I don't understand why this recursive program is looping, or its #s.

Question

i put in a println around the 7th line of code to see how it is working, i dont understand how to app is getting its numbers or why it is looping without a for statement. this demonstrates
recursion. the output i get is
value of n is 2
value of n is 3
Factorial of 3 is 6
value of n is 2
value of n is 3
value of n is 4
Factorial of 4 is 24
value of n is 2
value of n is 3
value of n is 4
value of n is 5
Factorial of 5 is 120
How is this possible when the only arguments passed in are 3, 4 , and 5, so that (n-1)*n would result in 6, 12, and 20. or at least thats what i thought it would produce, but instead,
it loops through 2 3 4 5 and i have no idea where its getting those numbers from. any help greatly appreciated.
public class Factorial {
      // this is a recursive method
     int fact(int n){
      int result;
     if(n==1) return 1;
     result = fact(n-1) * n;
     System.out.println("value of n is " + n);
     return result;
      class Recursion {
           public static void main(String args[]) {
                Factorial f = new Factorial();
                System.out.println("Factorial of 3 is " + f.fact(3));
                System.out.println("Factorial of 4 is " + f.fact(4));
                System.out.println("Factorial of 5 is " + f.fact(5));
      }

Answer 1

Recursion is closely related to the mathematical principle of proof by induction.
You use induction to see that your recursive method is correct. Take the factorial method
of the OP:
int fact(int n){
    if(n==1) {
        return 1;
    } else {
        return n * fact(n-1);
}Some peoples' heads explode when they see a method invoking itself, as though they were
seeing a snake swallowing its tail and disappearing! Yet nothing could be simpler or more
logical. These same people have no problem when f() calls g(), so suppose the previous
code had been written:
int fact(int n){
    if(n==1) {
        return 1;
    } else {
        return n * SOMEBODY_ELSES_FACTORIAL(n-1);
}Now we are just looking at ordinary code. Suppose method SOMEBODY_ELSES_FACTORIAL
does indeed compute the factorial function. In order to implement X you need to
know something about X, right? So we need to know a few things about factorial. These are:
1! = 1
n! = n * (n-1)!, for n>1
By inspection, our code handles the n=1 case, and because we assume SOMEBODY_ELSES_FACTORIAL works,
it handles the case n>1 successfully, too.
Finally, the principle of induction tells us SOMEBODY_ELSES_FACTORIAL can in fact be the method
we are defining, fact().
QED