 Proofs Without Words  Cut the Knot!, Alexander Bogomolny
Martin Gardner wrote, "There is no more effective aid in understanding certain algebraic identities than a good diagram. One should, of course, know how to manipulate algebraic symbols to obtain proofs, but in many cases a dull proof can be supplemented
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 Proof: The Sum of Prime Twins is Divisible by 12  Mathematics Dept., College of Marin
Prime twins are prime numbers that are two numbers apart. For example 41 and 43 are prime twins. Can you PROVE that the sum of prime twins is divisible by 12?
(This proof doesn't work for the prime twins 3 and 5.)
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 Public Domain Materials  Mike Jones
A collection of public domain instructional and expository materials from a USborn math teacher who teaches in China. Microsoft Word and PDF downloads include a monthly circular consisting of short problems, "The BowandArrow Problem," and "Twinkle
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 Punctured Polyhedra  Ivars Peterson (MathTrek)
A tetrahedron has four triangular faces, four vertices, and six edges. When a vertex of one tetrahedron pierces the face of a second tetrahedron it forms a new, more complicated polyhedron: one triangular face has a triangular "hole" where the face was
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 Pure Maths, Impure Thoughts  David A. Santos, Community College of Philadelphia
Includes lecture notes in elementary algebra, linear algebra, number
theory, probability, a problem seminar, plus review material in
precalculus and multivariable calculus. Selections available en castellano.
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 Purplemath  Your Algebra Resource  Elizabeth Stapel
Algebra modules provide lessons, practical tips, hints, worked examples, and point out common mistakes in basic triangle length (trigonometry) values, calculators (what model to buy), canceling units, domain and range, exponents, factoring quadratics,
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 Putnam Competition  Patrick J. Fitzsimmons
Postscript, Adobe Acrobat, and dvi downloads of problem sets from the William Lowell Putnam Mathematical Competition dating back to 1980. With links to solutions hosted by others, and information on participation.
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 Putnam Exams  Dave Rusin
Solution sets and comments compiled from postings on the sci.math newsgroup. Online resources and citations for exams dating back to 1987.
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 Put the Heart into Mathematics: Cardiac Output, Rates of Change and Accumulation (iMath Investigations)  Illuminations; National Council of Teachers of Mathematics (NCTM)
An online, interactive, multimedia math investigation. This activity explores the measurement of the amount of blood being pumped by a heart. The goal of this iMath investigation is a rich exploration of concentrations (mixture problems), rates of change
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 PuzzleBeast  James W. Stephens
A collection of computergenerated puzzles, some illustrated with Java applets, some with solutions included: The Fried Okra Perplexity, A Dozen Irritating Sliding Block Puzzles, The Bulbous Blob Puzzle, Meandering Monk Maze, The Kung Fu Packing Crate
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 Puzzle Choice
A wide choice of puzzles and games for all ages, including crosswords, wordsearches, brainteasers, number puzzles (a calculator is provided), logics, teacher resources and other online games, including mazes.
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 Puzzle Fun  Rodolfo Kurchan
An online newsletter edited by Rodolfo Kurchan (Buenos Aires  Argentina) about polyominoes and other puzzles. Back issues are on the Web.
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 Puzzlemaker  Discovery Education
A puzzle generation tool for teachers, students and parents. Create and print customized puzzles, including math squares and number blocks, online, or buy a CD with editing tools and other extras.
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 Puzzle of the Week  Ralph Grove
A small archive of math and word puzzles having to do with patterns, permutations, and probability, with solutions, from a professor of computer science at Indiana University of Pennsylvania.
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 Puzzle Page  The Black Stump Ltd.
More than 150 links to puzzle sites on the Web. A new site is added each week.
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 Puzzler  Franklin Institute Online
Puzzles from the inQuiry Almanack Puzzle Archive: e.g. Cryptograms, January, 1998. Coded messages hide famous quotations. A clue is offered to begin breaking the code, and an answer key is provided. One puzzle a month from March of 1995 through July of
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 Puzzlers  Bryan Clair, University of Chicago
A collection of Clair's favorite puzzles (no discussion or answers are provided): The Nine Dots; Weighing Coins; The Dollar Hotel, Planning an Orchard; The Math Department; Tea and Cream; Bugs on a Square; River Crossing; More Crawling Bugs; The Unexpected
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 Puzzles
A blog of math puzzles. Post your answers in the comments sections.
