Assignments, Mat115

There will be five homework assignments turned in, each worth 5% of your grade.

1. For Friday, 9/13: Explain the game of Fibonacci nim (FN), and particular explain
   1. how the game works, and
   2. how it is possible to win in every case, provided you are the first player and the initial number of tokens is not a Fibonacci number.
   3. Are there conditions under which player two is guaranteed to win?
   You might use any resource our book (or the web, or the library, etc.) provides (for example, you might want to look into the questions concerning FN to see if they give you any help.

   Keep your solutions to under five pages!

   If you want to practice, you might try this on-line version: http://www.math.ucla.edu/~tom/Game_Theory/fibonim.html.

2. For Friday, 9/27: Explain the game of Fibonacci nim (FN), and in particular:
   1. Explain mathematically how it is possible to win in every case, provided you are the first player and the initial number of tokens is not a Fibonacci number.
   2. Are there conditions under which player two is guaranteed to win a game of Fibonacci Nim, even if neither player makes a mistake?
   3. Show that every non-Fibonacci number can be written as a sum of non-consecutive Fibonacci numbers.
   Note:
   • Assume that both players know the rules of the game.
   • Mention the role of mistakes, but don't focus on them!
   • There is no rule that says that Fibonacci Nim must start on a non-Fibonacci number.
   • If I remove the smallest non-consecutive Fibonacci from a non-Fibonnaci's decomposition, does that guarantee that player 2 will not have a non-Fibonnaci whose smallest non-consecutive Fibonacci she could take? If so, player 2 would be in player one's shoes, and hence be able to win the game....
   You might use any resource our book (or the web, or the library, etc.) provides (for example, you might want to look into the questions concerning FN to see if they give you any help.

   Again, keep your solutions to under five pages!

3. For Friday, 10/25/02: hand in problem solutions for the problems of section 4.3.

4. For Friday, 11/08/02: hand in problem solutions for the problems of section 5.1.