February 2003 - mathfun@mailman.xmission.com

Re: [math-fun] Ulam(1,2)
by Dan Hoey 10 Feb '03

10 Feb '03

By the way, I get a period of about 10.1968 with Ulam(1,3). But the density waveform is bimodal: 0 40 96 105 112 95 48 0 0 0 0 0 0 29 89 104 111 102 65 4 Dan

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[math-fun] Conjecture
by David Wilson 09 Feb '03

09 Feb '03

Conjecture: For prime p, 3^p + 3^p is the smallest number of the form a^p +- b^p which is not of the form a^k += b^k for 2 <= k < p.

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[math-fun] Re: Hungarian prisoner switch puzzle
by The Purple Streak, Hilarie Orman 09 Feb '03

09 Feb '03

There are at least four solutions to the problem of the 23 prisoners and weird warden. The first three are efficient, the fourth is slow. 1. The prisoners realize that the room with switches is probably unused and dirty. They agree that the first time each of them is brought into the room, he will seem to spend a long time hesistating over which switch to change. While doing so, he will choose an unused spot on the floor and inconspicuously piss there. 2. The prisoners note that the warden has eight fingers, each with three sections (thumbs are not used in this solution). As each prisoner is taken from his cell to go to the switch room for the first time, he will quickly bite off one section of a finger. 3. The prisoners agree on to tell the warden the following: "We realize that you are a clever person who loves puzzles. We have one for you. We have selected a number that is the constant term of a degree 22 polynomial. Each day you can ask a prisoner the value of the polynomial at any argument except zero. The prisoner is allowed to either decline to answer, or he must evaluate the polynomial and tell you the result. If a prisoner asks you if you know the answer, you must answer truthfully. If you say you know the answer, you must tell him, and if you are wrong, you must release us and throw yourself into the alligator pit." 4. The prisoners ask the warden for 23 Palmpilots. One of the prisoners writes a program and beams it to the other prisoners. The program keeps track of the day and number of times the prisoner has been in the room; every 23 days is a "prison week", starting with week 0. As each prisoner enter the room, he enters the left switch position into the Palmpilot. If a prisoner goes into the room in prison week n and his total "been here" count is greater than n, he will be told to set the left switch up unless it is already up. He will do the same thing if his "been here" count plus the number of days left in the week is less than n. If there's nothing wrong with his count and it is the first day of the prison week, he will set the left switch down unless it is already down. If a prisoner has not changed the left switch then he will change the right switch. Before leaving the room, the prisoner will consult the Palmpilot. If it is the end of prison week, and if the left switch is down, the Palmpilot program will announce that it's time to tell the warden to release all the prisoners. They will throw the warden to the alligators, unless he has lost all his fingers.

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[math-fun] Fill in the blank
by Thane Plambeck 08 Feb '03

08 Feb '03

I had to look this up unary, binary, ____?_____, octal, hexadecimal Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm

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[math-fun] Ulam(1,2)
by David Wilson 07 Feb '03

07 Feb '03

I took a look at Ulam(1,2), the Ulam sequence starting with (1, 2) and including every subsequent number which is a unique sum of distinct earlier terms. This is Sloane's A002858. I took this sequence out quite a ways, and I noticed that after an initial flurry of numbers that are fairly uniform in distribution, the sequence starts to separate out into more or less regular clumps of numbers with a period that seems to be slighly more than 21.6. Between these clumps are spaces containing relatively few numbers. To see the phenomenon, let u(n) be the nth Ulam(1,2) number, and define f(n) = u(n) / 21.6 - [ u(n) / 2.16 ] f(n) is a number on [0, 1) which indicates "u(n) mod 21.6". If we compute f(n) for 1000 <= n <= 1999, (12336 <= u(n) <= 25511), we find the following distribution of f(n): range of f(n) count [0.00, 0.05) 110 [0.05, 0.10) 101 [0.10, 0.15) 120 [0.15, 0.20) 97 [0.20, 0.25) 90 [0.25, 0.30) 67 [0.30, 0.35) 58 [0.35, 0.40) 40 [0.40, 0.45) 31 [0.45, 0.50) 9 [0.50, 0.55) 4 [0.55, 0.60) 0 [0.60, 0.65) 0 [0.65, 0.70) 2 [0.70, 0.75) 7 [0.75, 0.80) 11 [0.80, 0.85) 22 [0.85, 0.90) 64 [0.90, 0.95) 66 [0.95, 1.00) 101 The values of f(n) are not uniformly distributed, the distribution indicate that u(n) values clump together with clumps occuring with a period of about 21.6. This all presupposes that my computations of Ulma(1,2) are correct. Can anyone confirm my data? Has this phenomenon been noted in the literature?

