Problem 1: A Bit of Fibonacci

• The Fibonacci sequence is a classic sequence whose \(n^{th}\) number can be defined recursively: \[F(n) = \begin{cases} 1 & \text{if } n \leq 1 \\ F(n-1) + F(n-2) & \text{otherwise}\end{cases} \]
• In partners or trios, define a function fib that returns the nth Fibonacci number

Problem 1b: Fibonacci Calls

• Switch who is typing!
• It can be interesting to track how many times the function is called with each \(n\) value
• Define a global dictionary which will have different values of n as the keys, and a counter associated with each
• Whenever your fib function is called, increment (or initialize) the correct count in your dictionary.
• When you call fib(5), how many times is fib(1) called?
  • You can just print it your counts after running fib(5)
• What about with fib(25)?

Problem 2: A-mazing Recursion

• We looked at one maze-solving algorithm earlier this semester
• Recursion can be another approach!
• The maze_solver.py program demonstrates one recursive approach, which also tracks the final path
• Running the program initially will just give you an interactive window with the maze
• Your task is to trace through the recursive algorithm by hand
• Keep in mind that each function call checks 4 directions! It can be useful to track which have been used so far

Problem 2b: A-mazing Solutions

• Add code to the bottom of the program to actually call the recursive solver and print out the found path
  • You know that the start point is always in the upper left corner
• Try it with larger mazes! Would our earlier approach worked with these mazes?

Sierpinski Carpet