 Appealing Numbers  Ivars Peterson (MathTrek)
A short history of amicable numbers  pairs in which each number is the sum of the proper divisors of the other. The smallest such pair is 220 and 284. The number 220 is evenly divisible by 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which add up to
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 Arbelos Internet Math Society  Bert Liberi and William F. Widulski, Westchester Communtiy College
A mathematics society dedicated to the pursuit of elegance in problemsolving. Every month, solve challenging problems that require "more imagination than memorization." Also, access old problems and solutions; and study a list of heuristics of problemsolving,
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 Arithmagic  Ivars Peterson (MathTrek)
Arthur T. Benjamin eschews the usual trappings of the magician's trade. Calling himself a mathemagician, he astonishes audiences with amazing feats of mental arithmetic. Behind the scenes, he reveals how you, too, can look like a genius without really
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 A Bibliography of Bernoulli Numbers  Karl Dilcher
A bibliography of Bernoulli numbers by Karl Dilcher and Ilja Sh. Slavutskii. The Bernoulli numbers are among the most interesting and important number sequences in mathematics. They are particularly important in number theory, especially in
connection
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 BrainBashers  Kevin N. Stone
A collection of logic, language and math puzzles, in a choice of easy/medium/hard. Rate the puzzles; show their answers. Logic puzzles include Einstein's Puzzle. Play games of strategy, such as "Four in a Row" (also known as "Connect Four") or games based
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 CalculationRankings  Youri Lieberton
Arithmetic problems to solve or number sequences to complete as a stopwatch counts down. Play the games to compare high scores against other speedy problem solvers; take the timed tests to see correct answers after submission.
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 Carl Offner's Home Page  University of Massachusetts, Boston
Expository papers by Offner on mathematics and computer science (levels: advanced undergraduate to 1styear graduate student): Repetitions of Words and the ThueMorse Sequence; Finite Fields and PseudoRandom Number Generation; Some Early Analytic Number
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 Coffee Hour Problem of the Week  Matthew McMullen
Classic and "found" problems, as well as original challenges, in number theory, logical reasoning, statistics, calculus, geometry, and algebra. McMullen has posted PDFs of the PoWs, as well as their solutions, since 2007.
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 A compendium of NP optimization problems  Crescenzi & Kann
A continuously updated catalog of approximability results for NP optimization problems. Because no NPcomplete problem can be solved in polynomial time (unless P=NP), many approximability results (both positive and negative) of NPhard optimization problems
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 Computing Science: A Question of Numbers  Brian Hayes; American Science
An article about the ISC (Inverse Symbolic Calculator) and numbers. "In my daydream, Neil Sloane and Simon Plouffe are contestants on "Jeopardy," the TV game show. Sloane picks the category "Integer Sequences" for $400, and Alex Trebek reads the answer:
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 Cubes of Perfection  Ivars Peterson (MathTrek)
...Six is the smallest perfect number. Twentyeight comes next. Its proper divisors are 1, 2, 4, 7, and 14, and the sum of those divisors is 28. Incidentally, if the sum works out to be less than the number itself, the number is said to be defective
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 David Sumner's Home Page  David Sumner
Study guides, exams, quizzes, problem sets, exam review materials, utilities, programs and simulations, syllabi, and other materials for courses such as calculus, sequences and series, graph theory, number theory and cryptography, and probability. The
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 Digits, Squares, and Cycles  Ivars Peterson (MathTrek)
Fascinating patterns lurk among the digits of whole numbers. Pick a positive integer, such as 57. Square each of the digits, then add the squares together: 5^2 + 7^2 = 25 + 49 = 74. Do the same thing with the digits of 74: 7^2 + 4^2 = 49 + 16 = 65. Keep
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 Edutest: Online Assessment Resource
Assessment testing based on nationally recognized standards in Math and Language Arts for grades 28. Scoring is instant and both parents and teachers can review the tests and the correct answers with their students. The site has some information for
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 Encyclopedia of Combinatorial Structures  Virginie Collette; INRIA
A site designed to be similar to Sloane's Encyclopedia of Integer Sequences but with an emphasis on sequences that arise in the context of decomposable combinatorial structures. Like the EIS, the database is searchable by the first terms in the sequence,
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 Enumeration of the Binary Trees (Catalan Numbers)  Ivan Galkin
For each number of nodes, n, there is a certain number of possible binary tree configurations. These numbers form a sequence of integers with respect to n. A useful way to describe an integer sequence is to construct a generating function...
