Work Backwards – MathCircles.org

Cup Stacking

By:

Topic(s):

Mathematical Games, Problem Solving / General

Supporting Resources:

Facilitator Guides, Handouts, Photos & Videos, References

Begin with a row of cups and end with all of the cups in a single stack. Rules: 1. Count the number of cups in a stack. That stack must jump that number of spaces. For example, 1 cup can only move 1 space; 2 cups have to move 2...

Lockers: An Open-and-Shut Case

By:

Elgin Johnston

Topic(s):

Number Theory

Supporting Resources:

Handouts

A classic Math Circle problem! At a large high school, there are 10000 lockers. The lockers are numbered, in order, 1, 2, 3, . . . , 10000, and to start, each locker is closed. There are also 10000 students, also numbered 1, 2, 3, . . . , 10000....

Magic, Latin, & Sudoku Squares

By:

Matthias Beck

Topic(s):

Algebra / Arithmetic, Combinatorics, Mathematical Games, Problem Solving / General

Supporting Resources:

Facilitator Guides

Squares and numbers, numbers and squares. There is something very satisfying about arranging numbers in a square formation, following specific rules, whether it is a Magic Square, Latin Square or Sudoku. This is probably why Sudoku puzzles are so popular. This session touches on some of the deep mathematics behind...

Match or No Match

By:

Sharon Lanaghan

Topic(s):

Combinatorics, Mathematical Games

Supporting Resources:

Facilitator Guides

In this session, participants will explore the Match-No Match game: two players each draw one chip out of a bag – if the color of the chips match Player 1 wins, if not Player 2 wins. Under what conditions is this a fair game? How do we know? How can...

Mathemagical Card Tricks

By:

Stephanie Santorico

Topic(s):

Algebra / Arithmetic, Mathematical Games, Problem Solving / General

Supporting Resources:

Facilitator Guides, References

There are many card tricks based on simple mathematics as opposed to sleight of hand. In this session, participants will play with a number of such tricks, test them out and work on discovering the math underneath, with a goal to formalize the mathematics that makes the trick work.

Mathematical Games

By:

Tatiana Shubin

Topic(s):

Mathematical Games, Problem Solving / General

Supporting Resources:

Facilitator Guides

This session includes 15 games using manipulatives or paper and pencil. The goal is to decide which one of the two players has a winning strategy. To solve a game means to find a winning, or a non-losing, strategy for one of the players. An answer must include a detailed...

Pigeonhole Principle and Parity Problems

By:

Brian Conrey, Corinne Cristiani

Topic(s):

Algebra / Arithmetic, Combinatorics, Geometry, Number Theory, Parity / Invariants, Problem Solving / General

Supporting Resources:

Facilitator Guides

The pigeonhole principle states that if n pigeons are put into m cubbies, with n > m, then at least one cubby must contain more than one pigeon. Parity problems deal with odd and even integers. Here is a collection of problems that can be used in a single problem...

Place Value Problems

By:

Harold Reiter

Topic(s):

Algebra / Arithmetic, Number Theory

Supporting Resources:

Facilitator Guides

In this session, we’ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should understand profoundly. Several example puzzles are followed by a rich selection of over 30 additional problems to explore. This...

Probability

By:

Elgin Johnston

Topic(s):

Combinatorics, Geometry, Probability and Statistics

Supporting Resources:

Facilitator Guides

Some probability problems can be solved by drawing a picture; this approach is sometimes called geometric probability. Other approaches can include experimentation, looking at smaller cases, looking at extreme cases, recursion, or carefully listing possibilities. This session includes ten problems that can be explored alone or in sets, providing material...

Tiling With Pentagons

By:

Judith Covington

Topic(s):

Geometry

Supporting Resources:

Facilitator Guides, Featured in MCircular, References

A pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. The plane cannot be tiled with regular pentagons. However, are there any convex pentagons that can tile the plane? This session explores various pentagons and their tiling abilities. From 1918...

Session Style: Work Backwards