Archives

Fall 2024

During the Fall 2024 semester, we will have the math circle meetings in Cupples I Hall Room 199 on Saturdays, 3-5pm, broken down into two sessions:

  • Saturday 3-4pm: middle school session (intended for middle school students and advanced elementary school students)
  • Saturday 4-5pm: high school session (intended for high school students and advanced middle school students)

Note: Any student is welcome to attend any session. The descriptions are only guidelines.

DateTimeSession LeaderTopic
Sep 28, 20243pmBrandon SweetingToothpick Tricks
Sep 28, 20244pmAlan ChangAM-GM inequality
Oct 5, 20243pmBrandon SweetingCryptic Cards
Oct 5, 20244pmHao ZhuangConvexity of functions
Oct 12, 20243pmBrandon SweetingChip Solitaire
Oct 12, 20244pmLily ZhangAn Introduction to Continued Fractions and the Magic Table
Oct 19, 20243pmAngel RomanEuler Characteristic of Convex Polyhedra
Oct 19, 20244pmAngel RomanEuler Characteristic of Any Polyhedra
Oct 26, 20243pmGreg KneseKing Chickens
Oct 26, 20244pmRJ AcunaMarkov numbers
Nov 2, 20243pmMartha PrecupMap Coloring
Nov 2, 20244pmAlan Chang, Brandon SweetingAMC 10 practice
Nov 9, 20243pmSwarup Dhar, Mohao YiDefeating the Hydra, Part 1
Nov 9, 20244pmSwarup Dhar, Mohao YiDefeating the Hydra, Part 2
Nov 16, 20243pmBrandon SweetingDots-and-Boxes
Nov 16, 20244pmParker EvansGeometry and Combinatorics of Catalan Numbers
Nov 17, 2024 (Sunday)2-3pmMichael Chen, Silas JohnsonARML Power Contest (the target audience is high school students, but middle school students are welcome)
Please fill out this form if you plan to attend
Nov 23, 2024[no meeting]
Nov 30, 2024[no meeting]
Dec 7, 20243-4pmAlan Chang, Nick DanisLanguage Puzzles, Part 1
Dec 7, 20244-5pmAlan Chang, Nick DanisLanguage Puzzles, Part 2
[The next meeting will be in mid/late-January. Stay tuned!]

Descriptions

Sep 28, 2024

Toothpick Tricks: In this activity, students use toothpicks to explore geometry by forming squares within a grid. They begin by counting squares of different sizes and then solve puzzles, like removing two toothpicks to leave exactly two squares without loose ends. They will also explore variations, such as moving toothpicks to form images, encouraging critical thinking and the development of spatial reasoning skills.

AM-GM inequality: We will investigate the relationship between the arithmetic and geometric means, and use this to prove inequalities and study optimization problems.

Oct 5, 2024

Cryptic Cards: In this game of deduction, one player chooses a secret rule (e.g., “only black cards” or “black cards must be even”). The rest of the group tries to figure out the rule by playing cards one at a time, with the rule-setter telling them whether each card follows the rule. As students gather more clues, they can narrow down the possibilities. The rules start simple and become more challenging, promoting strategic thinking and logical reasoning. (Note: please bring a deck of cards if you have one.)

Convexity of functions: Concave up or down, that is a straightforward observation but indicating surprising quantitative relations.

Oct 12, 2024

Chip Solitaire: In this puzzle, students start with blue and red chips on a row of squares. The goal is to remove all chips, following two rules: only blue chips can be removed, and removing a chip flips the color of its neighboring chips. Students explore different starting arrangements to determine if complete removal is possible and find patterns to predict outcomes, enhancing their problem-solving and strategic thinking skills.

An Introduction to Continued Fractions and the Magic Table: Real numbers are often approximated by rational numbers. The best rational approximation of a real number x is a rational number a/b that is closer to x than any other rational number with a smaller or equal denominator. In this week’s math circle, we will explore an iterative method for finding the best rational approximation of any irrational number using continued fractions.

Oct 19, 2024

Euler Characteristic of Convex Polyhedra: Convex polyhedrons are the familiar three dimensional versions of polygons, such as cubes and pyramids. These shapes always have faces, edges, and vertices and we will call the number of faces F, the number of edges E, and the number of vertex V. The Euler characteristic is then defined as F-E+V. We’re going to build many different convex polyhedra (maybe even some nonconvex ones) and calculate the Euler characteristic. What is the value we get? How are the Euler characteristic of these polyhedra related to each other? We’ll look for patterns and then we will try to explain, that is prove the pattern we find. To do this, we will also discuss planar graphs, which are sets of vertices and sets of edges between the vertices on the plane such that the edges do not overlap.

Euler Characteristic of Any Polyhedra: We will quickly discuss the Euler characteristic of regular polyhedra, which is F-E+V, where F is the number of faces, E is the number of edges, and V is the number of vertex. We will discuss the planar graph as a way of proof for the Euler characteristic. Then we will explore the Euler characteristic of nonconvex polyhedron. The go-to example will be a polyhedral version of a torus, which is the shape of a donut. What is the Euler characteristic of such a polyhedron? Can we come up with some proof? We’ll also explore other types of shapes.

