41. Kaprekar Number The 3n+1 Problem (collatz problem) top Take any natural number, from whichyou derive a sequence of numbers according to the following rules. http://www.mathematische-basteleien.de/kaprekar.htm

The Kaprekar Number and more playings with numbers Contents of this Page What is the Kaprekar Number? The 3n+1 Problem The Steinhaus Cyclus The 196-Problem The Number 1089 ... To the Main Page "Mathematische Basteleien" What is the Kaprekar Number? The number 6174 is called the Kaprekar number. The Indian mathematician D.R.Kaprekar made the following discovery in 1949. (1) Take a four-digit number with different digits (acbd with (2) Form the largest and the smallest number from these four digits (dcba and abcd). (3) Find the difference of these digits. Maybe this is 6174 (dcba - abcd = 6174?). If it is not, form the largest and the smallest number from the difference and subtract these numbers again. You may have to repeat this procedure. The end result is always 6174, but there are no more than 7 steps. 1st example: Take the number 1746. 1st step: 7641 - 1467 = 2nd example: Take the number 5644. 1st step: 6544 - 4456 = 2088 2nd step: 8820 - 0288 = 8532 3rd step: 8532 - 2358 = 3rd example: Take the number 7652. 1st step: 7652 - 2567 = 5085 2nd step: 8550 - 0558 = 7992 3rd step: 9972 - 2799 = 7173 4th step: 7731 - 1377 = 6354 5th step: 6543 - 3456 = 3087 6th step: 8730 - 0378 = 8352 7th step: 8532 - 2358 = The problem is solved. (Spektrum der Wissenschaft, Erstausgabe 1978)

42. Mathematical Problems - Problem Solving - Mathematical Competitions (Math Links The 3x+1 problem, also known as the collatz problem, the Syracuse problem, Kakutani sproblem, Hasse s algorithm, and Ulam s problem by Jeff Lagarias; http://mathres.kevius.com/problem.html

Mathematical Problems - Problem Solving Mathematical Competitions not a complete list, only what I happened to see... The 3x+1 problem and its generalizations The 3x+1 problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem by Jeff Lagarias 21st Century Problem Solving Solutions to solving word problems across the curriculum. (SureMath - hawaii.edu) MATHCOUNTS Foundation Algebra Story and Word Problems A collection of simple algebra word problems (SureMath: 21st Century Problem Solving) Algebrator - an Algebra Problem Solver for Students and Teachers ALVIRNE HIGH SCHOOL PROBLEM OF THE WEEK SITE Andrei Jorza's Home Page Calculus Problems Alvirne High School in Hudson, N.H. The Cereal Box Problem by George Reese A Collection of Math-Olympiad-Problems by Hans Vernaeve Competitions Canadian Mathematical Society (CMS) Countries with Mathematical Olympiad Websites - Links to Mathematical Olympiad Websites. CRUX Mathematicorium Online The Digital Supplement to the Canadian Mathematical Society's Problem Solving Journal Ken Duisenberg's Puzzle of the Week Elementary Problem of the Week by Ruth Carver (MathForum) Fermat's Last Theorem Flanders Mathematics Olympiad The Goal Angle Problem Hilbert's Tenth Problem by Karlis Podnieks Paul Erdos: An I nfinity of Problems - Ivars Peterson (MathLand) International Mathematical Olympiad / Olympiade internationale ... International Mathematics Olympiad (IMO) Eindhoven University of Technology Mathematic Olympiad Problems from around the world by Mohammad Katoozian

43. Heiner Marxen - Busy Beaver In einer Seite Prinzipielle Grenzen der Berechenbarkeit schreibt Arno Schwarzüber Turing Maschinen, Busy Beaver und das collatz problem (3n+1). Es gibt http://www.drb.insel.de/~heiner/BB/

