Algorithm: Recursion

The idea of calling one function from another immediately suggests the possibility of a function calling itself. Recursion is a powerful general-purpose programming technique, and is the key to numerous critically important computational applications, ranging from combinatorial search and sorting methods that provide basic support for information processing to the fast Fourier Transform for signal processing.

Induction: a method of reasoning in which you use individual ideas or facts to give you a general rule or conclusion.

Reasoning: the process by which you reach a conclusion after thinking about all the facts.

Recursive programming is directly related to mathematical induction, a technique for providing facts about discrete functions. Proving that a statement involving an integer N is true for infinitely many values of N by mathematical induction involves two steps.
step 1: the base case is to prove the statement true for some specific value or values of N(usually 0 or 1).
step 2: the induction step is the central part of the proof. For example, we typically assume that a statement is true for all positive integers less that N, then use that fact to prove it true for N.

Examples,
1. N!
base case: N = 1. Returns a value without making any subsequent recursive calls. it does this for one or more special input values for which the function can be evaluated without recursion.

Induction step: N* factorial(N-1) and N decreases by one for each call, so the sequence of parameter values converges to the base case of N=1;

Converge: approach a limit as the number of terms increases without limit.

public static int factorial(int N) { 
   if (N == 1) return 1; 
   return N * factorial(N-1); 
} 

2. Towers of Hanoi
Rules:
Move only one disc a time.
Never place a disc on a smaller one.

 public static void solve(int first_disc, char aTower, char bTower, char cTower) {
        if (first_disc == 1) {
            System.out.println("Disk 1 on tower " + aTower + " moving to tower " + cTower);
        } else {
            solve(first_disc - 1, aTower, cTower, bTower);
            System.out.println("Disk " + first_disc + " on tower " + aTower + " moving to tower " + cTower);
            solve(first_disc - 1, bTower, aTower, cTower);
        }
    }