# Mathematical Circles Topics

Last Updated: February 18, 2011.

Want to send me mail? tomrdavis@earthlink.net.

Mathematical Circles are for students of high school age or younger who want to increase their abilities to reason about mathematical problems. Students who study in circles learn to do mathematical olympiad-style problems (essay or proofs, not quick answer).

The web pages for a couple of the local circles are here:

An extensive discussion about how to lead a math circle can be obtained here:

So You're Going to Lead a Math Circle, or (in Postscript). (For some reason, the pdf version sometimes doesn't seem to load properly on an Apple Safari browser, but the Postscript version does.)

Here is the National Association of Math Circles website with links to existing circles, problems, lesson plans, and many other things.

For more lesson plans and for information on Teachers' Circles which are like math circles but aimed at teachers, go to the Math Teachers' Circle Network.

I have been volunteering time at three local circles (in Berkeley, San Jose, and Palo Alto, California) for the past nine years, and have prepared handouts for some of those sessions. Those handouts are almost all available here in both PostScript format (files with a .ps extension) or files in Adobe Acrobat format (files with a .pdf extension).

## Free Dynamic Geometry Software

I've written a program called "Geometer" similar to Geometer's Sketchpad or Cinderella that is freely available together with complete documentation, tutorials, and hundreds and hundreds of examples. It runs on a PC or Mac, and you can obtain a copy of it here.

## Rubik's Cube and Mathematics

Here is a web page that allows you to download a program that simulates Rubik's cube for the PC or Mac, complete documentation, and a paper on using the cube to teach something about the mathmatics of permutations and group theory.

## Math for Teachers

Of course all the handouts below are used for math circles so in a sense, all can be used by teachers. However, recently I have been adding some pedagogical advice to some of them, perhaps making them easier to use in a classroom setting. The handouts in this section all have at least a little of such advice and some have a lot. Each of the handouts in this section is listed twice: once here and once in the appropriate topic-sorted section.

There is a series of workshops for teachers, called "teacher's circles", that may be of interest. Here is the website:

The 1,2,3,4 problem:
plustimesminus.pdf(72 KB)

Exploring Pascal's Triangle:
pascal.pdf(148 KB)

An Introduction to Zome:
ZomeIntro.pdf (504 KB)

Julia Robinson Zome Exercises (used for Julia Robinson Festival at Google and Pixar):

An Unusual Way to Combine Numbers
numbercombine.pdf (216 KB)

A Game That's Not a Game
pilesubdivide.pdf (216 KB)

Mathematical Card Tricks
CardTricks.pdf (216 KB)

Conway's Rational Tangles
tangle.pdf (48 KB)

Huge numbers with short descriptions:
bignumbers.pdf (84 KB)

Geometry and Geography:
geography.pdf (60 KB)

Kenken:
kenken.pdf (92 KB)

## Counting and Combinatorics

Exploring Pascal's Triangle:
pascal.pdf(148 KB)

How to Count Things:
counting.ps (94 KB)
counting.pdf (108 KB)

Solutions to problems in "How to Count Things":
solnscount.ps (100 KB)
solnscount.pdf (134 KB)

Combinations and Permutations:
comb.ps (37 KB)
comb.pdf (39 KB)

Combinatorics Problems:
combprobs.ps (47 KB)
combprobs.pdf (39 KB)

Counting Derangements (by Brian Conrey and Tom Davis):
derange.ps (61 KB)
derange.pdf (65 KB)

Catalan Numbers:
catalan.ps (428 KB)
catalan.pdf (132 KB)

Counting Poker Hands:
poker.ps (28 KB)
poker.pdf (21 KB)

Pólya's Counting Theory:
polya.ps (248 KB)
polya.pdf (168 KB)

## Set Theory, Logic, Cardinal and Ordinal Numbers

Something from Nothing (Set Theory):
nothing.ps (117 KB)
nothing.pdf (152 KB)

Infinity (Cardinal and Ordinal Numbers):
Infinity.ps (168 KB)
Infinity.pdf (158 KB)

Computability and Recursive Functions:
Computability.ps (222 KB)
Computability.pdf (180 KB)

## Geometry and Transformations

Inversion in a Circle:
inversion.pdf (308 KB)

Practical Calculation of Areas of Polygons:
polyarea.pdf (140 KB)

Introduction to Matrices:
Matrices.ps (165 KB)
Matrices.pdf (165 KB)

Geometric Transformations with Matrices:
Xforms.ps (125 KB)
Xforms.pdf (177 KB)

Classical Geometric Construction:
construct.ps (90 KB)
construct.pdf (86 KB)

Geometry of the Circle:
circles.ps(45 KB)
circles.pdf (28 KB)

Four Points on a Circle:
fourpoints.ps(135 KB)
fourpoints.pdf (122 KB)

Geometry for contests:
geometry.ps(106 KB)
geometry.pdf (107 KB)

Geometry and Geography:
geography.pdf (60 KB)

Polygons:
Polygons.ps (34 KB)
Polygons.pdf (33 KB)

Pick's Theorem:
pick.ps (413 KB)
pick.pdf (182 KB)

Euler's Theorem:
euler.ps (149 KB)
euler.pdf (82 KB)

Projective Geometry:
projective.ps (167 KB)
projective.pdf (162 KB)

