June 2004 - mathfun@mailman.xmission.com

[math-fun] Least dense packings
by Jon Perry 07 Jul '04

07 Jul '04

Let C be a 2d container with height h and width w from the non-negative reals. Unit circles (radius 1) can be placed inside the container, and they obey the usual rules of gravity, and do not intersect. What is the minumum number of unit circles needed to place a circle such that its highest point is of height >= h? If w<2 then f(h,w)=oo as we cannot place any unit circles into a container of this width. At w=2, f(h,w)=ceil(h/2) but for w>2 f becomes much less defined. Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] sqrt(binomial) = hexanomial?
by R. William Gosper 30 Jun '04

30 Jun '04

I asked >Do I hear six? So far, only from trinomials: 1/3 sqrt(10 - sqrt(5) + 5) = 5/6 1/3 5 1/6 2/3 sqrt(5) 5 1/6 1/6 5 ---- - 2 5 + ------- - ---- + 2 2 5 + ------- 1/6 sqrt(2) 1/6 sqrt(2) 2 2 ---------------------------------------------------------, 3 2/3 1/6 1/3 sqrt(5 + 5 + 2 ) = 5/6 2/3 1/3 1/6 5 5 1/6 5 5 1/6 - ------- + ---- + 2 sqrt(5) + ------- + ---- + 2 2 sqrt(2) 1/6 sqrt(2) 1/6 2 2 ---------------------------------------------------------, 3 but I'll bet there are sqrt(binomials) = multinomials of any desired length, if you know how to look. Does anyone? --rwg neologisms <-> gloominess

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[math-fun] What does it mean?
by Jon Perry 30 Jun '04

30 Jun '04

If I consider a polynomial in integers, e.g. 4x^4 + 7x^2 - x + 1, then I can say that this is; (x+1)[1 - 2x + 9x^2 - 9x^3 + 13x^4 - 13x^5 + 13x^6 - ...] Does this mean anything? Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] Another sqrt(binomial) = pentanomial
by R. William Gosper 29 Jun '04

29 Jun '04

(c18) (sqrt(4-2^(2/3)*sqrt(3)*5^(1/6)),%%=denest_radicals(%%)) Time= 56792 msec. 2/3 1/6 (d18) sqrt(4 - 2 sqrt(3) 5 ) = 2/3 5/6 2/3 1/3 1/6 2 5 1/3 2 sqrt(3) 5 1/6 1/3 2 (- --------- + 2 sqrt(5) + ----------------- - 2 5 + 2 sqrt(3)) 2 2 ----------------------------------------------------------------------------- 3 or more prettily: 5/6 1/6 1/3 2/3 1/6 5 2 sqrt(5) 5 sqrt(4 - sqrt(3) 5 ) = - --------- + ------------ + --------------- 3 sqrt(2) 3 sqrt(2) sqrt(3) 5/6 1/6 1/6 2 5 2 - --------- + ------- 3 sqrt(3) Do I hear six? --rwg laminectomy <-> metonymical

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[math-fun] puzzles, some of which are geometric
by Simon Plouffe 28 Jun '04

28 Jun '04

hello, Here is a series of puzzles : http://wpc.puzzles.com/gt-IV/uspc-04.pdf sponsored by a company (the name begins with G). They ask for a password for the document : xcode6fire Some are interesting. Simon Plouffe

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[math-fun] Knuth Vol 4
by Thane Plambeck 27 Jun '04

