Society Investigating Mathematical Mind-Expanding Recreations

SIMMER at a glance
(for the 1999-2000 school year, with the most recent at the top)
For a more complete set of notes, see the ARCHIVES

• May 4, 2000

Title: “Split “P” Soup”
Presenters: Dr. Greg Martin and Emmanuel Knafo
(Mathematics, University of Toronto)

Partaking of Number Theory

The Purpose of this month's SIMMER Presentation is to Publicize the Primes, and Promote divisibility and modular arithmetic (their Partners in number theory), Pointing out the Properties they Possess that make them Pertinent to solving Problems that would otherwise Perhaps Perplex Poor People like us.  We Propose the following Plan of a SIMMER meeting split into two Parts.  Our Primary course will consist of a few Puzzles to Ponder, in which we must first Process the Problem to Perceive the underlying Pattern and then use a little number theory to Prepare a Persuasive Proof.  Then we will Proceed to a Period of Play, where the Population will learn two games Prompted by number theory and must Perfect their strategy (or face the Peril of Perishing when Pitted against one another!).

Please come Participate in this Pretty Popular Pastime:
“Split "P" Soup” …  in a SIMMERing Pot!

The two door Prizes, "After Math" by Prof. Ed Barbeau of the University of Toronto and "A Mathematical Mosaic: Patterns and Problem-solving" by Ravi Vakil were won by Bob Farrington and Pete Roulston. Similar books may be found at the MAA web site.

The May SIMMER topic featured the type of thing students will encounter at this summer's SOAR 2000, a mathematical sciences camp on Number Theory, to be held  July 24 - August 11.

Here are some of the problems and puzzles discussed:
"Locker Problems and Assorted Appetizers", including 'Locker chaos', 'Rocks and lockers', 'Counting change' and ' Monkey business'.
"Modular Arithmetic Games", including 'The Modular Addition Game' and 'The Modular Multiplication Game'.
"Solutions and Discussion" of the above two sets.

• March 30, 2000

Topic: “Dancing with Fractals:The Chaos Game (with Music!)"
Speakers:   Cynthia Church (Bishop Strachan School)
Dr. Randall Pyke (University of Toronto & Ryerson Polytechnic University)

One way to draw fractals is the so-called chaos game. Here one has a set of simple rules, each of which tells you how to draw the next point from a given point. One of the rules is chosen randomly at each step of the game and the resulting set of points are drawn. The remarkable fact is that the pattern that emerges is not random at all! If the rules are chosen properly, the pattern can be a highly complex and self-similar object; a fractal. The chaos game is an interesting mixture of probability, dynamics, and geometry.

If one assigns musical notes to parts of the fractal, then we can also listen to the chaos game as the points move around on the fractal. Different fractals sound differently!

The door prizes went to Dr. Chi Chan and Petra Najafee. Chi won the  "Generative Soundscapes I" CD (containing the piece called "Sierpinski #3" written by Cynthia Church) and Petra won the book "Fractals: Endlessly Repeated Geometrical Figures", by H. Lauwerier (Princton University Press). Congratulations!

Here are some notes and links you might want to check out. Here also is a link to Randall's Chaos Game applet:
http://www.math.toronto.edu/~pyke/mat335/Fractal.html

• February 24, 2000

Topic: "Escher and You"
Speaker: Dr. Bruce A. Cload (University of Toronto)

“I never got a pass mark in math. The funny thing is I seem to latch on to mathematical theories without realizing what is happening.”        M.C. Escher

Escher (1898-1972), the Dutch Graphic Artist, is well known for his captivating prints of impossible staircases and colourful mosaics. Certain of his diagrams, his tessellations of the plane, are rich in rotational, reflectional, and translation symmetries and provide many attractive examples of the mathematical objects known as the plane periodic groups. The three dimensional analogs of these groups are important in x-ray crystallography, which played a pivotal role in the discovery of DNA.

In this workshop, we will examine the concept of symmetry and compare, contrast, and classify the various symmetric structures in some of Escher’s diagrams. We will also discuss various computer and web teaching resources with which students can see, explore, and create their own Escher-like pictures. Part of the allure of this material is students can actually observe ‘mathematical representations’ in many everyday places—the stones on the library buildings, the tiles on the floor, the knitted pattern on a sweater. Through this, they can perceive mathematics as part of the world rather than as an abstract science.

The door prizes went to Cynthia Church, Doug Sloan and Gila Hanna. Congratulations!

Here are some 'Questions for Exploration', references and a short set of notes. The references include a number of links to very interesting web sites, computer programs and highly recommended books on the topic.

• January 27, 2000

Topic: "Problems and Puzzles in Babylonian Mathematics"
Speaker: Prof. Craig G. Fraser (IHPST, University of Toronto)

The ancient Babylonians (ca. 1800 B.C.) possessed a sophisticated mathematics based on a positional base-sixty number system. Records of their mathematical achievements are found on clay tablets first unearthed by European archaeologists in the nineteenth century. Among other achievements the Babylonians were able to solve what we would today call quadratic equations and possessed rules for generating Pythagorean triplets of numbers.

The lecture provided an overview of Babylonian mathematics and then considered, among other things, the methods that the Babylonians used to obtain approximations to the square root of 2. Facsimiles of original Babylonian tablets were circulated and sample problems from their texts were posed as challenges for the audience to solve.

Here are some notes, references, sample problems and solutions.

The door prize, "A Concise History of Mathematics" written by Struik, was won by Nahid Golafsani.

• November 23, 1999

• Topic: A Mathematical 'Hodgepodge'
Speaker: Prof. Mark Spivakovsky (Mathematics, University of Toronto)

Every professional mathematician is familiar with the following experience. You meet someone at a party and they ask you what you do for a living. You reply "mathematician" with some trepidation, for you already know the common reaction: "I have always hated math and was never any good at it".

The greatest challenge in my work as a teacher is trying to transmit my enthusiasm for solving problems to my students. In this session I would like to share with you some problems that might get people, young and old hooked on problem solving with reasoning. Here are a few you can try your hand at:

• Which is greater: 9950 + 10050 or 10150?
• Prove that for every integer n>2, (n!)2 > nn

The door prizes went to Doug Sloan, Larry Rice and Shaloub Razak.

• October 28, 1999

Topic: Pell's Equation
Speaker: Prof. E. J. Barbeau (Mathematics, University of Toronto)

Which triangular numbers are also squares? For which Pythagorean triples do the smallest numbers differ by 1? These and similar problems lead to a type of diophantine equation known as Pell's equation: x^2 - dy^2 = k, where d and k are given integers and we want solutions in positive integers x and y.

This equation has a long history that goes back to sixth-century India, and over the past four hundred years, European mathematics have studied it, eventually developing a coherent theory that can be extended to certain diophantine equations of higher degree.

We will look at the theory of this equation and some of its higher degree analogues. In preparation, you might try to get positive solutions in the special cases when d is between 2 and 100 inclusive, and k takes the values, 1, -1 and 4. In the case of 4, for which values of d are there solutions in odd integers?

The door prize, Ed Barbeau's book called "After Math", was won by Lesley Mason. Congratulations Lesley!