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Re: [math-fun] domino question
by Michael Reid 09 Jun '06

09 Jun '06

> For any n, does there exist a polyomino with exactly n distinct domino tilings? yes, consider the following sequence of polyominoes: +--+ | | +--+ | | +--+ +--+--+ | | | +--+--+ | | | +--+--+ +--+--+ | | | +--+--+ | | | +--+--+ | | | +--+--+ +--+--+ | | | +--+--+--+ | | | | +--+--+--+ | | | | +--+--+--+ +--+--+ | | | +--+--+--+ | | | | +--+--+--+ | | | | +--+--+--+ | | | +--+--+ +--+--+ | | | +--+--+--+ | | | | +--+--+--+--+ | | | | | +--+--+--+--+ | | | | +--+--+--+ +--+--+ | | | +--+--+--+ | | | | +--+--+--+--+ | | | | | +--+--+--+--+ | | | | +--+--+--+ | | | +--+--+ +--+--+ | | | +--+--+--+ | | | | +--+--+--+--+ | | | | | +--+--+--+--+--+ | | | | | +--+--+--+--+ | | | | +--+--+--+ ... (this sequence is surely well-known.) mike

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RE: [math-fun] Latin square question
by dasimov＠earthlink.net 09 Jun '06

09 Jun '06

I'm afraid I may not have expressed myself clearly. What I meant was, suppose a matrix is partially filled. These entries will never change in this question. CONDITION (*): Suppose that any *individual* row that already contains at least one filled square satisfies the condition that its unfilled squares can be filled without repetition so that there is also no repetition in any column. *And* the same thing after switching the roles of rows & columns. This is clearly a prerequisite to extending the originally partially-filled matrix to a Latin square. Question: Is this sufficient to know the original partially-filled matrix can be extended to a Latin square? (Of course the 2x2 matrix given below fails to satisfy the condition (*).) (I very doubt the answer to my question is Yes, but it would require a trickier counterexample.) --Dan --------------------------------------------------------------------------------------------- Christian wrote: << I am afraid that the answer to your question will be difficult -or impossible- to get, excluding of course one value: e With N=1, the answer will be always "yes" ;-) With N=2, the answer can already be "no" even with your prerequisite, with such a matrix: 1 ? ? 2 With small N (say N < 8 or 9), we can have fast algorithms, because the Latin squares are not too numerous, but looks more difficult for bigger N. Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr(a)mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de dasimov(a)earthlink.net Envoyé : mercredi 7 juin 2006 02:10 À : math-fun Objet : [math-fun] Latin square question Suppose an NxN matrix is partially filled with the numbers S_N = {1,2,...,N}. When can the blank squares be filled with elements of S_N so the result is a Latin square (each row and column contains each members of S_N) ??? An obvious prerequisite is that each nonempty row or column can *by itself* be filled with its missing numbers (even if there are none) in a way that leaves each number occurring at most once in any column or row, resp. Assuming this property, is there an algorithm to determine whether the original partial square can be completed to a Latin squre? (It's apparently a hard theorem just to prove that if the NxN matrix contains just N-1 entries so that no row or column has a repetition, then the fill can always be extended to a Latin square. So this question may be intractable.) >>

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[math-fun] Latin square question
by dasimov＠earthlink.net 08 Jun '06

08 Jun '06

Suppose an NxN matrix is partially filled with the numbers S_N = {1,2,...,N}. When can the blank squares be filled with elements of S_N so the result is a Latin square (each row and column contains each members of S_N) ??? An obvious prerequisite is that each nonempty row or column can *by itself* be filled with its missing numbers (even if there are none) in a way that leaves each number occurring at most once in any column or row, resp. Assuming this property, is there an algorithm to determine whether the original partial square can be completed to a Latin squre? (It's apparently a hard theorem just to prove that if the NxN matrix contains just N-1 entries so that no row or column has a repetition, then the fill can always be extended to a Latin square. So this question may be intractable.) --Dan

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[math-fun] order codes
by bobmayans＠aol.com 07 Jun '06

07 Jun '06

Without getting into the Risc/Cisc issues, let me add that it was a sheer pleasure to proram the PDP-10. It rewarded programmer skill and made for succinct, expressive code. I always appreciated the architecure of this system. Bob Mayans ________________________________________________________________________ Check out AOL.com today. Breaking news, video search, pictures, email and IM. All on demand. Always Free.

