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- What is game theory all about and where is it applied?
- 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].
- 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.
- 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.
- 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.
- 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].
- 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].
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