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# Applications

Number theory, the 15-puzzle, peg solitaire, the Rubik's cube, and more!

These quizzes take apply the fundamentals of group theory to interesting (and sometimes surprising) situations. For example, we'll use group theory to investigate the Rubik's Cube!

We'll see how many configurations there are, and gain deeper insights into some of the typical moves used to solve it.

Group theory also has applications in other branches of mathematics, and we'll explore how many of the most important observations in number theory can be easily explained in the language of group theory.

For example, have you ever seen a group theoretic proof of Fermat's Little Theorem, \[a^p \equiv a \text{ mod} \ p?\]

There are many games (especially those with geometric states) that have truly beautiful group theoretic interpretations. Besides the Rubik's Cube, we'll dive into the **15 puzzle** and **peg solitaire**:

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