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 Essay Ideas  Jim Wilson, Dept. of Mathematics Education, Univ. of Georgia
For a course on Using Computers in Mathematics Instruction. Friday the 13th (show there are at most three and at least one Friday the 13th in each year, consider when two consecutive months can be have Friday the 13th); Gingerbread Man (a Microsoft Excel
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 Excel Math  AnsMar Publishers, Inc.
Excel Math is a mathematics curriculum for Kindergarten through Sixth grade students. Students are repeatedly given practice in the use of the concepts, following their introduction, so that they retain them and develop a solid foundation on which to
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 Experiments with the 3n+1 Sequence  Alfred Wasserman
An online program tests numbers for convergence under the 3n + 1 sequence. Also available in German.
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 Four Numerical Triangles in Nexus  Len Smiley
Three wellknown combinatorial number triangles are tightly related to each other as coefficients in transforms of a seminal sequence of exponential generating functions (for n^{n+m}). A fourth triangle is a part of the picture in much the same way that
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 Friedman Numbers  Erich Friedman
A Friedman number is a number that can be written in some nontrivial way using its digits, the operations +  * / ^ and concatenation of digits. For example, 25 and 126 are Friedman numbers, since 25 = 5^2, and 126 = 6 * 21. Page includes a list of all
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 From Lewis Carroll to Archimedes  Cut the Knot!, Alexander Bogomolny
On March 29, 1879, Vanity Fair began offering its subscribers a new weekly puzzle invented by Lewis Carroll... on March 17, 1953 Frank Gray, a research scientist at Bell Labs, filed patent no. 2632058, for the Gray code encoding
the vacuum tube. An ndigit
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 Future School
Developer of educational software, online tutoring, and home schooling courses. Video tutorials. Sample lessons and ordering information available.
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 General Mathematics, NZ (Funk & Wagnalls Multimedia Encyclopedia)  Lycos Zone: Mathematics
Short articles on basic mathematics, including biographical information about mathematicians, from John Napier to Zeno of Elea.
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 Generating Functions  Interactive Mathematics Miscellany and Problems, Alexander Bogomolny
The definition and properties of generating functions with several examples, including variants of coin change, fast algorithm for computing Banzhaf and ShapleyShubik power indices, covering the chessboard with tromino and more. Many examples are illustrated
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 Handson Math: Activities for the Elementary Classroom  Janine Parker
Lesson plans for children in grades one through six about topology, number patterns, and geometry. Handouts for each lesson may be downloaded in PostScript form or as GIF files, and a zipped DOS program to display Platonic solids, Archimedean solids,
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 Home Run Numbers  Ivars Peterson (MathTrek)
The integers 61 and 62 have been very much in the news this summer. On the majorleague baseball front, Mark McGwire hit his 62nd home run on Sept. 8, and Sammy Sosa matched that total on Sept. 13, surpassing the record of 61 home runs in a season held
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 Identities for Linear Recurring Sequences  Kevin Brown
There are several methods for computing the Nth term (mod M) of a
linear recurring sequence of order d in log_2(N) steps, but most
such methods require d^2 full multiplications (mod M) per step.
The algorithm described below requires only d(d+1)/2
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 Integer Sequences and Arrays  Clark Kimberling; Dept. of Mathematics, Univ. of Evansville, Evansville, IN
Certain seemingly simple sequences of integers baffle the best mathematicians. Other sequences, less baffling, exhibit patterns  or absence of patterns  whose appeal shines beyond whatever applications these sequence might find outside mathematics.
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 IXL Learning
IXL provides free online practice, organized by grade levels K8 and aligned to state standards. Choose one math skill from among the hundreds offered, and submit your answer. Respond incorrectly, and click on the "explanation" button to learn more; get
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 Journal of Integer Sequences  Jeffrey O. Shallit, editorinchief
This is an electronic journal devoted to papers dealing with integer sequences and related topics. All papers are available as html documents.
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 Juggling by Design  Ivars Peterson (MathTrek)
The earliest known depiction of juggling is on the wall of an Egyptian tomb nearly 4,000 years old. The painting shows a woman keeping three balls aloft. It's only in the last decade or so, however, that juggling has become the subject of serious mathematical
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 Juggling by Number (MatheMUSEments!)  Ivars Peterson (Math Muse for Kids)
Mathematicians have invented a way to write down juggling patterns as sets of numbers. They look at the order in which balls are tossed into the air and then caught. Each toss or catch happens on a particular beat, as if the juggler were keeping time
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 K12 Math Problems, Puzzles, Tips & Tricks  Math Forum
Links to calculation tips (Beat the Calculator, Divisibility Rules, Multiplication Tips), Math Problem sets, and math number and line puzzles.