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Puzzles/Oneplayer Games  Jürg Nievergelt
Rubik's Cube and its variants. The 3*3*3* cube; Rubik's Revenge (4*4*4 cube); Pyramix (tetrahedron); Rubik's Clock; Rubik's Magic. Also Cryptequations, polyominoes, and links to a list of similar puzzles.
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 Puzzles, & Riddles  The Contest Center  Frank Rubin
Hundreds of puzzles and mathematical problems ranging from kindergarten through postdoctoral levels. For the easier puzzles, answers are provided on the site; names of those who submit correct solutions to harder problems are posted. See also Puzzles
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 The Puzzle TOAD  Tom Bohman, Oleg Pikhurko, Alan Frieze, and Danny Sleator, School of Computer Science, Carnegie Mellon University
Send the authors your solutions to their challenging graph theory, number theory, geometry, discrete mathematics, logic, and other puzzles. Revisit periodically to see the latest question, and download PDFs of solutions of past problems.
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 Puzzle Toons, The Citizen Scientist  Brian Mansfield
A comic strip devoted to puzzles, mind benders, and brain teasers. The subject matter of the puzzles ranges from mechanical devices and geometry to astrodynamics, chess, and billiards. Read back issues of Mansfield's column, dating to its inception in
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 Puzzle University  Amy Galitzer
A nonaccredited "university," created solely for fun. Everything is free to play and take part in. Register, pass a simple test, and advance from freshman to sophomore to junior to senior to grad student this way. The puzzles get harder as you progress.
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 PuzzleWise  Test Best International, Inc.
A supplemental program consisting of math crossword puzzles, each clue of which links directly to state and national gradelevel math skill expectations. The site offers samples, testimonials, prices, and a free newsletter.
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 Puzzle World  John Rausch
A searchable site with information and a discussion group (Puzzle Talk) about handcrafted mechanical puzzles. Most of the puzzles presented here are not the massproduced variety found in stores, but limited production puzzles produced by craftsmen (often
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 Puzzle Zapper  Alexandre Owen Muñiz
Original mathematical puzzles, some available in lasercut acrylic for sale: polyomino common superforms, pentomino coloring problems, miscellaneous polyomino explorations, flexible polyforms, square grid puzzles, visualizing sequences of numbers, Lissajous
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 Puzzling Lines  Ivars Peterson (MathTrek)
Sol LeWitt has often featured geometric and combinatorial themes in his sculptures, paintings, and drawings. In 1973 he composed "Straight Lines in Four Directions and All Their Possible Combinations," a grid of 15 squares, each inscribed with one or
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 Puzzling Primes  Isaac Reed
To fully comprehend our number system, mathematicians need to understand the properties of the prime numbers  but even finding them isn't so easy.
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 The Puzzling World of Barry R. Clarke  Barry R. Clarke
Brain teasers from Clarke's column in The Daily Telegraph (UK) and from his various puzzle books: liar problems, logic teasers, number conundrums, matchstick puzzles, lateral mindbenders, and more. With links to other puzzles sites and an article on
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 Pyramids and Algebra  David C. Miller
Four questions that explore the problem of creating a pyramid from a set of dominoes, dealing with the concepts of arithmetic series; derivation of square roots including a graphical method; scientific notation; exponential notation; quadratic equations;
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 Pythagoras' Haven  Jim Morey
Step through a geometrical Java applet proof of the Pythagoraean theorem  the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
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 Pythagorean Puzzle (SMILE)  Earl Zwicker, Illinois Institute of Technology
A lesson that models the phenomenological approach to enable students to prove the Pythagorean Theorem on their own using puzzle pieces and working together.
From the Geometry and Measurement and Algebra and Trigonometry sections of a collection of almost
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 Pythagorean Theorem (Manipula Math)  International Education Software (IES)
Java applets for experimenting with shifting, rotating, and moving pieces of quadrilaterals to prove the Pythagorean Theorem.
