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Mathematics

CONTENTS

General Mathematical Resources 
Puzzles, Problems-of-the-week, etc. 
Factoids and Oddities 
Worksheets, Example sheets 
Calculus 

Demonstration that a pair of linked circles at the ends of a rope are topologically equivalent to a pair of unlinked circles.

Math Puzzles

     *  Asterisk indicates free registration is required for this website, e.g. Project Euler

  1.  
    Nick's Mathematical Puzzles -- more than one hundred excellent puzzles of varying degrees of difficulty, each with hints, an answer, and a complete solution.

  2.  
     Ponder This is one math or logic puzzle a month.  All readers may send in solutions.  Solvers are honored by having their names appear on the puzzle page.  Past problems and solutions are archived on the website, going back to May 1998, so by April, 2007 there were nine full years -- 108 challenges -- archived for your enjoyment.
  3.  
     Project Euler * is a collection of challenging mathematical/computer programming problems that will require mathematical insights and computer programming skills to solve.  The website keeps track of problems you have solved, and can show you problems you haven't solved yet.  See also forum.projecteuler.net *  
  4.  
    Purdue University Problem of the Week A Purdue Math Department panel publishes a challenging problem once a week and invites college & pre-college students, faculty, and staff to submit solutions. Our objective is to stimulate and cultivate interest in good mathematics, especially among younger students.
  5.  
    The Problem Site Here you will find math problems, word games, word puzzles, mystery quests, and many other free educational resources.
  6.  
    Macalester College Problem of the Week Stan Wagon, a professor in the Mathematics and Computer Science Department at Macalester College, poses a mathematics problem to his students every week. The Problem of the Week tradition was started in 1968 by the late Professor Joe Konhauser. Professor Wagon took over in 1993. Since the problems are meant to be accessible to first-year college students, very little background is needed to understand or solve them.
  7.  
    Nrich

    Ask Nrich *

    The University of Cambridge.  For problems click "This Month" or the categories under it -- Stage 4 & 5 are the hardest problems.  Also, look for the Tough Nuts link.  For previous problems, choose a year and month under Past Issues.
    Nrich Problem of the Week -- select a year, month (*), and week-within-year
        Year: Month: Week:  
    (*) month seems to affect mostly the contents of the main page; same as selecting Year and Month from the dropdown on the main page.  But there is some very odd interaction between month and week-within-year.  Experiment.  Enjoy.
  8.  
    Puzzleup is a weekly puzzle competition.
  9.  
    Alma College's High School Math Challenge Alma College Hich School Math Challenge -- A new problem each month for students at participating high schools.  Students may submit their solutions.  The best solution will be published (giving credit to the solver, and earning the solver an Alma College t-shirt) and a list of all students who submit correct solutions will be published as well.
  10.  

    Problem of the Week

    College-level problems, posted weekly for the current semester by Professor Alberto L. Delgado, with archives of problems and solutions starting in spring 1997, for number theory, combinatorics, probability, calculus, etc.
  11.  
    ken.duisenberg.com/potw A weekly puzzle for advanced high school or college level, with a math, geometry, or game focus. The problems are easy to understand but tricky to solve. An archive of previous problems and solutions is included, together with a list of other puzzle sites Ken has enjoyed visiting.
  12.  
    A site devoted to original mathematical recreations. If you have a math puzzle, discovery, or observation, you are invited to e-mail the author. You can also submit answers to the problem of the month; questions and solutions are archived. Examples include: Divisor Chains; Integer Square Tilings; Angle Frequencies; Disconnected Polyominoes; Average Arrays; Convex Spectra; Surround Numbers; Palindrome Products, Fairy Chess Endgames, and Rectangles in Squares.
  13.  
    Problem Corner  - Les Reid; Southwest Missouri State University (SMSU).  High School, Advanced, and Challenge problems, updated monthly, plus links to other problem sites.  At the advanced and challenge levels, solutions are solicited and the names of those solving the puzzle are posted monthly along with one solution.
  14.  
             Maths problems
    "Kalva currently includes about 4,100 olympiad problems and about 1,700 other problems.  I have given solutions for about 2,300 Olympiad problems (and almost all the Putnam and AIME problems). My ultimate objective is to include a wide variety of interesting problems of all levels of difficulty. In general, I have no interest in pure bookwork problems."

    Pre-olympiad problems

    International/Regional Olympiad problems

    Undergraduate problems

    Miscellaneous problems

    AIME, Dudeney IMO, IMO shortlist, APMO, Austrian-Polish, Balkan, Junior Balkan, Centroamerican, Iberoamerican Putnam, Newman, IMC, Vietnam Misc

    National Olympiads

    Suggested books

    ASU, Brasil, Canada, Indian, Irish, Kürschák/Eötvös, Mexico, Polish, Russian, Spanish, Swedish, USA, Vietnam Elementary books
  15.  
    U.S. Military Academy Problem of the Week

    A current college-level Problem of the Week, with links to past problems and solutions, a project of the Pi Mu Episilon chapter of the United States Military Academy.  MathCad or Word are needed to view past problems and solutions.

