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- It can be said that “not all infinities are equal,” in other words, some infinite sets are “bigger” than others. Define the concept of a countable set. Give examples of such sets and show why they are countable. Then contrast this with examples of uncountable sets. In particular, choose a specific uncountable set and demonstrate why it cannot be countable.
- The Monty Hall problem

A probability puzzle that has appeared frequently in popular culture, the Monty Hall problem is a prominent example of how probability can be counterintuitive. Describe the classical version of the problem and the solution, justifying it with a probabilistic argument. Further illustrate your point by discussing other variants or generalizations of the problem. - The pigeonhole principle

Colloquially, the pigeonhole principle states that if n + 1 pigeons are roosting in n holes, there must be at least one hole with more than one pigeon. Though this is a simple concept, it’s a surprisingly deep theorem in mathematics with some unexpected ramifications. Describe the theorem mathematically and present several of its applications. - Logic gates and circuits

In computer architecture, circuit diagrams can be constructed from logic gates using a binary in- put/output system that draws from basic truth tables. Present the basics of logic gates and demon- strate their connection to symbolic logic. Discuss potential applications. - Topology

An extension of set theory and geometry, topology is a branch of mathematics which is concerned with continuous deformation (e.g. stretching or bending) of objects. Define basic concepts and results from topology such as the genus of an object, the torus and Klein bottle, topological equivalence, manifolds, etc. Given the nature of the subject, this type of project lends itself well to a very visual or hands-on presentation. - Tiling

Suppose you have a checkerboard with the top-left and bottom-right squares removed. Is it possible to cover the entire checkerboard with 2 × 1 dominoes so that no dominoes overlap and no square is left uncovered? This is a basic example of a tiling problem. Discuss multiple basic tiling problems and their solutions. This is another project that lends itself well to a visual or interactive presentation. - Fractals

A fractal can be thought of as a geometric figure that repeats a given pattern such that it has the same inherent structure however close you inspect it. Discuss basic fractals, examples of fractals in the physical world, and applications for fractals.

8. Graph theory

A graph is a mathematical structure consisting of nodes and edges that is used to model relationships between pairs of objects. Discuss the basic ideas of graph theory and some specific problems that can be solved using graph theory.

9. Game theory

An intersection of logic and probability, game theory is a field of study with applications spanning psychology, economics, computer science, biology, and more. Describe basic concepts of game theory and model a few different types of game.

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