Iteration:
In iteration, the loop body is repeatedly executed as long as the loop condition is true. Each time the loop body is executed, the variables are updated. However, there is also a property of the variables which remains unchanged by the execution of the loop body, This unchanging property is called the loop invariant. Loop invariant is the key to construct and to reason about iterative algorithms.

Recursion:
Recursion is another algorithm design technique, closely related to iteration, but more powerful. Using recursion, we solve a problem with a given input, by solving the same problem with" a part of the input, and constructing a solution to the original problem from the solution to the partial input.

(i) Establish the loop invariant at the start of the loop.
(ii) The loop body should so update the variables as to progress toward the end and maintain the loop invariant, at the same time.
(iii) When the loop ends, the termination condition and the loop invariant should establish the input-output relation

a loop variant, then it is true at four important points in the algorithm
(i) at the start of the loop (just before the loop).
(ii) at the start of each iteration (before loop body)
(iii) at the end of each iteration (after loop body).
(iv) at the end of the loop (just after the loop).

Let P(m, n) = m + 3n. Then
P(m, n) [m, n := m + 3, n -1]
= m + 3n [m, n := m + 3, n -1]
= (m + 3) + 3(n -1)
= m + 3 + 3n - 3
= m + 3n
= P(m, n)
Since (m + 3n) [m, n : = m + 3, n - 1] = m + 3n, m + 3n an invariant of the assignment m, n := m + 3, n-1.

Let P (p, c) = p - c. Then
P (p, c) [p, c := p + 1, c + 1]
= p - c [p, c := p + 1, c + 1]
= (p + 1)-(c + 1)
= p-c
= p(P, c)
Since (p - c) [p, c := p + 1, c + 1] = p - c, p - c is an invariant of the assignment p, c := p + 1, c + 1.

Recursion is another algorithm design technique, closely related to iteration. Using recursion, we solve a problem with a given input, by solving the same problem with a part of the input, and constructing a solution to the original problem from the solution to the partial input.

(I) In iteration, the loop body is repeatedly executed as long as the loop condition is true. Each time the loop body is executed, the variables are updated.
(ii) However, there is also a property of the variables which remains unchanged by the execution of the loop body.
(iii) This unchanging property is called the loop invariant. Loop invariant is the key to construct and to reason about iterative algorithms.

(i) Even though the action is the same, the state in which the action is executed is not the same. Each time we execute the action, the state changes.
(ii) Therefore, the same action is repeatedly executed, but in different states.
(iii) The state changes in such a way that the process progresses to achieve the desired input-output relation.

Let u - No. of tumblers right side up
v - No. of tumblers up side down
Initial stage: u = 0, v=7 (All tumblers upside down)
Final stage output: u=7, v=0 (All tumblers right side up)
Possible Iterations:
(i) Turning both up side down tumblers to right side up
u=u+ 2,   v=y-2   [u is even]
(ii) Turning both right side up tumblers to upside down.
u=u-2,  v=v+2  [u is even]
(iii) Turning one right side up tumblers to upside down and other tumbler from upside down to right side up.
u=u+1-1=u, v=y+1-1=v [u is even]
Initially u = 0 and continuous to be even in all the three cases. Therefore u is always even.
Invariant: u is even (i. e. No. of right side up tumblers are always even)
But in the final stage (Goal), u = 7 and v=0 i. e. u is odd.
Therefore it is not possible to reach a situation where all the tumblers are right side up.