ESP Biography

Given a chess position, can you find a game that leads to it? Can you find the shortest possible game? If you enjoy logic puzzles, and know the rules of chess, you should find this class fun! No particular skill level in chess is needed because we consider all possible games, not just "well-played" games. Many beautiful ideas and tricks will arise when we find a short (or shortest) game leading to a position.

In this class, we'll study wallpaper patterns. For two-dimensional wallpaper (the most common kind :) it turns out there are 17 distinct possible types of symmetries in a wallpaper pattern. We'll look at different patterns, see how to identify and classify their symmetries, how to construct patterns with particular symmetries, and briefly discuss why there are only 17 possible patterns.

Given a chess position, can you find a game that leads to it? Can you find the shortest possible game? If you enjoy logic puzzles, and know the rules of chess, you should find this class fun! No particular skill level in chess is needed because we consider all possible games, not just "well-played" games. Many beautiful ideas and tricks will arise when we find a short (or shortest) game leading to a position.

You've probably played it at some point: start with a board full of pegs except for one hole in the center, and remove pegs one by one by jumping over them. The object is to leave just a final peg in the center. But this "central" game is only one of numerous puzzles on a peg solitaire board. We can try to end the game in a hole other than the center, or to start the game with a different initial vacancy, or start the game with more than one vacancy, or end the game with some pretty pattern on the board... you get the point. Some of these puzzles are possible to solve, and others are unfortunately impossible. In this class, we'll look at some of the mathematics that help us either to solve peg solitaire puzzles, or to prove that certain puzzles are impossible.

An impartial combinatorial game is a game where both players choose from the same set of moves (unlike most games such as chess or checkers, where the players can only move their own pieces). Furthermore, the winner is the player who makes the last move. Impartial games are different from other games because there is no concept of score; the players are fighting over who gets the last move. We'll begin by studying the game of nim, which is the most fundamental impartial game. We will present a complete strategy for playing nim. Then we'll look at other impartial games and see how understanding nim can help us play them too.

Retrograde analysis chess problems are a type of chess logic puzzle. You are given a position, and asked to deduce something about it, such as: What were the last several moves played? Is there a game leading to this position? What is the shortest game leading to this position? We only assume that games are legal, not well-played, so no chess skill is required. If you are a chess player of any skill level that likes logic puzzles, you should find these problems enjoyable.

You've probably heard that it takes 7 shuffles to completely randomize a deck of cards. But did you know that 8 perfect shuffles will restore a (52-card) deck to its original state? In this class, we'll discuss the mathematics behind card shuffling, and take a look at some card tricks based on shuffling.

Symmetry is all around us, both in natural and man-made objects. A starfish has fivefold rotational symmetry, while a brick wall has translational symmetry. In this course, we'll discuss how mathematics helps us classify symmetries, and why there are 7 types of one-dimensional symmetries and 17 types of two-dimensional symmetries.

Let's play a game! In this course, we'll explore how a number system called the Surreal Numbers can help us understand many different games. Games that, at first glance, look very different turn out to be equivalent because they correspond to the same surreal number. We'll study a variety of games such as Nim, Hackenbush, Domineering, Amazons, and possibly others!

Dots and Boxes is a deceptively simple game you've probably played before. Starting with a grid of dots, two players take turns connecting two adjacent dots. A player that completes a box scores a point (by placing his or her initials in the box) and immediately takes another turn. When there are no more moves left, the player with more boxes wins. However, despite its simple rules, the game turns out to be quite rich in strategy. After playing several games, we will uncover several levels of strategies for the game. We will later relate the strategy of Dots and Boxes to the game of nim.