This Math Circle is for middle school (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.

Teachers and parents are welcome to join in.

You are encouraged to invite other middle and high school students who would enjoy this session. To help us organize, please send an email if you might attend this Math Circle meeting.

For further information about this Math Circle, contact: Daniel Shapiro.

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

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

• 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*.

__________

• 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.

__________

• 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.

Note: We discussed complex numbers in earlier sessions. This partial repetition seems justified because there are several new participants.

__________

• 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.)

__________

• 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?

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Here are topics of meetings held during 2017-2018:

• May 20, 2018: Complex Numbers.

Room: 154 Math Tower

A real number is represented as a point on the “number line.” Operations of addition and multiplication become certain geometric motions of that line. Introduce the new number i satisfying i^2 = -1. (Then i is a square root of minus-one.) *Complex numbers* have the form c = a + bi, for real numbers a, b. Addition and multiplication of complex numbers are defined in natural ways. By representing c as a point on the “number plane,” we will investigate those operations as geometric transformations of that plane.

__________

• April 22, 2018: V – E + F.

Room: 154 Math Tower

A cube has V = 8, E = 12, and F = 6, counting the number of **V**ertices, **E**dges, and **F**aces. For a pyramid with square base: V = 5, E = 8, and F = 5. Compute V, E, F for other polyhedra, calculate the number V – E + F, and check that it equals 2 for many examples. Does it *always* equal 2?

__________

• March 18, 2018: Whole Number Topics.

Room: 154 Math Tower

Some problems about whole numbers and their digits. Ideas used to solve them might motivate further questions. Example: Define a sequence f(n) by setting: f(1) = 7, f(2) = 7^7 = 823543, f(3) = 7^(7^7) = 7^823543, and generally f(n+1) = 7^f(n). Find the units digit of f(9).

__________

• February 25, 2018: Several Folds, One Cut.

Leader: Alex Fu, student at Dublin Jerome HS

Room: 154 Math Tower

Fold a piece of paper a few times, and make one straight cut through the folded paper. The unfolded pieces are plane polygons. What polygons can be made in this way? Given a triangle T, can we find a fold-pattern and cut to produce T ?

__________

• January 21, 2018: Triangulating a Triangle.

Room: 154 Math Tower

Every triangle can be cut into 4 pieces that are triangles all congruent to one another. (How?) Can every triangle be cut into 3 congruent triangle pieces? Or 5 pieces? If not, can * some* triangle be cut into 3 congruent pieces? Is there one that can be cut into 5 congruent pieces?

__________

• December 3, 2017: Tilings.

Room: 154 Math Tower

Suppose a 10×10 checkerboard has two opposite corners removed. Can that figure be tiled by dominoes? If only one corner is removed, can the resulting figure be tiled by trominoes?

__________

• October 22, 2017: Infinity.

Room: 154 Math Tower

How can we tell when a set is infinite? Are there different sizes of “infinity”? For instance, does the unit interval [0,1] have the same size as the entire line? Does the set of rational numbers have the same size as the set of irrational numbers?

__________

• September 24, 2017: Recursive Sequences.

Room: 154 Math Tower

Fibonacci numbers provide an example of a sequence where each term is a specific linear combination of earlier terms. What other sequences are there following similar rules?

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=