Homework 1:

1) Towers of Hanoi

a) Obtain a general expression for the minimum number of moves M as a function of the number of discs N.

b) Assuming one second per move, calculate the time T in seconds required to complete the N=64 problem ("Tower of Brahma Legend"), and compare it to the age of the universe (about 15 billion years).

c) What N-tower could a person complete working 8 hours a day for 50 years?

2) Read pages 81 to 118 of Gleick's book. Be ready to answer questions posted in:

http://www.sewanee.edu/physics/bridge/GLEICK.html

3 Draw by hand the first three iterations of the fractals generated by removal according to the following rules:

a) "FRACTAL **+**": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the four corner squares (to get the **+ **shape). Repeat for each remaining square.

b) "FRACTAL **H**": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the top and bottom mid- squares (to get the H shape). Repeat for each remaining square.

c) "FRACTAL **X**": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the top, bottom, left and right mid- squares (to get the X shape). Repeat for each remaining square.

d) "FRACTAL **O**": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the center square (to get the square O shape). Repeat for each remaining square. What is the "technical" name of this fractal?

Can you calculate the fractal dimension of the above objects?