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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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[math-fun] Pancake cutting problem
by lkmitch＠att.net 04 Jan '05

04 Jan '05

I was looking at the "pancake cutting" problem on MathWorld (more precisely, the problem of circle division by lines): http://mathworld.wolfram.com/CircleDivisionbyLines.html and its cousin, the problem of the plane divided by circles: http://mathworld.wolfram.com/PlaneDivisionbyCircles.html and I'm wondering if there are "nice" ways to draw the images for larger numbers of cuts. I realize that "nice" is horribly ill-defined, but I'm thinking of images that are aesthetically appealing, as well as mathematically accurate. Or, if anyone has any hints to offer about drawing my own, I'd be oh so appreciative. Thanks, Kerry Mitchell

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[math-fun] Freeman Dyson responds to the question "What do you think is true even though you cannot prove it"?
by Thane Plambeck 04 Jan '05

04 Jan '05

I thought this was interesting ---snip---- Since I am a mathematician, I give a precise answer to this question. Thanks to Kurt GÃ¶del, we know that there are true mathematical statements that cannot be proved. But I want a little more than this. I want a statement that is true, unprovable, and simple enough to be understood by people who are not mathematicians. Here it is. Numbers that are exact powers of two are 2, 4, 8, 16, 32, 64, 128 and so on. Numbers that are exact powers of five are 5, 25, 125, 625 and so on. Given any number such as 131072 (which happens to be a power of two), the reverse of it is 270131, with the same digits taken in the opposite order. Now my statement is: it never happens that the reverse of a power of two is a power of five. The digits in a big power of two seem to occur in a random way without any regular pattern. If it ever happened that the reverse of a power of two was a power of five, this would be an unlikely accident, and the chance of it happening grows rapidly smaller as the numbers grow bigger. If we assume that the digits occur at random, then the chance of the accident happening for any power of two greater than a billion is less than one in a billion. It is easy to check that it does not happen for powers of two smaller than a billion. So the chance that it ever happens at all is less than one in a billion. That is why I believe the statement is true. But the assumption that digits in a big power of two occur at random also implies that the statement is unprovable. Any proof of the statement would have to be based on some non-random property of the digits. The assumption of randomness means that the statement is true just because the odds are in its favor. It cannot be proved because there is no deep mathematical reason why it has to be true. (Note for experts: this argument does not work if we use powers of three instead of powers of five. In that case the statement is easy to prove because the reverse of a number divisible by three is also divisible by three. Divisibility by three happens to be a non-random property of the digits). It is easy to find other examples of statements that are likely to be true but unprovable. The essential trick is to find an infinite sequence of events, each of which might happen by accident, but with a small total probability for even one of them happening. Then the statement that none of the events ever happens is probably true but cannot be proved. ---snip---- more here: http://www.edge.org/q2005/q05_print.html -- Thane Plambeck http://www.plambeck.org/ehome.htm

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[math-fun] 2005
by Marc LeBrun 03 Jan '05

03 Jan '05

Happy MMV! From the OEIS we learn, among many other things, that 2005 is... Vertically symmetric [A053701] 6 4^5 + 5 4^4 + 4 4^3 + 3 4^2 + 2 4^1 + 1 4^0 [A059045] The value of 11 n^2 + 11 n + 3 for n=13 [A006222] The value of (14 n^3 - 21 n^2 + 13 n)/4 for n=10 [A071229] The value of ((2n - 1) 3^n + 1)/4 for n=6 [A014915] The coefficient of x^14 in (1-x)/(1-x-x^2-x^3-x^4+x^5) [A033305] Pick integers x, y and z between 1 and 32 inclusive. The expression xy+yz+zx can take on any one of exactly 2005 distinct values [A100440, recently contributed on 21 Nov 04]. Expand (p+1)^8-1 and then "umbrally" replace p^n by the n-th prime. The result sums to 2005 [A050513] Perhaps the most interesting and apropos is the following variant of the well-known "birthday paradox": Pick 53 years at random since the birth of Christ (ie any AD) and the chances are just better than half that two will coincide, but pick only 52 and the odds are just less than even. [A072829, if I've interpreted it aright] Hope you find these factoids of interest. Best wishes for the New Year! --Marc LeBrun

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Re: [math-fun] Learning computer programming for a bright 13-year old ?
by R. William Gosper 29 Dec '04

29 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 20 Dec '04

20 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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