Mathematical Experiment 4

  1. Construct a program that simulates the game of ``network tracing''.
    1. Construct a subprogram that reverses any segment of a given sequence.
    2. Construct a subprogram that generates a random permutation of 1,2,...,n.
    3. Combine the previous two subprograms to form a game in which the player is limited to perform certain permutations of an arbitrary permutation of 1,2,...,n. For example, he is allowed to reverse the tail of a sequence, or he is allowed to ``reflect'' the sequence with respect to a ``gap''. The goal of the game is to restore a randomly generated permutation back to the ascending order.
    4. Generalize the construction to a two-dimensional game.
  3. Every binary operation on the set S = {1,2,...,n} is uniquely determined by its multiplication table, which is just an n×n matrix with entries in S. With respect to any binary operation of S one can define the corresponding Fibonacci sequence a1 = 1, a2 = 1, an+2 = an+1·an(n³ 1). Discover mathematical properties of such sequences from observations of various binary operations. This experiment can be done with Symphony.
  4. Draw a torus.