GATE-IT-2004 Problem 20

This is a problem from the Graduate Aptitude Test in Engineering (GATE) for Information Technology (IT) in the year 2004. This problem requires solving a simple mathematical function using recursion.

Problem Statement

Given a function F(n) defined as follows:

F(n) = n - F(F(n-1))     for n > 0
F(0) = 1

Write a recursive function F(n) in Python that takes an integer n as an argument and returns the value of F(n).

Solution

The given function is defined recursively, so we can use the same approach to solve it. The base case is when n = 0, in which case the function returns 1. For n > 0, we call the function recursively with n-1, and pass the result to another call to the same function as an argument.

Here's the Python code for the function:

def F(n):
    if n == 0:
        return 1
    else:
        return n - F(F(n-1))

Testing

We can test our function by calling it with different values of n and checking if it returns the expected value. For example, we can test it for n = 4:

assert F(4) == 2

This assertion should pass, since F(4) = 4 - F(F(3)) = 4 - F(2) = 2.

Conclusion

In this problem, we learned how to solve a recursive function using recursion in Python. The problem tested our understanding of recursion and mathematical functions, and required careful analysis of the function definition in order to come up with the correct solution.