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 Pythagorean Theorem  Over 20 Proofs  Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny
A general exposition of the Pythagorean theorem, with proofs from Euclid to Garfield  and more. Also links to other sites on the Web with proofs.
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 Pythagorean Theorem  Warren F. Davis, Davis Associates, Inc.
An animated, nonmathematical, annotated proof of the Pythagorean Theorem, assumed to be the method of proof actually used by Pythagoras circa 550 BCE (see the Bronowski reference).
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 QPuzzle  XIAO Gang
A mathematical jigsaw puzzle: your goal is to recover a disordered photo by finding the inverse affine transformation. A French version is also available.
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 Quandaries and Queries (Math Central)  Dept. of Mathematics and Statistics, University of Regina, Saskatchewan, Canada
A math Question and Answer area for K12 students and teachers, with searchable archives of past questions and answers.
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 QuantumBrainBenders  Joel Klebanoff
A series of puzzles: find a series of numbers in a grid that add to the given sum. A new puzzle for each of three skill levels is posted daily. The site also includes an archive of previous puzzles.
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 The Quantum CyberTeaser  National Science Teachers Association (NSTA)
Archive of bimonthly problems posed online by Quantum, the math and science magazine. The magazine ceased publication with its July/August 2001 issue. Puzzles and most solutions are still available here.
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 The Quest for the Holy Value  Kjetil Valstadsve
Finding the correct value of pi. The likelihood of any random two integers having 1 as their greatest common divisor (no factors in common), has been proved to be 6 divided by pi squared  the numbers are prime in relation to each other. According to
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 Quia!  The Learning Group
Create your own Javaenabled quizzes, home pages, and learning games to play over the Internet: enter your material into a form, select a few options, and press a button. Games have been created and are available for browsing in mathematics, as well as:
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 Quick Trig Lessons  Matt Robinson
Brief lessons on ratios, triangles, identities, graphs, and formulas, with practice problems and a page of links to other math sites on the web.
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 Quiz Hub  Schmidel & Wojcik: Educational Web Weavers
A collection of online interactive learning quizzes that help K12 students enhance their core knowledge of English, languages, math, science, and social studies. Subscription required.
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 Quizma > Matemática Cálculo Robótica  Harold Suárez Guzmán
In Spanish: Clases y Orientación en Matemáticas Cálculo y Robótica. Fórmulas  Demostraciones  Ejercicios  Ejemplos  Cálculo Interactivo. Algebra, Logaritmos, Trigonometría, Geometría, Estadística, Ecuaciones, Problemas de Planteo, Funciones, Geometría
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 Quiz Me  D. Kell and Mercer County Community College
Interactive quizzes constructed using software provided by Addison Wesley Longman Publishers. Autoscoring, multiple choice selftests on whole numbers (place value, operations, applications, rounding, roots and order of operations); fractions (equivalencies,
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 QuizStar  Advanced Learning Technologies, Center for Research on Learning at The University of Kansas
Create online quizzes and organize the results. Features include the ability to attach multimedia files to questions, to make quizzes in multiple languages, and more.
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 QuizTree Math Games  Sierra Vista Software
A collection of online math quizzes in addition, subtraction, multiplication, and division.
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 Quizzes.cc
Use the menus and text boxes to create your own quizzes on addition, subtraction, multiplication, and division  or browse from among a selection. Click the appropriate button to reveal answers, grade responses, or generate new problems.
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 Radius and Diameter  NASA Lewis Learning Technologies Project
Mr. Jordan wants to put up a fence all the way across the middle of his round vegetable garden. If the radius of the garden is 3 feet, how many feet of fencing will Mr. Jordan need to buy? A detailed answer is provided. From NASA's 9th Grade Math Proficiency
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 Raise Smart Kids  Inna Shapiro
Download Powerpoint presentations of math problems, with answers: logic, basic arithmetic operations, natural numbers, pigeonhole principle, fractions, pouring, cutting, weighing, divisibility, and more. For Croatian translations of some of Shapiro's
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