Busy Beaver Currently Known Results The function Sigma (n) denotes the maximal number of tape marks which a Turing Machine (TM) with n internal states and a two-way infinite tape can produce onto an initially empty tape and then halt. The function S (n) denotes the maximal number of steps (shifts) which such a TM can do (it needs not produce many tape marks). The following table gives some known values: n Sigma (n) S (n) Source Lin and Rado Lin and Rado Lin and Rado Brady Marxen and Buntrock Marxen and Buntrock Note: The values for n=6 have been verified independently by Paul R. Stevens and by Clive Tooth. The exact numbers are found in the new list of 6-state record machines A general method due to Milton W. Green produces (computable, but not primitive recursive) lower bounds of Sigma for every n. He gives Sigma Of course, if you find an error in the above table, or can extend it... please let me know Local Resources (English) The author's paper Attacking the Busy Beaver 5 , Bulletin of the EATCS, Number 40, February 1990, pp. 247-251, [ISSN 0252-9742]

44. Atlas: The Relation Between Collatz Conjecture And Primes By Murad A. AlDamen the wellknown conjecture ,Collatz conjecture One of many conjectures still withoutproof a problem posed by L. Collatz in 1937 The collatz problem asks if http://atlas-conferences.com/c/a/f/t/16.htm

Atlas Document # caft-16 Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics December 21-24, 2000 University of Craiova Craiova, Romania Conference Organizers Minh Perez (American Research Press, Rehoboth, Box 141, NM 87301, USA) and Vasile Seleacu (University of Craiova, Department of Mathematics, Craiova, Romania) View Abstracts Conference Homepage The relation between collatz conjecture and primes presented by Murad A. AlDamen University of Jerash Date received: October 4, 2000 The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc.

45. Internet Center For Mathematics Problems collatz problem. Famous Problems in Mathematics The Four Colour Theorem;The Trisection of an Angle. Which are the 23 Hilbert Problems? http://www.mathpropress.com/mathCenter.html

Internet Center for Mathematics Problems On this page, we try to identify and list all sources of mathematics problems on the Internet and related information. Please advise us if you know of any sources that we are missing. Table of Contents Problem Columns from Mathematics Journals The Fibonacci Quarterly Mathematics and Informatics Quarterly (coming soon) Missouri Journal of Mathematical Sciences Problem Columns on the World Wide Web Mathematics Problems on the World Wide Web Newsgroups ... Sign-in We invite you to visit the MathPro Press home page Problem Columns from Mathematics Journals Many mathematics journals contain problem columns. They invite readers to send in solutions to the problems. The best solution submitted gets printed in a future issue along with the names of all the solvers. We list below only a few of the journals that are currently running problem columns. You are welcome to try your hand at solving these problems. Solutions should be sent to the appropriate problem column editor ( not to MathPro Press).

46. Introduction To Artificial Intelligence Selmer Bringsjord Hyperproof file.); A picture of the collatz problem; More on the CollatzProblem and Related Problems; Is English Compositional? (picture http://www.rpi.edu/~brings/intai.html

The Intro to AI Show Your Host, Selmer Bringsjord and Selmer as.... Spock with your two Co-Hosts The Terminator 7+MB quicktime movie Selmer himself Meet the Cast Video Highlights Handouts ... Supporting Links Handouts Syllabus ( html dvi postscript pdf Practice Exams For Practice Final Summer 99 postscript pdf Practice Midterm Summer 99 ( html postscript dvi pdf Practice Final Su 97 ( html postscript dvi pdf Practice Midterm Su 97 ( html postscript pdf Midterm Su 97 ( html dvi postscript pdf Problems/Puzzles for Smart Agents Godelian Puzzle 1 ( html dvi postscript Godelian Puzzle 2 ( html dvi postscript The Mystery of the Missing Peraflop Machine ( html postscript The Dreadsbury Mansion Mystery ( documentation graphic problem (hqx) The Wise Man Puzzle ... possible formalizations The Lottery Paradox ( html postscript The Schubert Steamroller Problem ( html dvi ps Projects Project 3 Finish Shapiro's ELIZA, i.e., (p1). Due by 5pm on day of final exam for studio audience. Due on same day as final for home audience (emailed to Selim as ASCII attachment).