Oct 26, 2024

King Chickens: In a flock of chickens, between any two chickens there will be one chicken that always bosses (or pecks) the other. How do we decide which chicken is the king or the mega-boss chicken in a flock? We will give a reasonable notion of “king chicken” and then figure out how many kings we can end up with.

Markov numbers: We’ll introduce Markov numbers, and explain how they’re used in Diophantine Approximation.

Nov 2, 2024

Map Coloring: When creating a map, we want to make sure that regions sharing a boarder have distinct colors. An assignment of colors to the regions satisfying this rule is called a proper coloring. In this activity, we will explore proper colorings and look for patterns that explain how many colors are needed.

AMC 10 practice: We will discuss problems from the 2023 AMC 10B (AoPS page with problems and PDF version). Students should work on the problems before coming to the math circle. At the session, we will discuss problem solving strategies. Students will have the opportunity to present solutions to some questions from the 2023 AMC 10B. For more on the AMC (American Mathematics Competitions), see the Mathematical Association of America website.

Nov 9, 2024

Defeating the Hydra: We find ourselves facing off against a hydra, and not just any hydra a mathematical hydra. This mathematical creature has a tree-like body with heads branching out of its body. Whenever we defeat one of its heads it seems to grow more of them according to peculiar rules. Students go on an adventure to defeat this mathematical creature by exploring various strategies and starting configurations. Students answer questions about what kinds of hydras can be defeated and what are the optimal strategies for defeating said hydra, encouraging them to think algorithmically and challenging them to construct abstract arguments.

Nov 16, 2024

Dots-and-Boxes: Dive into the classic game of Dots-and-Boxes, where simple rules lead to complex strategy! In this interactive session, participants will learn the basics of the two-player game played on a rectangular grid of dots. Taking turns, each player draws an edge between two adjacent dots to create boxes. When a box is completed, the player claims it by marking their initial inside. The game continues until no moves are left, and the player with the most boxes wins. Through hands-on play, we’ll explore tactics, strategic planning, and ways to stay ahead of your opponent.

Geometry and Combinatorics of Catalan Numbers: In this activity, we will explore the Catalan numbers, a beautiful sequence that can be described as enumerating many different interesting geometric objets. We will focus on one such geometric perspective, and explore both a recursive way to count the Catalan numbers, as well as possible methods to explicitly count the numbers.

December 7, 2024

Language Puzzles: We will look at some questions from the North American Computational Linguistics Open (NACLO) Competition. No prior knowledge of linguistics or foreign languages is necessary. As we will see, many of these language puzzles are very similar to mathematical brainteasers. In the high school session (4pm), we will see how ideas from number theory and group theory show up in some linguistics olympiad problems.

(Note: WashU is hosting the NACLO 2025 Open Round on Thursday, January 23, 2025, 9am-12pm. Interested students can register here: https://naclo.org/ The top 10% qualify for the Invitational Round, and the top performers are invited to represent the USA in the International Linguistics Olympiad: https://ioling.org/)

Spring 2024

The regular time and place for the circle event is 3:30 pm-5pm every Sunday in Room 199, Cupples I, on the Danforth Campus at Washington University in St. Louis. All sessions are in-person unless otherwise notified. Please see the schedule for Spring 2024 below.

DateSpeakerTopicMentors
Jan 28, 2024Pooja JoshiVectors in Geometry (handout)Israel Fulton, Sydney Mayer, Vedul Palavajjhala
Feb 4, 2024Prof. Greg KneseSteps vs Squares
and Continued Fractions (handout)		Israel Fulton, Heidi Tamm
Feb 11, 2024Prof. Mladen WickerhauserPigeons and Rams(ey) (handout)Vedul Palavajjhala, Carlee Auld, Gavin John
Feb 18, 2024Devin AkmanElliptic Curves (handout)Sydney Mayer and Heidi Tamm
Feb 25, 2024Dr. Silas JohnsonARML Power ContestVedul Palavajjhala, Sydney Mayer, Carlee Auld
Mar 3, 2024Dr. Blake ThortonThe Island of the Blue Eyed Islanders
Mar 10, 2024Ricardo J. AcunaEquivalence relations (handout)
Mar 24, 2024Prof. Matt KerrOf Cattle and Cryptanalysis (handout)Nathan Johnson, Sydney Mayer
Mar 31, 2024Prof. Martha PrecupMathematical GamesNathan Johnson, Sydney Mayer
Apr 7, 2024Prof. Nan LinPurple Comet Math MeetAaron Lin
Apr 14, 2024Dr. Silas JohnsonBidding Games (handout)Nathan Johnson, Sydney Mayer
Apr 21, 2024Prof. Xiang TangCubic Roots and Diophantine ApproximationNathan Johnson

Important dates for math competitions

AMC8: Jan 18 – 24, 2024

MATHCOUNTS:

  • Chapter: February 1 – 29, 2024
  • State: March 1 – 31, 2024
  • National: May 12 – 13, 2024

MCTM Middle School Contest:

  • State: March 2, 2024

Purple Comet: April 2 – 11, 2024

Fall 2023

The regular time and place for the circle event is 3:30 pm-5pm every Sunday in Room 199, Cupples I, on the Danforth Campus at Washington University in St. Louis. All sessions are in-person unless otherwise notified. Please see the schedule for Fall 2023 below.

