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.

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• 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. (So i is a square room of minus-one.) A “complex number” has the form c = a + bi, for real numbers a, b. Complex addition and multiplication are defined by simple formulas. By representing c as a point on the “number plane,” we view those operations as geometric motions of that plane.

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• 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?

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• 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).

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• 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 ?

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• 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?

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• 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?

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• 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?

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• 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?

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Here are topics of meetings held during 2015-2016:

• April 23, 2017: Integer Equations.

Room: 154 Math Tower

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• March 26, 2017: Counting and Probability.

Room: 154 Math Tower

In twenty coin flips, how likely is “three heads in a row”? What’s the probability of getting a pair in a five card hand? If a stick is cut in two places, how likely is it that the three pieces can form a triangle?

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• February 12, 2017: Area.

Room: 154 Math Tower

Can you find the area of a plane polygon if you are given its vertices? For instance, if A = (1, 2), B = (5, -1), and C = (6, 8) are points in the plane, find the area of triangle ABC. These ideas lead to investigations of polygons with lattice point vertices. (A “lattice point” is a point that has integer coordinates).

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• January 22, 2017: Simple Doodles.

Room: 154 Math Tower

Draw a closed curve (with some self-crossings) on a piece of paper.

What mathematical questions come to mind when you contemplate that picture?

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• December 4, 2016: Congruence by scissors.

Room: 154 Math Tower

Two polygons A and B in the plane are called “scissors-equivalent” if A be cut apart (with some straight cuts) and the pieces rearranged to make B. In that case, the areas of A and B must be equal.

Is every polygon scissors-equivalent to a square?

If so, then any two polygons of equal area must be scissors-equivalent.

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• November 13, 2016: Differences.

Room: 154 Math Tower

If f is a sequence of numbers, form its “difference-sequence” Δf by taking differences of successive terms. For instance, if s is the sequence of squares 0, 1, 4, 9, 16, 25, . . . then Δs is 1, 3, 5, 7, 9, . . . We will work out some examples and investigate different patterns.

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• October 16, 2016: Infinity.

Room: 154 Math Tower

We will talk about infinite processes, with the first focus on cardinality (counting). We might also discuss infinite decimals, sums of infinite series, etc.

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