  16.  
    MathPuzzle.com is an absolutely ginormous compendium of puzzles of all kinds (math, logic, word, mechanical -- you name it) along with a huge number of links to websites with background information and more puzzles.
  17.  
    An eclectic mix of popular puzzles, short films, the shakespeare puzzle, articles, biography, books, guestmap, fun page, forum, and links.  This is a great site to explore.  Go to the "fun page" and look for links to a version of the Lunar Lander game, for example.  (Barry's old site was homepage.ntlworld.com/barry.r.clarke)
  18.  
    The Harbour Puzzle -- Oskar's Harbour Puzzle: M. Oskar van Deventer
    5 January 2002 (Original design: 16-11-1989).  Object: Move all containers plus cranes from the quay onto the ship.  
    Rule 1: A crane can only move itself and things below it (including other cranes)
    Rule 2: For stability, there can never be empty space below a crane.
  19.  
     pqrst means puzzles, quarterly, rate, solve, ten.  The idea was that every quarter there would be ten puzzles that people would rate as to how enjoyable they are, and they would solve.  It's been a few years since there were any new puzzles on this site, but the archives are extensive. 
  20.  
     Aha! Puzzles are 168 very clever puzzles, each with its own very well-done graphic.  To find them, click the "puzzles" link, then click the links at the bottom of each page to advance through the puzzles.  There are answers to all but the last three, as of 1/20/2008.
  21. These should be better categorized and checked for duplicates.  They came from http://www.stetson.edu/~efriedma/mathmagic/ 
    Weekly Puzzle Sites: |  | Ken Duisenberg | NPR | Puzzability | Nick's |
    Other Puzzle Sites: | IBM | Retrograde Analysis | Puzzle Palace | Terry Stickels |
    Interactive Puzzle Sites: | Sliding Blocks | Mazes | Puzzle Beast | Lunar Lockout |
    | Joker Games | Mazeworks | Paint By Numbers | Theseus & Minotaur |
    Colleges with Puzzles: | Macalester | S. Missouri | Bradley | U. Mass. | Purdue | Hamline |
  22.  

Math Test Links

This section has sample test papers from math tests that are used to challenge (and in some cases, select) the brightest high school students.

  1.  
    STEP Mathematics: a resource for teachers and students;
    Supermaths: Mathematics' workshops for young students in Newcastle upon Tyne, UK;
    Examinations' solutions' Library: for GCSE Mathematics (MEI) and GCE A/AS-level Structured Mathematics (MEI). 
    A proof of pi's irrationality;
    A proof of e's irrationality;
    Meikleriggs' homepages;
    Other links
  2.  
    STEP (sixth term examination paper)
    Links from Wikipedia: Sixth Term Examination Paper 
    bulletCambridge Assessment page on admission tests
    bulletA booklet of STEP Mathematics questions called Advanced Problems in Core Mathematics, by Stephen Siklos
    bulletSolutions to STEP Maths papers (forum)
    bulletSolutions to STEP Maths papers (database)
    bulletSolutions to STEP Maths papers (requires login)
    bulletSTEP syllabus
  3.  
    OCR -- Oxford, Cambridge, and RSA Examinations
  4.  
    BMO (British Math Olympiad) past papers -- (pdf) -- This is a 3˝-hour paper with 5 problems, taken by students in their own schools.  Selection is based on performance in the UK Senior Mathematics Challenge (UKSMC).
  5.  
    International Mathematics Olympiad 
  6.  
  7.  

 

Riddles and other non-mathematical puzzle sites

  1. samuelloyd.com — click the sam lloyd's puzzles link on the main page.
  2. www.dansmath.com/probofwk/probofwk.html — suitable for grade school kids.
  3. colstate.edu/mathcontest/ Columbus State University Math Contest, suitable for grade school kids; a new problem every Monday

Puzzle Links

  1. The Math Forum @ Drexel lists hundreds of sites that have math puzzles.  Also, though the Drexel POW site requires a "library card", it lists a good number of other problem sites by the level of math required. 

Old sites that, alas, no longer exist:

  1. www.shepherd.edu/mathweb/problems.html still exists, but it is no longer updated with new problems.
  2. www.olemiss.edu/mathed/pow is no longer updated, but has an archive of puzzles that will be fun for elementary school children.  This site moved to www.umassd.edu/mathcontest, and then will move again to Columbus State University in Georgia.
  3. MARP -- Mark's Archive of Riddles and Puzzles, by Michael Winckler