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[math-fun] Indian series for pi
by Richard Schroeppel 07 Feb '03

07 Feb '03

I came across a Math History web site. The authors take seriously the Indian claims of inventing the Gregory series for arctangent, and the power series for sin and cos, circa 1400. http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Madhava.html Madhava's math works are lost, but are described in Nilakantha's writings, including proofs of the series. I suppose the whole thing could be a hoax, but it's a pretty long running one, involving several people. Assuming it's all true, we must ask "What happened to Indian Calculus?" How did it disappear? Why has it only come to Western attention recently? Rich rcs(a)cs.arizona.edu

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[math-fun] A New Kind of Science * 4:15PM, Wed February 12, 2003 in Gates B03
by Michael B Greenwald 07 Feb '03

07 Feb '03

FYI, for those of you out in the Bay Area who may not have seen this: ------- Forwarded Message Date: Thu, 06 Feb 2003 15:11:19 -0800 From: allison(a)Stanford.edu To: colloq(a)CS.Stanford.EDU Subject: [CSL Colloq] A New Kind of Science * 4:15PM, Wed February 12, 2003 in Gates B03 COMPUTER SYSTEMS LABORATORY COLLOQUIUM 4:15PM, Wednesday, February 12, 2003 NEC Auditorium, Gates Computer Science Building B01 and B03 http://ee380.stanford.edu[1] Topic: A New Kind of Science Speaker: Dr. Stephen Wolfram Creator of Mathematica, author of A New Kind of Science, and CEO of Wolfram Research, Inc. About the talk: Starting from a few computer experiments, Stephen Wolfram has spent more than twenty years developing a new approach to science, described for the first time in his book A New Kind of Science. Basic to his approach is the idea of studying not traditional mathematical equations but instead rules of the kind embodied in the simplest computer programs. A key discovery is that such rules can lead to behavior that shows immense complexity and mirrors many features seen in nature. Wolfram has built on this to tackle a remarkable array of fundamental problems in science, from the origins of apparent randomness in physical systems, to the development of complexity in biology, the ultimate scope and limitations of mathematics, the possibility of a truly fundamental theory of physics, the interplay between free will and determinism, and the character of intelligence in the universe. When Wolfram's book was released on May 14, 2002, it became an instant bestseller, and is now showing many signs of initiating a major paradigm shift in science. Wolfram's presentation will cover some of the key ideas and discoveries in his book, outlining their implications, and discussing their personal and historical context. An extended question and answer period will be included. About the speaker: Stephen Wolfram was educated at Eton, Oxford, and Caltech, receiving his Ph.D. in theoretical physics in 1979 at the age of 20. His early work in physics and computer science was recognized by a MacArthur award in 1981. In the early 1980s he made a series of now-classic discoveries about systems known as cellular automata, leading to numerous applications in physics, mathematics, computer science, biology, and other fields. In 1986 he founded Wolfram Research, Inc. and began the creation of Mathematica---now the world's leading software system for technical computing and symbolic programming (and tool which made A New Kind of Science possible). Over the past decade Wolfram has divided his time between leadership of his company and pursuit of basic science. The results of Wolfram's fifteen years of work were presented for the first time in his book, A New Kind of Science (May 2002). An instant bestseller, A New Kind of Science constituted international science news and quickly emerged as one of the most-discussed science books in decades. +-----------------------------------------------------------------+ | *** ROOM CHANGE *** | |Because we anticipate a large audience, we will use Gates B1 for | |the live presentation and Gates B3 as a video overflow room. | |Students will have precedence for seating in the Gates B1 room. | | | | *** OTHER WOLFRAM TALKS *** | |Stephen Wolfram will also be speaking in several other venues.[2]| |He will speak on the Stanford campus at 7:30PM on Monday, | |February 10, in Dinkelspiel Auditorium. For details see | |http://scil.stanford.edu[3]. This is a free public event | |presented under the auspices of the Stanford Center for | |Innovations in Learning. Following the CSL Colloquium (Wednesday,| |February 12) he will head to San Francisco to speak in the Herbst| |Theater starting at 8PM. See Wolfram Talk[4] for more | |information. This is a public event sponsored by California | |Academy of Arts and Science; tickets are $18.00. | +-----------------------------------------------------------------+ Contact information: Dr. Stephen Wolfram 100 Trade Center Drive Champaign, IL 61820-7237 sw(a)wolfram.com[5] Embedded Links: [ 1 ] http://ee380.stanford.edu [ 2 ] http://www.wolframscience.com/appearances/ [ 3 ] http://scil.stanford.edu [ 4 ] http://www.cityboxoffice.com/eventperformances.asp?evt=552 [ 5 ] mailto:sw@wolfram.com +----------------------------------------------------------------------------+ | This message was sent via the Stanford Computer Science Department | | colloquium mailing list. To be added to this list send an arbitrary | | message to colloq-subscribe(a)cs.stanford.edu. To be removed from this list,| | send a message to colloq-unsubscribe(a)cs.stanford.edu. For more information,| | send an arbitrary message to colloq-request(a)cs.stanford.edu. For directions| | to Stanford, check out http://www-forum.stanford.edu | +-------------------------------------------------------------------------xcl+ ------- End of Forwarded Message

