**Some previous years of the Columbus Math Circle.**

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Here are topics discussed 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.

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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 x 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?

Topics of meetings held during 2016-2017:

• 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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Here are topics of meetings held during 2015-2016:

• April 10, 2016: Polynomials.

Room: 154 Math Tower

Algebra has developed a lot since the __cubic formula__ was discovered in the 1540s. We will discuss polynomials and some of their properties, especially those connected with roots and factors. Those ideas motivate the study of infinite series, as illustrated by some of Euler’s work dating from the 1700s.

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• March 20, 2016: Moebius bands, and surfaces.

Room: 154 Math Tower

We will discuss Moebius strips, make some paper models, and draw more abstract models on the board. Those pictures lead us to think about closed surfaces like the torus and Klein’s bottle. Hexaflexagons provide interesting models of Moebius bands. We will fold a few hexaflexagons and then investigate some of their mathematical properties.

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• February 14, 2016: Modular Arithmetic.

Room: 154 Math Tower

Impose a new rule on the integers stating: 12 equals 0. To avoid confusion we use the symbol ≡ (three parallel dashes) instead of = for equality in this new system. From that new rule we deduce that 13 ≡ 1, 20 ≡ 8, and -3 ≡ 9.

How can we tell whether given integers a and b are equal here? Is 1841 ≡ 2093 ?

Can you solve equations like 5x ≡ 3 and 3x ≡ 5 ?

This new system of “integers modulo 12” contains exactly 12 different elements. By posing standard problems in this new system and its relatives, we obtain insights into algebra.

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• January 17, 2016: Game Theory.

Leader: Prof. Crichton Ogle

Room: 154 Math Tower

Two-person games and mixed strategies.

We expect to have a second meeting on this topic the following Sunday, Jan 24.

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• November 15, 2015: Pascal’s Triangle.

Leader: Matisse Peppet (a high school student)

Room: 232 Cockins Hall

Pascal’s Triangle is an array of numbers built recursively using an addition rule. The numbers have many properties involving combinatorics, factorial formulas, and expansion of powers of (1 + x). We will discuss various aspects and generalizations of that triangle of numbers.

The triangle’s name refers to Blaise Pascal, a French mathematician who wrote about this “arithmetical triangle’ in the 1650s. However many other scholars investigated those patterns in earlier centuries.

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• October 25, 2015: Centers of Mass.

Where can two people sit on a seesaw and balance each other even though they have different weights? If weights of 3 and 5 pounds are placed one meter apart on a rigid wire, where is their equilibrium point, the spot where a fulcrum will balance them? Where is the center of mass of three several weights on a line? Or in the plane, or 3-dimensions?

Archimedes (who died in 212 BCE) wrote about those ideas, and his work is still worth investigating today.

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• September 27, 2015: Combinatorial Games.

Two players, named L and R, take turns playing a game (a finite game with complete information). A game G is “positive” if it is a win for L (no matter who goes first). A game is “negative” if it is a win for R. Games can be added and subtracted, and we consider the algebra of games.

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• August 30, 2015: 2:00 – 4:00 PM: Pythagorean Theorem and Pythagorean Triples.

Hundreds of different proofs of the Pythagorean Theorem have been published. Which one is the simplest? Which is the most elementary? Which of the proofs do you like the best?

Sometimes a right triangle has sides of integer length. For instance, the lengths 3, 4, 5 belong to a right triangle because 3^2 + 4^2 = 5^2, (that is: 9 + 16 = 25). How can we generate more examples of this type?

We seek whole numbers a, b, c satisfying a^2 + b^2 = c^2.

How is this question is related to lists of rational points on the unit circle?

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Here are topics of meetings held during 2014-2015:

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• May 3, 2015: 2:00 – 4:00 PM: Permutations and symmetry.

A cube C is symmetric in several ways. A quarter-turn about a face-center moves C back to its original position. Similarly, a (1/3)-turn about an axis connecting opposite corners also returns C to the same place. Those rotations are examples of “symmetry operations” of C.

How many different symmetry operations does C have?

A permutation of set A = {1, 2, 3} is a mixing of the elements. For instance, one permutation switches symbols 1 and 2 while leaving 3 fixed. Another one sends symbol 1 to 2, sends 2 to 3, and sends 3 to 1.

