February 2003 - mathfun@mailman.xmission.com

[math-fun] parity of arithmetic sequences
by Mike Stay 04 Feb '03

04 Feb '03

The low bit of x^n over GF(2^3) mod x^3+x+1 equals the parity of (n+8), 0<=n<7. Is it reasonable to expect that there will always be a pair (a,b) such that the low bit of the powers n of a generator of GF(2^k) match the parity of an+b for 0 <= n < 2^k-1? -- Mike Stay staym(a)clear.net.nz

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[math-fun] Hungarian prisoner switch puzzle
by Scott Huddleston 03 Feb '03

03 Feb '03

The Cartalk Puzzler this week looks interesting. Puzzle and URL: <http://cartalk.cars.com/Radio/Puzzler/Transcripts/200305/index.html> RAY: I got this from my pal, Stan Zdonik, who teaches at Brown University. It was given to him by a colleague who failed to provide him with the answer -- maybe because he didn't know it? Here it is. This puzzle has been making the rounds of Hungarian mathematicians' parties. The warden meets with 23 new prisoners when they arrive. He tells them, "You may meet today and plan a strategy. But after today, you will be in isolated cells and will have no communication with one another. "In the prison is a switch room, which contains two light switches labeled A and B, each of which can be in either the on or the off position. I am not telling you their present positions. The switches are not connected to anything. "After today, from time to time whenever I feel so inclined, I will select one prisoner at random and escort him to the switch room. This prisoner will select one of the two switches and reverse its position. He must move one, but only one of the switches. He can't move both but he can't move none either. Then he'll be led back to his cell. "No one else will enter the switch room until I lead the next prisoner there, and he'll be instructed to do the same thing. I'm going to choose prisoners at random. I may choose the same guy three times in a row, or I may jump around and come back. "But, given enough time, everyone will eventually visit the switch room as many times as everyone else. At any time anyone of you may declare to me, 'We have all visited the switch room.' "If it is true, then you will all be set free. If it is false, and somebody has not yet visited the switch room, you will be fed to the alligators." Here's the question: What is the strategy the prisoners devise?

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[math-fun] Greg's series
by R. William Gosper 03 Feb '03

03 Feb '03

I know a kid named Greg with four fingers on one hand, and "pi" on the other, i.e., three plus just the tip of a fourth. Coincidentally, if you start Gregory's series for 4 arctan(1) one term prematurely, block([fancy_display:false], intosum(4*sum((-1)^k/(2*k+1),k,-1,inf)),print(%%=closedform(%%))) inf ==== k \ 4 (- 1) > -------- = %pi + 4 . / 2 k + 1 ==== k = -1 Greg could become a regular J. von Neumann if he learns to sum infinite series by counting on his fingers. --rwg

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[math-fun] Inverter program for Maple. (access to the plouffe's inverter from maple).
by Simon Plouffe 03 Feb '03

