\scr A}, |G|\le x} t(G).$ There are constants $A,B,$ and $C$ such that $\Delta(x)\colon =H(x)-Ax\log x - Bx -C\sqrt x=o(\sqrt x).$ In this paper, the author shows that $\Delta(x)\ll x^{3/8} \log^{7/2}x.$ The previous best estimate for $\Delta(x)$ had been $\Delta(x)\ll x^{31/82} \log^2 x$; this was given by H. Menzer \ref["Exponentialsummen und verallgemeinerte Teilerprobleme", Habilitationsschrift, Friedrich-Schiller-Univ., Jena, 1992; per bibl.]. Further references from the Wu (1998) paper: Many accented letters have been dropped! # R. C. Baker and G. Harman, Numbers with a large prime factor, Acta Arith. 73 (1995), 119--145. MR1358192 (97a:11138) # E. Cohen, On the average number of direct factors of a finite abelian group, ibid. 6 (1960), 159--173. MR0118764 (22 #9535) # E. Fouvry and H. Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989), 311--333. MR1027058 (91b:11097) # S. W. Graham and G. Kolesnik, Van der Corput's Method of Exponential Sums, Cambridge Univ. Press, Cambridge, 1991. MR1145488 (92k:11082) # C. H. Jia, The distribution of square-free numbers, Sci. China Ser. A 36 (1993), 154--169. MR1223084 (94f:11095) # G. Kolesnik, On the number of abelian groups of a given order, J. Reine Angew. Math. 329 (1981), 164--175. MR0636451 (83b:10055) # E. Krtzel, On the average number of direct factors of a finite abelian group, Acta Arith. 51 (1988), 369--379. MR0971087 (89m:11091) # H.-Q. Liu, The greatest prime factor of the integers in an interval, ibid. 65 (1993), 302--328. MR1259341 (95d:11117) # H.-Q. Liu, On some divisor problems, ibid. 68 (1994), 193--200. MR1305200 (96a:11089) # H.-Q. Liu, Divisor problems of 4 and 3 dimensions, ibid. 73 (1995), 249--269. MR1364462 (96k:11118) # P. G. Schmidt, Zur Anzahl unit\"arer Faktoren abelscher Gruppen, ibid. 64 (1993), 237--248. MR1225427 (94k:11108) # P. Sargos and J. Wu, Multiple exponential sums with monomials and their applications in number theory, Prpublications 97/ n37 de l'Institut lie Cartan, Universit Henri Poincar (Nancy 1). # J. Wu, On the distribution of square-full and cube-full integers, Monatsh. Math., to appear. cf. MR1657818 (2000a:11125)

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[math-fun] Learning computer programming for a bright 13-year old ?
by Daniel Asimov 05 Jan '05

05 Jan '05

My very bright 13-year-old nephew would like to learn computer programming (from scratch). He's quite familiar with using software, especially having used Midi music software for several years now. At this point I don't think he has any specific goal of building a particular kind of software; he just wants to learn what computer programming is all about. Has anyone had success with a particular language and/or book in similar circumstances? All suggestions are welcome. At this point I'd like to give him something simple (so probably Java is out) but not too simple (so maybe Alice is out). But my mind's not made up about these, either. Many thanks for your ideas. --Dan

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Re: [math-fun] Learning computer programming for a bright 13-year old ?
by R. William Gosper 30 Dec '04

30 Dec '04

I said > For a certain concrete thinker, I'm > planning the following first program: Combine Macsyma's string > manipulation functions, especially CARDINAL_STRING (12 -> "twelve") and > ORDINAL_STRING (12 -> "twelfth") to make a FRACTION_STRING function > (11/12 -> "eleven twelfths"). There are nice opportunities for bugs > ("one twelfths", 1/2 -> "one second", 3 -> "three firsts") This went great!! After he got those fixed and was sure it all worked, 3/2 -> "three halfs" and after the obvious fix, "three halvess". He *really* enjoyed that session. Richard Fateman added >You might be surprised to learn that speaking fractions is ambiguous. 30/400 and 34/100 are quite different numbers, but each can be pronounced thirty four hundredths Luckily, we dodged that one. It probably can't happen with reduced fractions anyway, but just for kicks: block([gcd:false,fancy_display:false], for x in [34/100,30/400] do print(x=number_string(x)))$ 34 --- = thirty-four one hundredths 100 30 --- = thirty four hundredths 400 --rwg TrimSpa armpits imparts partism RATSIMP