How many different permutations does A have?

Those two examples can be viewed with the same lens: an object (cube C or set A) and its symmetry group. We will discuss several examples, and talk about compositions of symmetry operations.

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• April 19, 2015: 2:00 – 4:00 PM: Exploding dots.

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• April 5, 2015 (Easter !! ): 2:00 – 4:00 PM: Measure, dimension, and fractal sets: Part 2.

This session is a continuation of the previous one, building on ideas of cardinality and measure.

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• March 22, 2015: 2:00 – 4:00 PM: Measure, dimension, and fractal sets.

A key example is the “Cantor set” built from the interval [0,1] on the number line by “removing middle thirds”. Here’s the procedure:

Step 1. Remove the open interval (1/3, 2/3). This leaves two separated intervals [0, 1/3], [2/3, 1].

Step 2. Remove the middle third of each of those two remaining intervals (i.e., remove (1/9, 2/9) and (7/9, 8/9) ). This leaves 4 smaller intervals, each of length 1/9.

Repeat this process:

Step n. Remove the middle third of each of the intervals left at the end of Step n – 1.

The collection of points remaining after all (infinitely many) steps is the Cantor set S.

**Question**: How big is S ?

The answer is a bit tricky. If we compute the total length of our set after n steps, and take the “limit” as n gets large, we conclude that S must have length 0 (or “measure zero”). On the other hand, S has infinitely many points.

For a better answer, we investigate how to measure sizes of subsets of the number line. This is closely related to the idea of *dimension*. A point, line, and plane have dimensions 0, 1, and 2. Our set S has a “dimension” that is between 0 and 1. The exact dimension of S (its *Hausdorff dimension*) equals

Dim(S) = log(2)/log(3) = .630929753571457 (approximately).

If we *measure* S according to its correct dimension, it turns out to have “length” equal to 1!

[Historical Note: The dimension of the middle-third Cantor set was first computed by Hausdorff himself in 1919, and appeared in the same paper in which he introduced what is now called Hausdorff dimension. However, the calculation of the Hausdorff length of S was found 82 years later, in 2001.]

We will explore different ways of computing the Hausdorff dimension of “fractal” sets (like the Cantor set). Participants should be familiar with some of the geometry of the line and plane. It’s also useful to know how to add up a geometric series. We will develop the rest of the ideas as we go along.

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• March 1, 2015: 2:00 – 4:00 PM: Sequences and Differences.

The sequence s: 0, 1, 4, 9, 16, 2.5, 36, . . . is made by squaring the counting numbers. This infinite sequence is described by a simple rule s(n) = n^2. For instance, s(3) = 9 and s(5) = 25. That compact formula contains the same information as the long sequence of numbers.

Given a sequence f, build its difference-sequence Δf by subtracting successive terms

Δf(n) = f(n+1) – f(n).

The square sequence s above has difference-sequence Δs: 1, 3, 5, 7, 9, 11, . . . Repeat the process to find the second-difference sequence is constant: ΔΔs: 2, 2, 2, 2, 2, . . . That constant sequence is pretty simple, and ΔΔΔs is all zeros.

Perform similar difference calculations for the cube sequence c: 0, 1, 8, 27, 64, 125, . . . Is ΔΔc a constant sequence?

We will investigate these types of questions, and the converse idea: Given a sequence of numbers, can we construct a polynomial formula that describes it?

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• January 18, 2015: 2:00 – 4:00 PM: The Fourth Dimension.

Led by Brad Findell.

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• November 9, 2014: 2:00 – 4:00 PM: Non-Standard Digits.

We use “positional notation in base ten” (the Hindu-Arabic number system) with standard digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Every whole number is represented uniquely as a string of those digits. You learned to count with this notation.

Maybe you have seen some other bases, like binary (base two) and hex (base sixteen). Can you do standard arithmetic in other bases? For instance, find (26) times (23) in base seven, without converting any numbers to base ten.

Sticking to base ten, let’s eliminate the digit 0, replacing it with the new digit X, where X represents “ten”. Is every whole number represented uniquely in this “X system” using digits {1, 2, 3, 4, 5, 6, 7, 8, 9, X}? For instance, the number after 79 (seventy-nine) is 7X (seventy-ten).

We will explore this X-system for number names, and some variations.

