# Math Circle:  2019 – 2020

This Math Circle is for middle and high school students who enjoy thinking about mathematical ideas, problems, and puzzles. We hope to get students involved in investigating some mathematical questions, talking about their ideas, coming up with guesses and suggestions, and arriving at answers and explanations.

Meetings will be held on occasional Sundays, from 2:00 to 4:00 PM, usually in room 154 of the Math Tower (231 W. 18th Ave) on the Columbus campus of the Ohio State University.
For directions, click the link on the left of this page.  Room 154 is the first door to the left after you enter the building from 18th Ave.

Teachers and parents are welcome to join in.
You are encouraged to invite other students who would enjoy this session. To help us organize, please send an email if you might attend this Math Circle meeting.

Info on floor tilings in the OSU Math Tower:  Tessellation Project.

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Autumn Semester 2019:

• December 8, 2019:  Vectors and dots.
Room:  154 Math Tower

What are vectors?  How to compute the angle between given vectors v and w ?  Or the area of the parallelogram they span?

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• November 10, 2019:  Complex numbers (and primes).
Room:  154 Math Tower

Standard integers are viewed as discrete dots on the number line.  “Complex integers” are numbers a + bi  (where  a  and  b  are standard integers), often viewed as dots on the number plane.  The prime number 5 is not a complex-prime, because 5 = (2 + i)(2 – i).  Which standard primes can be factored this way?

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• October 20, 2019:  Prime Numbers.
Room:  154 Math Tower

Prime numbers are the “atoms” for multiplication of numbers, studied since ancient times.  We will discuss several of their properties.

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• September 22, 2019:  Trigonometry.
Room:  154 Math Tower

Following student requests, we will discuss triangles and standard trig functions.  If time allows, we might mention polar coordinates and complex numbers.

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• August 25, 2019:  Permutations and symmetry.
Room:  154 Math Tower

Aspects of groups, combinatorics, and geometry.

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Topics in 2018-2019:

• April 28, 2019:  Cutting & pasting, Moebius, and hexaflexagons.
Room:  154 Math Tower

We will cut some Moebius strips, and discuss ways to build closed surfaces.
You might bring a pair of scissors to help with the constructions.
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• March 24, 2019:  Probability.
Room:  154 Math Tower

What is the probability of rolling a 7 with two dice?  Of getting a pair in a 5-card poker hand?
In a class of 20 people, how likely is it that two of them have the same birthday?
Flip a coin repeatedly until you get “heads”.  On average, how many flips are needed?

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• February 10, 2019:  Vectors and matrices.
Room:  154 Math Tower

What is a matrix? Why is matrix multiplication defined in that strange way?  Is matrix theory really a part of geometry?  Why is “linear algebra” so important in modern mathematics?   We will discuss those questions.

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• January 27, 2019:  Congruence with scissors.
Room:  154 Math Tower

For figures  A  and  B  in the plane,  when is it possible to cut  A  into pieces and then rearrange those pieces to make  B ?  We restrict attention to polygonal figures and straight cuts.
How do those ideas extend:  to non-straight cuts, to polyhedra in space, etc.

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• December 2, 2018:  Sequences.
Room:  154 Math Tower

We’ll discuss the difference operator and polynomial sequences.  We might continue with recursive sequences (generalizing Fibonacci), described using the shift operator.

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• November 11, 2018:  Modular Arithmetic.
Room:  154 Math Tower

Can you find integers  x, y  satisfying  95x – 58y = 1 ?  Euclid’s algorithm provides a systematic way to solve such equations.  Given integer  k,  we will see how those methods provide a tool for solving “congruences” in the system of integers modulo  k.

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• October 14, 2018:  Complex Numbers.
Room:  154 Math Tower (or to a larger room: Cockins 240)

Elementary school students learn about rational numbers (fractions).  The larger system of “real” numbers includes many irrationals, like sqrt{2}  and  pi.   Since real numbers correspond to points on the number line, the reals are useful for geometry.   “Complex” numbers are built from the reals together with the “imaginary” number  i = sqrt{-1}.  Complex numbers correspond to points in the plane, leading to interesting connections with plane geometry.

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• September 16, 2018:  Sizes of infinity.
Room:  154 Math Tower

Given two sets, how can we tell whether they have the same size?
(We discuss this topic every year, but there are different variations each time.)

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• August 26, 2018:  Stern’s Triangle.
Room:  154 Math Tower

Any fraction  a/b  has a left-child  L(a/b) = a/(a+b)  and a right-child  R(a/b) = (a+b)/b.
Starting from 1/1, apply  L  and  R  to obtain 1/2  and  2/1.  The operations  L  and  R  applied to those yield four new fractions.  Applying L and R repeatedly, which fractions  x/y  appear in this list?

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