Homogeneous Coordinates for Computer Graphics:
cghomogen.ps (149 KB)
cghomogen.pdf (182 KB)

Homogeneous Coordinates (slightly obsolete -- see above):
homogen.ps (50 KB)
homogen.pdf (53 KB)

Spline Curves:
curves.ps (173 KB)
curves.pdf (147 KB)

## Zome

An Introduction to Zome:
ZomeIntro.pdf (504 KB)

The Mathematics of Zome:
zome.pdf (3 MB)

Julia Robinson Zome Exercises (used for Julia Robinson Festival at Google and Pixar):

## Games

Hackenbush:
hackenbush.pdf(168 KB)

Kenken:
kenken.pdf (92 KB)

A Game That's Not a Game
pilesubdivide.pdf (216 KB)

Mathematical Card Tricks
CardTricks.pdf (216 KB)

Mathematics of Sudoku:
sudoku.pdf (130 KB)

Mathematics of Mastermind:
mastermind.pdf (130 KB)

Binary Guessing (A two-color version of Mastermind):
bitmaster.pdf (130 KB)

## Miscellaneous

Huffman Encoding:
compression.pdf (72 KB)

Units:
units.pdf (105 KB)

Iterated Functions:
iterated.pdf(664 KB)

Coloring:
coloring.pdf(96 KB)

An Unusual Way to Combine Numbers
numbercombine.pdf (216 KB)

Conway's Rational Tangles
tangle.pdf (48 KB)

An Introduction to the Riemann Integral
riemannint.pdf (92 KB)

Bin Packing Problems
binpacking.pdf (64 KB)

Introduction to Linear Diophantine Equations
diophantine.pdf (64 KB)

Practical Probability: Casino Odds and Sucker Bets
suckerbets.pdf (148 KB)

Graph Theory Problems and Solutions:
graphprobs.pdf (132 KB)

Mathematics of Geodesic Domes:
geodesic.pdf (520 KB)

Optics and Geometry with Applications to Photography:
optics.pdf (188 KB)

Huge numbers with short descriptions:
bignumbers.pdf (84 KB)

The relationship between decimals and fractions:
fractions.ps (189 KB)
fractions.pdf (197 K)

Permutation Groups and Rubik's Cube:
perm.ps (99 KB)
perm.pdf (111 K)

Mathematical Induction:
Induction.ps (129 KB)
Induction.pdf (183 KB)

Mathematical Induction Problems and Solutions:
indprobs.pdf (95 KB)

General Encryption:
crypto.ps (115 KB)
crypto.pdf (147 KB)

The RSA Encryption Algorithm:
RSA.ps (53 KB)
RSA.pdf (61 KB)

Elliptic Curve Cryptography:
ecc.pdf (61 KB)

Mathematical Biology:
mathbio.ps (292 KB)
mathbio.pdf (92 KB)

San Jose State Circle Contest:
contest.ps (34 KB)
contest.pdf (28 KB)

Hints for Competing in Olympiad-Style Contests
solvit.ps (54 KB)
solvit.pdf (57 KB)

I prepared a following document for people who are interested in leading math circles. If you wish to make a nice version to hand out to prospective volunteers, here it is in PostScript format: MathCircles.ps, and in Acrobat format: MathCircles.pdf. Here it is in HTML:

## How to Lead a Math Circle Session

Math circles are for high school or younger students. Circle sessions concentrate on problem solving techniques applicable in many areas. Sample circle topics include: symmetry, the pigeon-hole principle, divisibility, counting, probability, invariants, graphs, induction, plane geometry, or inversion in a circle.

Here are some differences between a math circle and the usual math club:

• One school teacher usually runs all math club sessions. Leaders of math circles rotate. Circle leaders don't burn out, the kids see different approaches to math, and the leaders only need to prepare a few sessions that can be repeated at multiple circles. Circle leaders include teachers, professors, graduate or even undergraduate students, and other professional mathematicians.
• Circle sessions are focussed on a particular topic. ``Here are a bunch of unrelated old AIME problems.'' is usually not a suitable circle topic.
• There is homework, but exciting and seductive homework.
• Math clubs often prepare the math team for multiple-choice or short-answer competitions, without going through the problems in depth. Math circles prepare students for Olympiad-style problems like those of BAMO, the Bay Area Mathematical Olympiad. Circles teach kids to be mathematicians who solve essay-style problems requiring proof.

Make sure your circle session goes as well as possible:

• Hand out a set of problems a week before your session. Not too many, but seductive. Include an easy one and a challenging one.
• Try not to lecture. Even though introducing new theory and techniques is an integral part of math circles, your sessions should be as interactive as possible. Score yourself: 1 point per minute you talk; 5 points per minute a student talks; 10 points per minute you argue with a student; 50 points per minute the students argue among themselves.
• Divide students into groups of 2-4 to solve problems. Have them present their own solutions.
• Be encouraging, even about wrong answers. Find something positive in any attempt, but don't be satisfied until there is a rigorous solution. Wrap up each problem by reviewing the key steps and techniques used.
• If the kids don't answer your question immediately, don't just tell them the answer -- let them think. If they're still stuck, give hints, not solutions.

For details on Math Circles or on BAMO, see:

``Mathematical Circles (Russian Experience)'', by Fomin, Genkin, and Itenberg, American Mathematical Society, 1993.

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