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• October 19, 2014: 2:00 – 4:00 PM: Groups and Symmetry.

Geometric figures exhibit various types of symmetry. For instance, a square rotated by ninety degrees looks the same as before. There seem to be eight different symmetries of that square (rotations and reflections). The set of all those symmetrical motions forms a “group” of eight elements. We will examine some small symmetry groups and discuss ways to represent similar groups.

Note: This topic requires knowledge of high school math. It’s not intended for students in middle or elementary school.

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• September 21, 2014, 2:00 – 4:00 PM: Tiling with dominoes and trominoes.

We investigate some old tiling puzzles. An 8-by-8 checkerboard can easily be tiled by dominoes. (Each domino covers two squares). A board with two opposite corners removed cannot be tiled by dominoes. [Why?]

If some other two squares were removed, could the resulting board be tiled by dominoes?

If one square is removed from the board, can the remaining 63 squares be tiled by straight trominoes?

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Here are topics of meetings held during 2013-2014:

• June 8, 2014 from 2:00 to 4:00 PM: Algebraic Numbers.

A complex number s is “algebraic” if it satisfies g(s) = 0, for some nonzero polynomial g(X) with integer coefficients. The “degree” of s is the smallest degree of a nonzero polynomial that kills s. For instance, rational numbers are algebraic of degree 1, while any irrational square root is algebraic of degree 2.

Can you prove that the cube root of 5 is algebraic of degree 3 ?

Is \sqrt{2} + \sqrt{3} algebraic? If so, what is its degree?

If s and t are algebraic, must s + t be algebraic?

cos(0) and cos(60) are algebraic of degree 1, while cos(30) and cos(45) have degree 2.

Are cos(20) and cos(72) algebraic?

Euclidean geometry provides construction tools: straight-edge and compass. Starting with a segment of length 1 we can use those tools to construct segments of other lengths, like 3, 1/6, and \sqrt{5}. In the 1800s mathematicians proved:

Every constructible length is an algebraic number whose degree is a 2-power.

This Theorem implies the impossibility of solving some famous ancient problems: trisecting a given angle, duplicating a cube, and constructing a regular heptagon.

• May 18, 2014 from 2:00 to 4:00 PM: Finding areas.

How can we measure the area of a region D drawn on paper? We might draw a grid of small squares on that paper and estimate area by counting the number of squares that are inside D. If D is a polygon with lattice point vertices (i.e. they have integer coordinates), “Pick’s Theorem” says that the area of D can be found by counting the number of lattice points in the interior and on the boundary.

For general D, is there a mechanical tool that can calculate the area of D after we use the tool’s pointer to trace around the boundary of D?

• April 13, 2014 from 2:00 to 4:00 PM: Farey Fractions.

The n-th Farey sequence F(n) is the list of fractions between 0 and 1 having denominator at most n. Those fractions are expressed in lowest terms, and listed in order of size. For instance, the sequence F(5) starts out: ( 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, . . . ). Students will explore those sequences and investigate their properties. Then we will try to explain (prove) why those observations hold true.

• March 9, 2014 from 2:00 to 4:00 PM: What is Pi?

For any circle, C = 2 pi R or A = pi R^2. These formulas for circumference and area are familiar, but lead to hard questions. If the number pi is defined using that circumference formula, why must that area formula hold true?

If we draw a circle of the size of Ohio on the earth’s surface and measure its area and circumference, will we get an accurate calculation of pi ?

Other shapes: Are there definitions for pi and R so that those two formulas will still work for a square? For a triangle? What shape has the largest value for pi? The smallest pi?

• February 9, 2014 from 1:30 to 3:30 PM: Fractions and repeating decimals.

Led by Brad Findell

The Division Algorithm comes in two flavors, producing either: (1) a quotient and remainder; or (2) a decimal. For instance, 18 divided by 7 is either: (1) quotient 2 and remainder 4; or (2) 2.57142 . . . Why do those algorithms make sense? Why does every fraction have a decimal expression that eventually repeats itself? What does this have to do with modular arithmetic?

• January 12, 2014 from 2:00 to 4:00 PM: Flexagons and Moebius bands.

Participants will fold flexagons and investigate some of their mathematical properties. This may lead to discussion of Moebius strips, and some other surfaces that arise from gluing the edges of a rectangular region.