DateSpeakerTopicMentors
Sep 17, 2023Prof. Nan LinPlane Geometry in AMC Tests: Angle Chasing (handout)Allison Berman, Carson Brame, Gavin John
Sep 24, 2023Nathan LesnevichCryptography: The math of making and breaking codes (handout)Carson Brame, Sophia Dykstra, Gavin John
Oct 1, 2023John NaughtonIntroduction to Knot Theory (handout)Carson Brame, Sophia Dykstra, Gavin John
Oct 8, 2023Pooja JoshiGroups and Symmetries (handout)Gavin John, Aaron Lin
Oct 15, 2023Akshata PisharodyUsing Topological Invariants to Win at GamesCarson Brame, Gavin John, Lexy Sokolowski
Oct 29, 2023Prof. Nan LinARML Power ContestGavin John
Nov 5, 2023Prof. Nan LinAMC 10 mock testGavin John, Aaron Lin, Sydney Mayer
Nov 12, 2023Prof. Wanlin LiQuadratic number fieldsCarson Brame, Sophia Dykstra, Lexy Sokolowski
Nov 19, 2023Devin AkmanFinite FieldsCarson Brame, Sophia Dykstra
Dec 3, 2023Prof. Alan ChangMathematics of the Rubik’s cubeCarson Brame, Sophia Dykstra

Important dates for math competitions

AMC10/12: Nov 8, 2023 (Test A) and Nov 14, 2023 (Test B)

Spring 2023

DateSpeakerTopicModeMentors
March 5, 2023Prof. Nan LinCalculate π by throwing Buffon’s needle (handout, computer demo)in-personIsaac Anderson, Sydney Mayer, Aaron Lin
March 12, 2023Prof. Aliakbar DaemiCutting and pasting polygonsin-personBowen Peng, Aaron Lin
Mar 26, 2023Devin AkmanHow to Hang Pictures Poorly Using Math? (handout)in-personGyrie Gu, Junyi Yao, Bowen Peng
Apr 2, 2023Prof. Xiang TangHow to approximate √2? (handout)in-personGyrie Gu, Jason Yang, Bowen Peng
Apr 9, 2023Rachel WuDoubling the cube (straightedge and compass constructions) (handout)in-personJason Yang, Isaac Anderson, Lexy Sokolowski
Apr 16, 2023Shibashis MukhopadhyayDivisibility in integers: Primes, Euclidean Algorithm, and more (handout)in-personIsaac Anderson, Lexy Sokolowski, Junyi Yao
Apr 23, 2023John NaughtonChaotic Dynamics (handout)in-personSydney Mayer, Isaac Anderson, Gyrie Gu
Apr 30, 2023Prof. Matt KerrMathematical Recreationsin-personJason Yang, Lexy Sokolowski, Leah Qin

Before 2023

Spring 2022

  • Mar 6: Silas Johnson, ARML Power Contest
  • Mar 13: Martha Precup, Graph theory and Ramsey numbers
  • Mar 20: Shuhao Cao, Mathematics and Logic through Puzzles
  • Mar 27: Xiang Tang, How to calculate $\pi$?

Fall 2021

  • Oct 17: Karl Schaefer, Tic-Tac-No! Strategy and Symmetry
  • Oct 24: Jeet Sampat, An introduction to Sudoku and its different variants
  • Oct 31: Ben Wormleighton, Patterns from cluster algebras
  • Nov 7: Silas Johnson, ARML Power Contest
  • Nov 14: Shuhao Cao, The game of Go, the most mathematical game human ever invented
  • Nov 21: Michael Landry, Gigantic Numbers

Spring 2021

  • Feb 14: Victor Wickerhauser, Error Correcting Codes
  • Feb 28: Silas Johnson, ARML Power Contest
  • Mar 7: Xiang Tang, The Field of Cubic Roots
  • Mar 14: Aliakbar Daemi, Cutting and Pasting Polygons
  • Mar 28: Shuhao Cao, How to gamble like a mathematician if we must
  • Apr 11: Gregory Knese, The 100 prisoner problem and other combinatorial surprises
  • Apr 18: Adeli Hutton and Hyojeong Son, Chip-firing on Graphs

Fall 2020

Spring 2020

Fall 2019

Spring 2019

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

​Spring 2016

Fall 2015

Previous Semesters