03 Feb '03

hello, I made a program called inverter, it can be fetched at http://www.lacim.uqam.ca/~plouffe/inverter.txt what it does : from within a Maple (7 or 8) session you can access the plouffe's inverter and make some request to the web site, of course you need an internet connection to do it and maple. The program is self-contained and anyone can use it as long as you have Maple, it has been tested on a PC, a linux machine and a variety of other platforms. So far everything is according to design. there are 3 functions : pilook( ) , it does a lookup to the inverter and returns 1 answer (the shortest if any). pismart () will do 140 lookups with 1 number if K is the constant then a variety of variations on K are made and tested against the database (201 million entries). pidev( ) will try to find a rational polynomial for the expansion of K in 17 kind of expansions (like binary, fibonacci representation, continued fractions and egyptian fraction expansion). -------------------------------------------------------------------------------- pilll ( ) is NOT yet implemented but it will do a variety of integer relation searchs with K, I am trying to tune a fast program with only maple (wishful thinking). Also : the O.E.I.S. is imbeded into the database of constants and some answers will point to some sequences or a transformed sequence, so far I did not made a link to it but it can be made if Neil Sloane agrees to that (ok Neil ?). Finally, this first version does NOT use GFUN for trying to find a generating function when pidev is called but my intention is to include the source later. I had a prototype of a program for the OEIS that does the same and could be included, the function is eislook ( ). -------------------------------------------------------------------------- - it is free. - you can make many requests to the server as long as it is not abusive, the machines can take a good load of requests. - the program can be copied any number of time as long as you mention the source. --------------------------------------------------------------------------- Instructions to use : 1) download the file to your pc (rename it inverter). 2) put the file in the bin.wnt or bin.win folder of your PC, it is located under C:/program files/Maple .../ 3) Under Unix just put it where you put your maple programs usually, same on the macintosh. 4) in a maple session, type --> read inverter; 5) type tan(1.0); pilook(%); to test if your internet connection is working. 6) the program accepts real numbers OR any maple expression as long as it can be evaluated to a real floating point number, it is converted in the program BEFORE to be sent to the web site. -------------------------------------------------------------------------------- Simon Plouffe ps ; have fun!

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[math-fun] Efficient tournaments
by asimovd＠aol.com 01 Feb '03

01 Feb '03

Suppose a tennis (or any other 2-player game whose outcome is always a win for one player) tournament is held for N players, every pair of whom will play one match, that is, one game (i.e., a round-robin tournament). Further assume each player's ability is rated by one (unknown) number selected independently from the uniform distribution on (0,1), and that no two players' ability ratings are equal. Finally, assume that in a match between players with ability ratings a and b, their probabilities of winning are a/(a+b) and b/(a+b), respectively. Given the outcome of all the matches (which can be encoded in a skew-symmetric NxN matrix T with all off-diagonal entries being +-1): 1. What calculation based on T will yield a guess for which player has the greatest ability, with the maximum likelihood of being correct ? (I'm also interested in other questions, like the following, but let's think about #1 first.) 2. What calculation based on T will yield an estimate for the ranking of all N participants according to the order of ability rating (i.e., 1st, 2nd,..., Nth), minimizing the expected sum of squared errors ? 3. What calculation based on T will yield an estimate for the numerical ability rating of each participant, minimizing the expected sum of squared errors ? . --Dan

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Re: [math-fun] cell phone math (delayed reaction)
by asimovd＠aol.com 01 Feb '03

01 Feb '03

<< >From a Reuters news story today, "Cell Phones 'Blind' Drivers"... "It is like studies that show 90 percent of people think they are better-than-average drivers. Forty percent of them are wrong." >> I'd have to say that this is a low estimate. Clearly, somewhere between 44 4/9 % and 55 5/9 % of them are wrong. Oh, well. This is bad, but no worse than the ubiqitous price tags one sees these days for items going for less than a penny, like .99¢ . --Dan

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[math-fun] "Proofs from the Book" book
by asimovd＠aol.com 01 Feb '03

01 Feb '03

This is a book which Paul Erdõs participated in the planning of before his death in 1996. It's based on Erdõs's notion that there is a book, maintained by God, which contains the most elegant proof of every theorem. The math here is the type of math Erdõs liked to work on; a number of his own proofs and theorems are among those discussed. It came out with its second edition in Dec., 2000, but I've just got my hands on it now. It's really a great book that I'm sure all active participants in Math-Fun would enjoy a lot. Even without reading the proofs, the theorems -- in number theory, graph theory, geometry, analysis, and combinatorics -- are fascinating, and (unsurprisingly) often ones I'd never heard of before. It's extremely readable, using almost exclusively lower division undergrad math as background, and with abundant sidebars explaining anything that might be less than obvious, and some things that are. IMHO, quite a few of these proofs, however, do not strike me as elegant -- even if they may well be the *most* elegant proofs of their theorems. They sometimes involve fairly intricate 3-4 page arguments where it's hard to see the light at the end of the tunnel. It's $32.xx at amazon.com -- money well-spent. --Dan

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