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 Keith Numbers  Mike Keith
A Keith number is an ndigit integer N with the following property: If a Fibonaccilike sequence (in which each term in the sequence is the sum of the n previous terms) is formed, with the first n terms being the decimal digits of the number N, then N
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 Lehmer's Conjecture  Michael Mossinghoff; Dept. of Mathematical Sciences, Appalachian State Univ.
A summary of what is known today about Lehmer's Conjecture, including descriptions of algorithms, histories of searches performed, and various
lists of polynomials with small measure. References and a thesis available for downloading.
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 Le site des maths à petites doses  SaintMartin Arnaud
Supplementary information and explanations of various mathematical topics, often illustrated with figures and JavaScript, as well as some sample problems and multiplechoice quizzes. In French.
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 Letter/Number Puzzles  Math Forum, Ask Dr. Math Common Question
A selection of answers to questions about cryptograms, alphametrics, letter sequences, and other letter/number puzzles, such as SEND + MORE = MONEY.
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 License Codes  Ivars Peterson (MathTrek)
Some states use complicated schemes for establishing driver's license identification numbers. The numbers may include check digits to detect errors or fraud, or they may encode such personal data as the month and date of birth, year of birth, and sex.
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 LSL Math Traits  Gary T. Leavens
A guide to an archive of mathematical traits written in the Larch Shared Language (LSL), constituting a small LSL handbook. The traits do not duplicate the traits in Guttag and Horning's book, but add to them. Real numbers; Polynomial Functions and Polynomials;
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 Marc Chamberland
A mathematician at Grinnell College interested in differential equations and dynamical systems. Resources for the 3x + 1 problem and the Jacobian Conjecture include papers to download in PostScript format and information and proceedings for related conferences.
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 MarisMcGwireSosa Numbers (Or: Mathematics Plays Baseball, Again)  Mike Keith
An article in the spirit of Playing with RuthAaron Pairs by Ivars Peterson, wherein in honor of the 62nd season home run hit by Mark McGwire on 8 Sept 1998 and Sammy Sosa on 13 Sept 1998, which eclipsed the longheld record of 61 by Roger Maris, Keith
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 Mastering Mastermind  Ivars Peterson (MathTrek)
"Games offer wonderful playing fields for developing mathematical problemsolving skills. For Mathew Mitchell of the School of Education at the University of San Francisco, the preferred training ground is a classic game called Mastermind. "Students young
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 The Mathematical Works of Rajesh Ram  Rajesh Ram
A collection of formulas and identities: Fibonacci, Pell, square and triangular numbers, and sums of cubes and other powers, with an identity related to Ramanujan's Number.
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 The Mathman  Don Cohen
Materials for sale for K12 students, teachers and parents; precalculus. Materials include Get Ready for Calculus (Calculus By and For Young People book, CDROM, worksheet book, videotapes and map) and Changing Shapes with Matrices. Patterns, visualization,
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 Medieval Harmony  Ivars Peterson (MathTrek)
Harmonic numbers. The history of mathematics contains many gems of mathematical reasoning and thought. At a Mathematical Sciences Research Institute meeting last October, number theorist Hendrik W. Lenstra Jr. of the University of California at Berkeley
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 A Nameless Number  Len Smiley
Trianglefree polygon dissections are a variant of the dissection questions of Catalan, Schroeder, and others: In a convex (n+2)gon with n+2 labeled nodes, what is a(n), the number of ways of drawing nonintersecting diagonals so that no triangles are
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 Names for Numbers  Ivars Peterson (MathTrek)
Once someone discovers an interesting pattern or type of behavior, those particular numbers are likely to earn a collective name. So we have perfect numbers, amicable numbers, lucky numbers, Mersenne numbers, Fermat numbers, Fibonacci numbers, Keith numbers,
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 Neil J. A. Sloane
Neil Sloane's research ranges far and wide and includes coding theory, sphere packing, lattices and quadratic forms, packing lines, and planes, spherical codes and designs, quantizing, geometry, combinatorics, the design of experiments, integer sequences,
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 Next in Line  Ivars Peterson (MathLand)
Number sequences present all sorts of intriguing puzzles and patterns. Consider the sequence of counting numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .... Now, take out every second number, leaving: 1 3 5 7 9 11 13 15 ...; form the cumulative totals
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