### Fold & Cut

What shapes can result from the following fold-and-cut process?

Take a piece of paper.

Fold it flat.

Make one complete straight cut.

Unfold the pieces.

Are all shapes possible?

Skip to content # Session Style: Try a Smaller Problem

### Fold & Cut

### Lockers: An Open-and-Shut Case

### Magic, Latin, & Sudoku Squares

### Mathemagical Card Tricks

### Mathematical Games

### Pigeonhole Principle and Parity Problems

### Probability

### Recruiting Change for a Dollar

### Simplex Locks

### The Futurama Theorem

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What shapes can result from the following fold-and-cut process?

Take a piece of paper.

Fold it flat.

Make one complete straight cut.

Unfold the pieces.

Are all shapes possible?

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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. The students walk the length of the corridor, opening and closing lockers according to a set of rules. How many lockers remain open? Which lockers? What if the rules were slightly different? Can you manipulate the rules to obtain specific outcomes? This collection of nine locker problems is suitable for...

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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 these special squares.

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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.

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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 description of such strategy, and you have to explain what the winning player should do so that this player wins regardless of his opponent’s moves.

These games may be presented as a single circle session, or individually in a circle or classroom.

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

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 solving session, or as individual teaser questions.

Problems are suitable for a math circle or classroom.

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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 for several circle sessions or the classroom.

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How many different ways are there to make change for a dollar? As mathematicians we often search for patterns in a problem. However, for this problem, there is no simple, predictable pattern to build to an answer, encouraging participants to reach outside their comfort zones and ponder alternative strategies in order to make progress.

This monetary problem is engaging, and classroom adaptable with multiple entry points.

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A simplex lock is a type of combination door lock that involves pushing-in buttons. Given the set of rules for using a 5-button simplex lock, how many different combinations are there?

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In the television show Futurama, Professor Farnsworth and Amy decide to try out their newly finished “Mind-Switcher” invention on themselves. When they try to switch back, they discover a key flaw in the machine’s design: it will not allow the same pair of bodies to be used in the machine more than once. Is there a way to restore their minds back to their original bodies?

The Futurama theorem is a real-life mathematical theorem invented by Futurama writer Ken Keeler (who holds a PhD in applied mathematics), purely for use in the Season 6 episode “The Prisoner of Benda”.