Games

1. What is game theory all about and where is it applied?
2. Study games and winning strategies - maybe explore a game where the winning strategy is not known. Analyze subtraction games (nim-like games in which the two players alternately take a number of beans from a heap, the numbers being restricted to a given subtraction set). References: [Ber], (this book contains hundreds of other games for which the complete analysis is unknown eg. Toads and Frogs), [Guy] (pay special attention to the last section where lots of questions are asked), Volume 1 of [Gard3].
3. Ten frogs sit on a log - 5 green frogs on one side and 5 brown frogs on the other with an empty seat separating them. They decide to switch places. The only moves permitted are to jump over one frog of a different colour into an empty space or to jump into an adjacent space. What is the minimum number of moves? What if there were 100 frogs on each side? Coming up with the answers reveals interesting patterns depending on whether you focus on colour of frog, type of move, or empty space. Proving it works is interesting also - it can lead to recursion. There is also a simple proof that is not immediately obvious when you start. Look for and explore other questions like this - one of the most famous is the Tower of Hanoi.
4. Try the ``Monty Hall'' effect. Behind one of three doors there is a prize. You pick door #1, he shows you that the prize wasn't behind door #2 and then gives you the choice of switching to door #3 or staying with #1, what should you do? Why should you switch? Make an exhibit and run trials to ``show'' this is so. Find the mathematical reason for the switch.
5. A graph is a mathematical structure made up of dots (called vertices) and lines joining pairs of dots (called edges). There are many games that can be played on graphs, and much mathematics involved in finding winning strategies. See the web site [MegaMath] for ideas.
6. Investigate card tricks and magic tricks based in mathematics. Some of the best in the world were designed by the mathematician/statistician Persi Diaconis. References: [Alb], [Gard3].
7. All forms of gambling are based on probability. Investigate how much casinos anticipate winning from you when you play black-jack, roulette, etc. Study a variety of lotteries and compare them. Should one ever buy a lottery ticket? Why does three of a kind beat two pairs in poker? Discover why the different types of hands are ranked as they are. References: [Gard1], [Col].