47. From Gerry@mpce.mq.edu.au (Gerry Myerson) Subject Re Is Collatz seem to recall that Conway may have established this fact.) Conway defined a familyof problems, a natural generalization of the collatz problem, and showed http://www.math.niu.edu/~rusin/known-math/99/collatz_undec

From: gerry@mpce.mq.edu.au (Gerry Myerson) Subject: Re: Is Collatz Conjecture (3N+1 Problem) Undecidable? Date: Wed, 30 Jun 1999 13:05:01 +1100 Newsgroups: sci.math Keywords: Conway's generalizations include some undecideable problems In article

48. From Robert Harrison Mongrels@streamlinerec. Nospamprestel.co. odds. The 2^23^1 = 1 values for s and t is related to the know attractor of the collatz problem the 1- 4- 2- 1 cycle. Robert http://www.math.niu.edu/~rusin/known-math/98/collatz_catalan

From: "Robert Harrison" Newsgroups: alt.fan.hofstadter,alt.tanaka-tomoyuki,sci.math Subject: Re: Fermat, Goldbach, open problems Date: Wed, 23 Dec 1998 19:59:27 -0000 steiner wrote in message ... >In article

49. Puzzle 73.- A Collatz-like Sequence. (A Puzzle Suggested By Jack Brennen ) This is obviously derived from the collatz problem, and in fact, all even or primenumbers = 3 have the same successor as they do in the Collatz sequence http://www.primepuzzles.net/puzzles/puzz_073.htm

Puzzles Puzzle 73.- A Collatz-like sequence. (A puzzle suggested by Jack Brennen Consider the sequence of integers B[] defined by a starting point B[0] and the rules: If B[n] is prime, B[n+1] is defined as 3*B[n]+1. If B[n] is composite, B[n+1] is defined as the largest proper divisor of B[n], which can also be described as B[n]/P, where P is the smallest prime factor of B[n]. The question is this : Does every starting point eventually end up entering the cycle The same question can be posed with the rule changed so that prime numbers go to 3*B[n]-1 instead of 3*B[n]+1. Do all starting points then eventually enter this cycle? Solution Felice Russo got the following interesting results: " Question 1 . I have checked (with an Ubasic program) that all the integers between and terminate in the cycle: 2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2. Moreover for the first 1000 integers I calculated the number of steps needed to end up in the cycle. I indicate this quantity with T(n) and we can call it the "transient time". Then I calculated the ratio T(n)/n ("Normalized Transient time") which behavior is reported in the below graph.

50. The Collatz Problem, Also Known As The 3x+1 Problem The collatz problem, Also Known as The 3x+1 Problem Ilan Vardi The Collatz mapis taken to be x x/2 if x is even and x - (3x+1)/2 if x is odd. http://www.cs.ucla.edu/~klinger/col.html

The Collatz problem, Also Known as The 3x+1 Problem Ilan Vardi The Collatz map is taken to be x -> x/2 if x is even and x -> (3x+1)/2 if x is odd. ... I. Vardi, Computational Recreactions in Mathematica, Addison-Wesley 1991, Chapter 7 ... the 4 known cycles.... Discussion: This package computes the iterates of the Collatz map x -> x/2 if x is even, x -> (3x+1)/2 if x is odd, until an iterate reaches one of the 4 known cycles (the program runs on positive and negative integers): An efficient algorithm is used to compute how many iterations there are up to a cycle (the total stopping time). This algorithm is discussed in detail in Computational Recreations in Mathematica, Chapter 7. BeginPackage["Examples`Collatz`"] <= Abs[n]

51. F. Conjectures (Math 413, Number Theory) Conj All integers greater than 1 have occur as multiplicities. (Sierpinski). RefsUnsolved Problems in Number Theory, Guy, 1994, Sect B.37. The collatz problem. http://www.math.umbc.edu/~campbell/Math413Fall98/Conjectures.html