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Re: [math-fun] Chinese Mathematicians Prove Poincare Conjecture
by dasimov＠earthlink.net 06 Jun '06

06 Jun '06

Joshua wrote: << On 6/6/06, Ray Tayek <rtayek(a)comcast.net> wrote: > http://science.slashdot.org/article.pl?sid=06/06/05/0440258 As some of the slashdot comments say, the conjecture was (probably) proved a couple years ago. The Chinese mathematicians appear to have done nothing more than explain the (already-published on ArXiV) proof a little more clearly. . . . . . . >> One reason to suspend judgment is that the Chinese Xinhua News Agency quoted the Harvard differential geometer and Fields medalist Shing-Tung Yau as having said that these Chinese mathematicians have put the finishing touch on the Poincare conjecture. As far as I know -- and I've made frequent inquiries -- this is the first time that an eminent U.S. mathematician has asserted that the Poincare conjecture has been proven. (On the other hand, Yau grew up in China and may not be entirely unbiased.) --Dan

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[math-fun] Chinese Mathematicians Prove Poincare Conjecture
by Ray Tayek 06 Jun '06

06 Jun '06

http://science.slashdot.org/article.pl?sid=06/06/05/0440258 --- vice-chair http://ocjug.org/

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Re: [math-fun] Mathematical art
by N. J. A. Sloane 06 Jun '06

06 Jun '06

> > On 6/5/06, Jason Holt <jason(a)lunkwill.org> wrote: > >> The number paintings inspired me to do some artwork of my own: > >> http://lunkwill.org/tmp/mystery.png > >> Any guesses what the plot represents? > >> > > Collatz problem? > > Yes indeed, showing the number of steps to reach 1, log-scaled in both x and y, > for n up to a billion. I assume that is sequence A006577 in the OEIS, and indeed if you click on "graph" there you see a somewhat similar picture (look at the second plot). Indeed, many classic sequences have dramatic graphs - see e.g. A005132, or Pascal's triangle A007318. Note that you can usually get to an entry like A006577 simply by typing A006577 in the URL window (!) of your broswer. It is also amusing to listen to some of these sequences - see the "Listen" button at the foot of the OEIS pages. Neil Sloane

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Re: [math-fun] Mathematical art
by dasimov＠earthlink.net 05 Jun '06

05 Jun '06

Jason wrote: << On Mon, 5 Jun 2006, Mike Stay wrote: > On 6/5/06, Jason Holt <jason(a)lunkwill.org> wrote: >> The number paintings inspired me to do some artwork of my own: >> http://lunkwill.org/tmp/mystery.png >> Any guesses what the plot represents? >> > Collatz problem? Yes indeed, showing the number of steps to reach 1, log-scaled in both x and y, for n up to a billion. . . . >> Mike, you are brilliant for guessing that so fast! Jason, that's a really nice mathematical graphic! --Dan

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[math-fun] Mathematical art
by Jason Holt 05 Jun '06

05 Jun '06

The number paintings inspired me to do some artwork of my own: http://lunkwill.org/tmp/mystery.png Any guesses what the plot represents? -J

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[math-fun] Alan Kotok
by Eugene Salamin 05 Jun '06

05 Jun '06

Alan Kotok died May 26, 2006. He was 64 years old. Alan was one of the original hackers and a contributor to Spacewar, the world's first video game, which ran on a PDP-1 computer at MIT. His obituary is in the June 3, 2006 New York Times. __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com

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