We assume that the poles are arranged in a row, and that each instruction to move a disc specifies its number and whether to move it left or right. To solve the problem, our goal is to issue a sequence of instructions for moving the discs. The task is to move the stack of discs to another pole, while obeying the following rules: The discs differ in size and are initially arranged on one of the poles, in order from largest (disc n) at the bottom to smallest (disc 1) at the top. We have three poles and n discs that fit onto the poles. No discussion of recursion would be complete without the ancient Towers of Hanoi problem.

The function gcd() in euclid.py is a compact recursive function whose reduction step is based on this property.

We can efficiently compute the gcd using the following property, which holds for positive integers p and q: If p > q, the gcd of p and q is the same as the gcd of q and p % q. For example, the greatest common divisor of 102 and 68 is 34 since both 102 and 68 are multiples of 34, but no integer larger than 34 divides evenly into 102 and 68. The greatest common divisor (gcd) of two positive integers is the largest integer that divides evenly into both of them. Such a proof suffices to show that the statement is true for all n: we can start at the base case, and use our proof to establish that the statement is true for each larger value of n, one by one. For example, we typically assume that a statement is true for all positive integers less than n, then use that fact to prove it true for n.

The induction step is the central part of the proof.

The base case is to prove the statement true for some specific value or values of n (usually 0 or 1). Proving that a statement involving an integer n is true for infinitely many values of n by mathematical induction involves two steps. Recursive programming is directly related to mathematical induction, a technique for proving facts about discrete functions. For factorial(), the reduction step is 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. Furthermore, the sequence of parameter values must converge to the base case. It relates the function at one (or more) inputs to the function evaluated at one (or more) other inputs.

The reduction step is the central part of a recursive function.

It does this for one or more special input values for which the function can be evaluated without recursion.

The base case returns a value without making any subsequent recursive calls.

Our factorial() implementation exhibits the two main components that are required for every recursive function. We can trace this computation in the same way that we trace any sequence of function calls. Then it properly computes the value n! = n × ( n-1)! = n × ( n-1) × ( n-2) ×. You can persuade yourself that it produces the desired result by noting that factorial() returns 1 = 1! when n is 1 and that if it properly computes the value N! is easy to compute with a for loop, but an even easier method, used in factorial.py is to use the following recursive function: The 'HelloWorld' program for recursion is to implement the factorial function, which is defined for positive integers n by the equation: n! = n × ( n-1) × ( n-2) ×. 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 methods that provide basic support for information processing (Chapter 4) to the Fast Fourier Transform for signal processing. The function-call mechanism in Python supports this possibility, which is known as recursion. The idea of calling one function from another immediately suggests the possibility of a function calling itself.

Hanoi towers n bits space