F. Conjectures Number Theory, Math 413, Fall 1998 A collection of easily stated number theory conjectures which are still open. Each conjecture is stated along with a collection of accessible references. The Riemann Hypothesis Fermat Numbers Goldbach's Conjecture Catalan's Conjecture ... The Collatz Problem The Riemann Hypothesis Def: Riemann's Zeta function, Z(s), is defined as the analytic extension of sum n infty n s Thm: Z( s )=prod i infty p i s , where p i is the i th prime. Conj: The only zeros of Z( s ) are at s s Thm: The Riemann Conjecture is equivalent to the conjecture that for some constant c x )-li( x c sqrt( x )ln( x where pi( x ) is the prime counting function. Refs: Unsolved Problems Elementary Number Theory http://www.utm.edu/research/primes/notes/rh.html (notes in Chris Caldwell's Prime Pages) http://match.stanford.edu/rh/ (Dan Bump's notes) Def: n is perfect if it is equal to the sum of its divisors (except itself). Examples are 6=1+2+3, 28, 496, 8128, ... Def: The n th Mersenne Number, M n , is defined by M n n Thm: M n is prime implies that n n is perfect. (Euclid)

52. E. Akin Why Is The 3x+1 Problem So Hard? Math. Magazine (ca. Eichsta t Conference 1999, 5p. 13469 Stefan Andrei/Manfred Kudlek/Radu Stefan NiculescuSome results on the collatz problem. Eichsta t Conference 1999, 15p. http://felix.unife.it/Root/d-Mathematics/d-Number-theory/b-3x 1

53. !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!! Translate this page Antwort *grins*. Schauma mal, wer das zusammengebracht hat Jens löst das Rätselam 11.3.2003 um 1202 Uhr Es handelt sich dabei um das collatz problem. http://www.astro.univie.ac.at/~strv/Puzzle/old_56.html

oder: Des Pudels Kern lautete: The 3n+1 problem Eingeschickt von niemandem Antwort: *grins* Schauma mal, wer das zusammengebracht hat ... Jens < 3 * 2^53 = 2.7...*10^16 (Oliveira e Silva 1999) und Thwaites (1996) has offered a £1000 reward for resolving the conjecture. Das heisst, du willst die 1000 Pfund haben und derjenige, der die Arbeit macht bekommt nur ein Bier - so nicht ;-) Flo F Legende zur Bewertung Lili - guggenberger@NO.SPAMastro.univie.ac.at Freitag, 26-Sep-2003 22:16:59 UTC

54. [The Collatz Problem Or 3n+1 Problem Is As Follows. Take A Natural The collatz problem or 3n+1 problem is as follows. Take a natural numbern. If it s even, halve it; if odd, triple it and add one. http://www.hevanet.com/cristofd/brainfuck/collatz.b

http://www.hevanet.com/cristofd/brainfuck/)]>,[ [ [ >>>[>>>>]+[[-]+ <, ] [The Collatz problem or 3n+1 problem is as follows. Take a natural number n. If it's even, halve it; if odd, triple it and add one. Repeat the process with the resulting number, and continue indefinitely. If n is 0, the resulting sequence is 0, 0, 0, 0... It is conjectured but not proven that for any positive integer n, the resulting sequence will end in 1, 4, 2, 1... See also http://www.research.att.com/projects/OEIS?Anum=A006577 This program takes a series of decimal numbers, followed by linefeeds (10). The entire series is terminated by an EOF (0 or "no change"). For each number input, the program outputs, in decimal, the number of steps from that number to zero or one, when following the rule above. It's quite fast; on a Sun machine, it took three seconds for a random 640-digit number. One more note. This program was originally written for Tristan Parker's Brainfuck Texas Holdem contest, and won by default (it was the only entry); the version I submitted before the contest deadline is at http://www.hevanet.com/cristofd/brainfuck/oldcollatz.b Daniel B. Cristofani

