Purpose
The programming assignment will provide an exercise in using recursion. There are two problems which need to be solved. The first asks for a recursive function to calculate a Catalan number. The second is to solve a path-finding problem in two dimensional space. The assignment will require the usage of recurrence as well as good class design.
Problem 1: The good Mr. Catalan.
The Catalan number form a sequence of natural numbers that occur in many counting
problems. They are named after the mathematician Eugene Charles Catalan
(http://en.wikipedia.org/wiki/Eug%C3%A8ne_Charles_Catalan) and are defined by the
following recursive formula:
Write a program called Catalan which takes one argument and calls a recursive function which computes the nth Catalan number. The program then prints out the result to the console.
For instance,
python catalan.py 4
Would print out: 14
python catalan.py 10
Would print out: 16796
Problem 2: “The Greedy Robot” or “Lost in the Supermarket”
A robot is positioned on an integral point in a two-dimensional coordinate grid (x1, y1). There is a treasure that has been placed at a point in the same grid at (x2, y2). x1, x2, y1, and y2 are all integers. The robot can move up (North), down (South), left (West), or right (East). Commands can be given to the robot to move one position in one of the four direction. That is, “E” moves a robot one slot East (to the right) so if the robot was on position (3, 4), it would now be on (4, 4). The command N would move the robot one position north so a robot at position (4, 4) would be at (4, 5).
Write a recursive program which determines all the unique shortest paths from the robot to the treasure.
The input to the program will be the position of the robot (x1, y1) and the position of the treasure (x2, y2): four integers. For instance, an input of 1 3 -2 4 corresponds to the robot starting at position (1, 3) and needing to get to position (-2, 4). Do not worry about error conditions on input as we will assume the input is well formed. Read the input from the console.
The output of the program should be the listing of the unique paths followed by the number of unique paths. A path should be output as a string of characters with each character corresponding to a direction the Robot should move. For instance, NNENE corresponds to having the robot move North, North, East, North and East. This would be one answer to the input: 3 3 5 6, which corresponds to (3,3) - (5,6).
For the input 1 2 3 5 which corresponds to (1,2) - (3,5) the output should be:
NNNEE
NNENE
NNEEN
NENNE
NENEN
NEENN
ENNNE
ENNEN
ENENN
EENNN
Number of paths: 10
Turn-In
In a .ZIP file
A python program called catalan.py which computes the catalan number
The needed .py drivers and modules to solve the robot problem (recursively). Please use appropriate factoring.