• December 1, 2013 from 2:00 to 4:00 PM: Graphs.

We discuss the graphs that are made of vertices and edges. Think of a vertex as a “dot” and an edge as a path between two vertices.

Which graphs can be drawn on paper, tracing every edge without repetition and without lifting pencil from paper? Can this be done with a path that doesn’t cross itself?

Suppose a closed curve is drawn in the plane, perhaps with several “simple” self-crossings. Is there a way to color each of the resulting regions red or blue, so that adjacent regions have different colors? What does that situation have to do with graphs?

• November 10, 2013 from 2:00 to 4:00 PM: Complex numbers.

Represent a complex number a + bi as the point (a, b) in the plane. For numbers z = a + bi and w = c + di, their sum z + w is represented geometrically as the “vector sum”: the parallelogram diagonal. The product zw also has geometric meaning, involving polar coordinates. These ideas provide a powerful connection between algebra and plane geometry.

• October 13, 2013 from 2:00 to 4:00 PM: How can a square be covered by squares?

This question has different interpretations, depending on how terms are defined. We will discuss some possibilities and work toward answers.

• September 8, 2013 from 2:00 to 4:00 PM: Fibonacci numbers and related sequences.

The sequence of Fibonacci numbers is a popular elementary topic, involving interesting properties and generalizations. You may have worked with such sequences before, so we will discuss some of their less familiar properties. I hope some participants will bring calculators to this session.

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**ADVANCED MATH CIRCLE **:

(assuming some calculus background)

• March 23, 2014 from 2:00 to 4:00 PM: Topology of Closed Surfaces.

A square is made of thin, flexible rubber, and each edge is assigned a “directed-label”. The label is one of the letters a or b, and the direction is indicated by an arrow along that edge. Assume that each letter appears exactly twice. Now let’s glue together the two “a” edges, matching their arrows. Similarly glue the two “b” edges. This produces a surface with no edges. Different surfaces arise, depending on how the directed-labels were chosen. For example, we could get a Torus, or a Klein Bottle.

How many “essentially different” surfaces are possible?

What surfaces can arise if we put directed-labels on edges of a hexagon,

using letters a, b, c ? Can every surface be build by gluing edges of some polygon with directed-labels?

• February 23, 2014 from 2:00 to 4:00 PM: Generating functions.

A sequence {a_n} of numbers can be studied by analyzing the associated power series: \sum a_n x^n.

For instance, the constant sequence 1, 1, 1, 1, … corresponds to the function 1 + x + x^2 x^3 + . . . = 1/(1 – x).

What function corresponds to the sequence of Fibonacci numbers?

What do “partial fractions” have to do with this topic?

• January 26, 2014 from 2:00 to 4:00 PM: Cyclotomic polynomials.

This topic arises from questions about “roots of unity” and their algebraic properties. Those properties also relate to some interesting questions involving linear algebra, geometry, and combinatorics.

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Here are topics of meetings held during 2012-2013:

• May 19, 2013 from 1:00 to 3:00 PM: Non-standard digits.

Led by Daniel Shapiro

Standard base ten notation uses the first ten numbers as the digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Could a different collection of numbers be used as digits, instead of those ten?

If so, is every number expressed uniquely in that system? Do the usual algorithms for addition and multiplication still work?

• April 28, 2013: Cutting and pasting.

Led by Daniel Shapiro

Call two plane figures “equi-decomposable” if there is a way to break one figure into several pieces, and rearrange those pieces into the other figure.

Is every polygon equi-decomposable with some square? Is there a 3-dimensional version of this theory?

• April 7, 2013: Aspects of counting.

Led by Brad Findell and Daniel Shapiro

There are many ways to count the number of things in a collection. We will discuss one or two aspects.

The group of students might split in two, with one discussion aimed at younger students and a more advanced topic for high school students.

• February 24, 2013: What’s the deal with zero?

Led by Brad Findell

Why are students not allowed to “divide by zero”? Is there some good reason for that prohibition?

• November 18, 2012: What do means mean.

Led by Brad Findell

A “mean” of two numbers is some sort of average, right? But different contexts require different kinds of averages. We will explore several different means and the relationships among them.