55. Mathsoft: Mathsoft Unsolved Problems: Unsolved Problems On Other Sites also Keith Matthew s The Generalized 3x+1 Mapping (University of Queensland) and1999 Conference on the collatz problem Proceedings (Eichstätt, Germany); http://www.mathsoft.com/mathresources/problems/article/0,,1999,00.html

search site map about us  + news  + ... Unsolved Problems Unsolved Problems Links On a Generalized Fermat-Wiles Equation Zero Divisor Structure in Real Algebras Sleeping Habits of Armadillos Mathsoft Constants ... Math Resources Unsolved Problems on Other Sites Jeff Lagarias' 3x+1 problem and related problems The Generalized 3x+1 Mapping (University of Queensland) and 1999 Conference on the Collatz Problem Proceedings Alex Lopez-Ortiz's sci.math FAQ on Famous Problems in Mathematics (University of New Brunswick) Chris Caldwell's Riemann Hypothesis (University of Tennessee at Martin); also Daniel Bump's Riemann Hypothesis (Stanford University) and Barry Cipra's A Prime Case of Chaos MathPro Press Unsolved Math Problem of the Week column and General References Twin Primes Conjecture and Goldbach's Conjecture, discussed in Prime Numbers: Some Unsolved Problems , part of MacTutor History of Mathematics (University of St Andrews); also Chris Caldwell's Prime Conjectures and Open Questions (University of Tennessee at Martin) Jan Otto Munch Pedersen's Known Amicable Pairs (Vejle Business College, Denmark) and Chris Caldwell's

56. Index Of /muneer_ilyas/school/Maths Problems/Collatz Problem (£1000)_files Index of /muneer_ilyas/school/Maths problems Parent Directory 09Apr-2003 0815 - collatz problem (£10.. 21-Mar-2003 0414 36kcollatz problem (£10.. 21-Mar-2003 0346 - Help files/ 21-Mar-2003 0406 http://homepage.ntlworld.com/muneer_ilyas/school/Maths problems/Collatz Problem

Index of /muneer_ilyas/school/Maths problems/Collatz Problem (£1000)_files Name Last modified Size Description ... Parent Directory 21-Mar-2003 05:04 - book.gif 21-Mar-2003 02:50 1k c2_253.gif 21-Mar-2003 02:55 1k c2_26.gif 21-Mar-2003 02:53 1k c2_39.gif 21-Mar-2003 02:54 1k c2_531.gif 21-Mar-2003 02:55 1k c2_532.gif 21-Mar-2003 02:56 1k c2_533.gif 21-Mar-2003 02:56 1k c2_534.gif 21-Mar-2003 02:57 1k c2_535.gif 21-Mar-2003 02:58 1k c2_536.gif 21-Mar-2003 02:58 1k c2_537.gif 21-Mar-2003 02:59 1k c2_538.gif 21-Mar-2003 03:00 1k c2_539.gif 21-Mar-2003 03:01 1k c2_540.gif 21-Mar-2003 03:03 1k c2_541.gif 21-Mar-2003 03:04 1k c2_542.gif 21-Mar-2003 03:06 1k c2_543.gif 21-Mar-2003 03:07 1k c2_544.gif 21-Mar-2003 03:08 1k c2_545.gif

57. EDUCATION PLANET - 1228 Web Sites For Problem Solving 12. The 3x+1 problem and its generalizations The 3x+1 problem, also known as thecollatz problem, the Syracuse problem, Kakutani s problem, Hasse s algorithm http://www.educationplanet.com/search/Math/Problem_Solving?startval=10

58. Math Trek: Dangerous Problems, Science News Online, June 29, 2002 has baffled mathematicians for more than 70 years. Also known as thecollatz problem, it concerns a sequence of positive integers. http://www.sciencenews.org/articles/20020629/mathtrek.asp

