A loop invariant need not be true

We wish to cover a chessboard with dominoes,ロロ the number of black squares and the number of white squares covered by dominoes, respectively, placing a domino can be modeled by

If m x a + n x b is an invariant for the assignment a, b: = a + 8, b + 7, the values of m and n are

Which of the following is not an invariant of the assignment? m, n := m+2, n+3

If Fibonacci number is defined recursively as
\(F(n)= \begin{cases}0 \ \mathrm{n}=0 \\ 1 \ \mathrm{n}=1 \\ \mathrm{~F}(\mathrm{n}-1)+\mathrm{F}(\mathrm{n}-2) \text { otherwise }\end{cases}\)
to evaluate F(4), how many times F() is applied?

Using this recursive definition
\(a^n= \begin{cases}1 & \text { if } n=0 \\ a \times a^{n-1} & \text { otherwise }\end{cases}\)
how many multiplications are needed to calculate a¹⁰?

What is an invariant?

Define a loop invariant.

Does testing the loop condition affect the loop invariant? Why?

What is the relationship between loop invariant, loop condition and the input-output recursively?

What is recursive problem solving?

Define factorial of a natural number recursively

There are 7 tumblers on a table, all standing upside down. You are allowed to turn any 2 tumblers simultaneously in one move. Is it possible to reach a situation when all the tumblers are right side up? (Hint: The parity of the number of upside-down tumblers is invariant.)

A knockout tournament is a series of games. Two players compete in each game; the loser is knocked out (i.e. does not play anymore), the winner carries on. The winner of the tournament is the player that is left after all other players have been knocked out. Suppose there are 1234 players in a tournament. How many games are played before the tournament winner is decided?

Power can also be defined recursively as
\(a^n= \begin{cases}1 & \text { if } \mathrm{n}=0 \\ \mathrm{a} \times \mathrm{a}^{\mathrm{n}-1} & \text { if } \mathrm{n} \text { is odd } \\ \mathrm{a}^{\mathrm{n} / 2} \times \mathrm{a}^{\mathrm{n} / 2} & \text { if } \mathrm{n} \text { is even }\end{cases}\) 
Construct a recursive algorithm using this definition. How many multiplications are needed to calculate a¹⁰?