• October 21, 2012: Tangent Lines*.*

Led by Bart Snapp

When is a line “tangent” to a given curve? We will draw lots of tangent lines to a given curve and explore some of the patterns geometrically and algebraically.

Participants are invited to bring a ruler (and pencil) to this session.

• September 23, 2012: What’s the Difference?

From a given row of numbers, construct a new row by taking differences of consecutive numbers. For instance, if the first row is 0, 1, 4, 12, 16 the row of differences is 1, 3, 8, 4.

If the initial row is the list of squares, 0, 1, 4, 9, 16, … , then the derived row is 1, 3, 5, 7, … . Repeating the process with that row yields the second derived row 2, 2, 2, … . We investigate various sorts of sequences using such differences.

The *Four Numbers Game* is a related situation but using “absolute differences” in a cyclic fashion, as described in: 4 Numbers Game [pdf]

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Here are topics of meetings held during 2011-2012:

• May 6: Triangulations.

Many polygonal figures can be built from triangular pieces. Can *every* polygon in the plane be broken into triangles? We will investigate triangulated figures in different ways, leading to results in geometry and number theory.

• April 15: Paper Folding Constructions.

Led by Bart Snapp.

In high school geometry, students learn to construct different objects using two “Eudlidean tools”: a straight-edge and a compass. For instance, given three points A, B, C, can you construct the circle that passes through them? Given a circle and a point P outside that circle, can you construct a line through P and tangent to that circle?

Today we will use “paper folding” instead of those other tools. You are invited to bring tracing paper, a ruler, and a marker, to help work out some of those constructions.

• March 25: Extremal Geometry.

Led by Professor Matthew Kahle.

We’ll begin with some problems involving length, and then move on to some involving area.

• Warmup problem #1: Given ten points in a unit square (side length one), show that some pair of points is no more than distance 1/2 apart.

• Warmup problem #2: Given nine points in a triangle of area one, show that some three of the points must form a triangle of area 1/4 or less. If you solve that — can you do better than nine? What is the smallest number of points for which this is true?

• March 11: Roots of Unity.

Complex numbers can be represented as points in the plane, and there are geometric ways to look at their addition and multiplication. A complex “root of unity” is a number z having some power equal to 1. That is: z^{n} = 1, for some positive integer n. We will discuss some of their properties and uses.

• February 26: The Fourth Dimension and Beyond.

Visitor Chris Altomare will lead this session.

232 Cockins Hall.

• February 5: Polyhedra and Symmetry.

How many rotations of space carry a given cube to itself? What other geometric figures are as symmetric as a cube?

• January 15, 2012: Combinatorial Games.

The two players, L and R, play a finite game with no hidden information. A game G that is a win for L (no matter who goes first) is called a “positive” game. A game that is a win for R is “negative”. If G is a win for the second player it is a “zero game”. Games can be added and subtracted, and we consider the algebra of games.

• December 18: Tilings.

An 8-by-8 checkerboard can easily be tiled by dominoes (where each domino covers two squares). A simple argument proves that a checkerboard with two opposite corners removed cannot be tiled by dominoes. So if two squares are removed from the board, when is there a domino tiling.

What about tromino tilings?

• December 4: Infinities.

Even if I don’t know how to count, I can detect whether two sets have the same size by matching their elements. This idea allows us to compare sizes of infinite sets.

• November 20: Bases.

We all express positive integers in base ten and own calculators that use that notation. But can you calculate with base B expressions, for another number B? Even if those are familiar, can you find (1/3) in base 5? Or (3/4) in base 7? Which “base B decimals” terminate?

This session will meet in 232 Cockins Hall.

• October 23: Vectors.

Some ways that vectors apply to elementary geometry.

This session will meet in room 154 (first floor).

• October 2: Mass point geometry.

Led by visitor Max Warshauer from Texas State University.

For further information on this topic see the article by Tom Rike posted at the San Jose Math Circle.

Meet in room 724.

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Here are topics of meetings held during 2010-2011:

October 24: *Walking the dog. Comparing lengths of paths.*

December 5: *Taxicab geometry.*

January 30: *Symmetry: Tilings and polyhedra.*

March 6: *Mobius strips: Ideas of topology.*

April 3: *Impossible scores.*

April 17: *Hyperbolic geometry: Five squares at each corner?*