Math Trek Dangerous Problems Food for Thought Slugging It Out with Caffeine Science Safari Remember Typewriters? TimeLine 70 Years Ago in Science News Week of June 29, 2002; Vol. 161, No. 26 Dangerous Problems Ivars Peterson Some mathematical problems are easy to describe but turn out to be notoriously difficult to solve. Nonetheless, despite their reputed difficulty and repeated warnings from those who had failed to solve them in the past, these infamous problems continue to lure mathematicians into hours, days, and even years of futile labor. In a presentation this week on "mathematical problems between order and chaos," Jeffrey C. Lagarias of AT&T Labs–Research highlighted three such notorious unsolved problems. His was a cautionary tale, aimed at an audience that included 12 high-school students who had already shown their problem-solving proficiency by topping the 2002 U.S.A. Mathematical Olympiad (USAMO). Susceptible himself to the lure of these tantalizing conundrums, Lagarias admitted that he could have subtitled his talk "some problems I wish I could solve." In recent years, Lagarias has made important research contributions in a variety of mathematical fields, including work on the randomness of pi's digits, number patterns related to circles nested within circles, and the problem of distinguishing knots from unknots. Billiards in triangles In a game of mathematical billiards, a ball moves across a table of a given shape at a constant speed forever. There's no friction to slow the ball down. It simply travels in a straight line until it hits a cushion and rebounds according to the rule that the angle of reflection equals the angle of incidence.

59. Practical Foundations Of Mathematics function pList({ a,b})\rightharpoonup N with p( ) = 1, p(cons(a, l)) = a(p(l))and p(cons(b,l)) = b(p(l)). The collatz problem asks whether p is surjective. http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/html/s62.html

Practical Foundations of Mathematics Paul Taylor Well Formed Formulae Free algebraic theories provide a useful ``scaffolding'' which can be used during the building of more complicated linguistic structures, such as dependent type theory in Chapters VIII and IX . In this section we shall describe the recursive aspects of arguments about such structures, which are for example used in the construction of the interpretation functor [[-]]: Cn L S In practice, additional side-conditions are required of the terms which are to be admitted to the language. Some of these, such as the number of arguments taken by each operation-symbol, can be enforced in advance, but others must be stated by simultaneous recursion together with the expressions themselves. The terms which do satisfy the conditions are traditionally known as wffs well formed formulae D EFINITION 6.2.1 A wff-system is a set X of terms for a free theory ( W ar ) such that if r u X j X for all j ar r ]. Therefore parse X TX is a total function on X , and is injective (since ev is a partial inverse).

60. Science: Mathematics: Number_Theory: Open_Problems - Open Site 23 5 giving A = 1.6299. collatz problem. Also known as the 3n+1 problem,Syracuse problem, Thwaites problem. Define a function on http://open-site.org/Science/Mathematics/Number_Theory/Open_Problems/

Open Site The Open Encyclopedia Project home submit content become an editor the entire directory only in Number_Theory/Open_Problems Top Science Mathematics Number Theory : Open Problems Riemann Hypothesis ABC Conjecture We define the radical R(n) of an integer n to be the the product of its distinct prime divisors. ABC Conjecture: For every k>1, there is a constant C k such that if a,b,c are coprime positive integers satisfying a+b=c, then c < C k R(abc) k The conjecture was proposed by Osterle and Masser in 1985. At present the best that can be proved is that c < exp(R(abc)^f) for a suitable f. Mason has proved the analogue of the conjecture for polynomials. If true, the conjecture would have numerous important consequences: among them would be another proof of Fermat's Last Theorem. We define the ABC ratio for a triple (a,b,c) to be A = log(c) / log(R(abc)). The conjecture implies that A is bounded, so it is of interest to find large values of A. The current best is: a=2, b=3 .109, c=23 giving A = 1.6299 Collatz Problem Also known as the 3n+1 problem, Syracuse problem, Thwaites problem.

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