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[math-fun] Article on math education in Sunday's NY Times
by Daniel Asimov 29 Dec '04

29 Dec '04

Here's the link: http://www.nytimes.com/2004/12/12/weekinreview/12mcne.html; it's valid through most of next Saturday, Dec. 18. (Note: If they make you divulge your e-address and choose a userid, I can assure you from almost 10 years' experience that -- unless you opt in for receiving various e-mails from the Times -- you won't get any e-mail from them or anyone else as a result of registering.) I'm curious what people's reactions are to this opinion piece. --Dan Asimov

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[math-fun] Possibly new sequence related to dodecahedron
by N. J. A. Sloane 28 Dec '04

28 Dec '04

The following is an interesting question raised by Wouter Meeussen <wouter.meeussen(a)pandora.be> on Dec 27 2004: He said: >> Based on the symmetry point-group, how many 'different' sets of 1, 2, .., 10 vertices >> exist on a dodecahedron? >> >> If I count different norms of sums of 1, 2, .., 10 unit vectors, I find: >> 1,5,12,22,34,50,65,78,78,86 >> but there might be 'different' sets that accidentaly sum to the same resultant's norm. >> So the given integers are a lower bound. We can rephrase the question as follows. Let G be the full icosahedral group, of order 120. Let v_1, ..., v_20 be the vertices of the dodecahedron. Let S(n) be the set of vectors v_{i_1} + v_{i_2} + ... + v_{i_n} where 1 <= i_1 <= i_2 <= ... <= i_n <= 20. Then what is s(n), the number of orbits of G on S(n)? (so s(1) = 1, s(2) = 6, ...) Presumably this is different from A039742: ... %S A039742 1,6,50,475,4881,52835,593382,6849415,80757819,968400940,11773656517, %T A039742 144791296055,1797935761182 %N A039742 Lattice animals in the fcc lattice (12 nearest neighbors), connected rhombic dodecahedra, edge-connected cubes. ... Repeat for icosahedron, etc.! NJAS

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[math-fun] Re: math-fun Digest, Vol 22, Issue 8
by Gary McGuire 21 Dec '04

21 Dec '04

Victor Miller wrote: >one looks at relative harmonic density. There is a nice paper by >DIA Cohen and Talbot Katz that explains this and other >generalizations. > >MR0746863 (85j:11014) >Cohen, Daniel I. A.(1-RCF); Katz, Talbot M.(1-RCF) >Prime numbers and the first digit phenomenon. >J. Number Theory 18 (1984), no. 3, 261--268. >11A63 (11B05) .... >The authors prove that for a fairly general class of sequences >$A$ the relative supernatural density of the subsequence of $A$ whose >first digit (in the decimal expansion) is $k$ is $\log \sb >{10}((k+1)/k)$. So the set of primes with first digit 1 has no natural density, but has supernatural/Dirichlet density log_{10} (2) ~= 0.3, the primes with first digit 2 have (supernatural) density log_{10} (3/2) ~=0.176, .... and the primes with first digit 9 have density log_{10} (10/9) ~= 0.046. This would seem to explain the first digit phenomenon all right. Nice that sum_{k=1}^9 log_{10} (k+1)/k = 1. Gary McGuire

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[math-fun] more densities
by Gary McGuire 21 Dec '04

21 Dec '04

A puzzle on natural density appeared in the Fall 2004 newsletter of MSRI: Find two sets of natural density 1/2 whose intersection does not have a natural density (i.e. the limit does not exist). Here I mean natural density in the positive integers, density(S) = lim (number of elements in S that are <= n) / n Gary McGuire

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[math-fun] Article on math education
by James Propp 21 Dec '04

21 Dec '04

>I wonder how those prospective teachers would have fared > >comparing 7^sqrt(8) vs. 8^sqrt(7) instead of 1/13 vs. 0.13 I originally misread this as "comparing 7*sqrt(8) vs. 8*sqrt(7)", which is actually a pretty good question for diagnosing underlying math knowledge. (I'm afraid that many of them, if told "You can do this in your head", would object "You can't expect us to compute square roots without a calculator!") Jim Propp

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[math-fun] triangular sums
by William Thurston 19 Dec '04

19 Dec '04

There's a nice, well-known theory of which integers can be expressed as the sum of two squares. What can be said about numbers that are sums of two triangles, i.e. 1 or 3 or 6 or 10 or 15 ... + 1 or 3 or 6 or 10 or 15 ...? Bill

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