27 Jun '04

This is probably old news to many people, but I just noticed that this page (scroll down a bit) http://www-cs-faculty.stanford.edu/~knuth/taocp.html has quite a bit in the way of "Knuth Vol 4" preview material on it. There's easily several hundred pages of stuff there embedded in downloadable .ps.gz files. Here's a puzzle from it [rated 25 on the Knuth scale] What is the 7.2.1.2.26: What is the unique alphametic solution to NIIHAU \pm KAUAI \pm OAHU \pm MOLOKAI \pm LANAI \pm MAUI \pm HAWAII = 0? (\pm : "plus or minus") * * * * * Knuth writes: The material will first appear in beta-test form as fascicles of approximately 128 pages each, issued approximately twice per year. These fascicles will represent my best attempt to write a comprehensive account, but computer science has grown to the point where I cannot hope to be an authority on all the material covered in these books. Therefore I'll need feedback from readers in order to prepare the official volumes later. The publishers have pledged to make the fascicles available in an inexpensive form, essentially at cost. (Of course, the paper and binding will probably be designed to self-destruct in a few years, so that you will have to buy the real book after it is debugged:-) If all goes as planned, Volumes 4A, 4B, and 4C will be ready in the year 2007. Some "pre-fascicles" are now available for alpha-testing: Pre-fascicle 2a (Generating all n-tuples); Pre-fascicle 2b (Generating all permutations); Pre-Fascicle 3a (Generating all combinations). Pre-Fascicle 3b (Generating all partitions). I've put them online primarily so that experts in the field can check the contents before I inflict them on a wider audience. But if you want to help debug them, please go right ahead. Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org

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[math-fun] sqrt(binomial) = pentanomial
by Bill Gosper 25 Jun '04

25 Jun '04

1/5 1/5 sqrt(4 - 3 ) = 3/5 4/5 3/5 2/5 2/5 4/5 1/5 1/5 - 2 3 + 3 + 2 2 3 - 2 3 + 2 --------------------------------------------------- 5 ~ 0.27161916566652. Also, I lied: 1/3 9 - 3 4 atan(----------) 1/2 4/3 4/3 1/3 4/3 5 3 (4 + 2 - 7) - 6 4 - 2 + 10 tan(----------------) = ------------------------------------------- 3 5 It isn't even trivial to *verify* this symbolically. Actually, you can just plug this into the triple angle formula and compare the two algebraic expressions. --rwg

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[math-fun] Primes in First N Digits of Pi
by Robert Baillie 23 Jun '04

23 Jun '04

These are prime: 3 31 314159 31415926535897932384626433832795028841 (first 38 digits of Pi). The first 16208 digits of Pi are a probable prime. Ed T. Prothro found this in 2001. I just finished checking up through 40000 digits. There are no other numbers in this sequence for which Mathematica's PrimeQ[ ] is TRUE. Has anyone checked beyond this limit? Bob Baillie

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Re: [math-fun] normal numbers and factorial base
by Daniel Asimov 23 Jun '04

23 Jun '04

Michael Kleber's comment << . . . (Pretending that pi is a string of random digits and other such falsities...) >> got me thinking. Suppose we have a probability distribution D on the set S of all countable sequences of decimal digits. And suppose that for all *finite* sets of indices {i_1 < ... < i_n}, the corresponding marginal distribution is uniform on {0,...,9}^n. 1. Does it then follow that D is uniform on S ??? 1a. Is there a way countably many variables can be correlated even if no finite subset of them are? Related question: I like considering numbers' "digits" in "factorial" base. For x in [0,1], I mean by this writing x = a_1/2! + a_2/3! + ... + a_n/(n+1)! + ... where each a_n is an integer such that 0 <= a_n <= n. Luckily, 1/2! + 2/3! + 3/4! ... = 1/2 + 1/3 + 1/8 + 1/30 + 1/144 +... = 1. 2. Is pi normal to the factorial base? Here I mean only the fractional part of pi. I.e., for any finite set of indices {i_1 < i_2 < ... < i_n}, does the joint distribution on a_(k + i_1)/(k + i_1+1)! , , ... , a_(k + i_n)/(k+i_n+1)! approach a uniform cdf on [0,1]^n as k -> oo ? N.B. It's easy to check that e would definitely NOT be normal to the factorial base. --Dan

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[math-fun] Double transit
by Eugene Salamin 23 Jun '04

23 Jun '04

If you look at http://iss-transit.sourceforge.net/IssVenusTransit.html you will see a photo of the International Space Station transiting Venus while Venus transits the Sun. __________________________________ Do you Yahoo!? Yahoo! Mail is new and improved - Check it out! http://promotions.yahoo.com/new_mail

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