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[math-fun] Fibonacci power triangle
by Richard Schroeppel 06 Feb '03

06 Feb '03

(Everything that follows is unproven conjecture. It's probably old news to Fibonacci aficionados.) It occurred to me that squares of Fibonacci numbers should have a simple recurrence relation. After a bit of futzing, I found 2 2 2 2 F' = 2 F + 2 'F - ''F (I use ' to mean "next term" and prefix ' to mean "previous term".) So 169 = 2*64 + 2*25 - 9. Another way to say this is that the sequence of Fibonacci squares can be dotted with the vector (-1 2 2 -1) to produce 0. The vector for the plain Fibonacci sequence is (1 1 -1). Experimenting showed the vector for Fibonacci cubes to be (-1 -3 6 3 -1). I worked out fourth and fifth powers, and developed the Fibonacci-Pascal triangle, built from the vectors. 1 -1 1 1 -1 -1 2 2 -1 -1 -3 6 3 -1 1 -5 -15 15 5 -1 1 8 -40 -60 40 8 -1 -1 13 104 -260 -260 104 13 -1 The rule for the signs seems clear enough. The magnitudes turn out to be Fibonacci-Binomials. The outer diagonals are 1 1 1 1 ...; the next diagonals are just the plain Fibs; and the next diagonals are products of adjacent Fibs: 104 = 8*13, etc. The next diagonals turn out to be products of three adjacent Fibs divided by 2: 260 = 5*8*13/2. The right way to look at this seems to be 5*8*13/(1*1*2), a kind of binomial coefficent with the regular integers replaced by Fibonacci numbers. The next diagonals will be products of four adjacent Fibs divided by 1*1*2*3, and so on. This can probably be connected with q-binomial coefficients. Some other curiosities: (-1 2 2 -1) also works for F*F' (-1 -3 6 3 -1) works for F*F'*F'' (1 -5 -15 15 5 -1) works for F*F'*F''*F''' Fib(2N+1) = (1 1) dotted with F^2; Fib(2N) = (-1 0 1) dot F^2. Fib(3N) = (-1 1 1) dotted with F^3. Since (-1 2 2 -1) works with both F^2 and F*F', it will also work for squares of the Lucas sequence. Continuing this idea a bit further, the vectors work with powers of generalized Fibonacci sequences, where the starting pair of numbers is A,B, and the terms continue A+B, A+2B, 2A+3B, ... Rich rcs(a)cs.arizona.edu

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Re: [math-fun] Hungarian prisoner switch puzzle
by Dan Hoey 04 Feb '03

04 Feb '03

Scott Huddleston writes: > The Cartalk Puzzler this week looks interesting. Puzzle and URL: > <http://cartalk.cars.com/Radio/Puzzler/Transcripts/200305/index.html> The puzzle previously appeared in the IBM Research "Ponder This" puzzle section. See <http://www.research.ibm.com/ponder/> under July 2002. Dan Hoey(a)AIC.NRL.Navy.Mil

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[math-fun] digits to powers
by Erich Friedman 04 Feb '03

04 Feb '03

it is well known that there exist numbers that are the sum of their digits to some fixed power: 153 = 1^3 + 5^3 + 3^3 4151 = 4^5 + 1^5 + 5^5 + 1^5 consider instead the problem of expressing a number as the sum of its digits to different powers, similar to david wilson's recent post on energetic numbers. we would like to make the minimum m of those powers is as large as possible. for example, here is an example where the smallest power is m=7: 3212 = 3^7 + 2^9 + 1^9 + 2^9 it seems like there should be no upper bound on possible values for m, but i have not been able to find any larger than m=7. can any of you? erich

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