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.
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 we construct a fair game? What variations of this game are possible? Participants will explore these questions to determine how this game connects to other mathematical problems.
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.
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.
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.
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?
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”.
A Mad Veterinarian has created three animal transmogrifying machines…
While grappling with the posed questions, players will explore a set of problems, figuring out how and if the machines can complete a given transformation. Connections can be made to invariants, abstract algebra, graph theory, and Leavitt path algebra.
College students need to be matched with a roommate. They each make a list of who they prefer to room with. Given the preference lists for each individual, can we find a matching that is stable? That is, would any pair ask to change rooms because they would rather room together than with their current roommates? Explorations lead to new questions or new avenues to investigate using various mathematical methods including, but not limited to, combinatorics, graph theory, or matrices.
The purpose for having students play the game of Criss-Cross is to motivate them to explain the underlying mathematical reason governing who wins or loses. This exploration should lead the students to form, test, and ultimately prove conjectures about how to win at Criss-Cross. The game illustrates a beautiful application of the Euler characteristic and gives them practice at elementary counting techniques as well.
