Big-O notation is defined as follows: The function f(n) is O(g(n)) if there are positive integers c and N such that f(n) ≤ c · g(n) for all n > N. Is the relation “f(n) is O(g(n))” reflexive? symmetric? transitive? Justify your answers. “Θ(g(n))” is defined as follows: f(n) is Θ(g(n)) if and only if (f(n) is O(g(n)) & g(n) is O(f(n))). Is the relation “f(n) is Θ(g(n))” reflexive? symmetric? transitive? Justify your answers

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Eight puzzle problem

The N-puzzle problem contains N tiles (N+1 including empty tiles) where the value of N can be 8, 15, 24, etc. The puzzle is divided into the square root of (N+1) rows and the square root of (N+1) columns. These problems have an initial state or configuration and a goal state or configuration. The puzzle can be solved by shifting the tiles one by one in the empty space. The objective is to place the numbers on the tiles to match the final state using the empty space. The other name for the N-puzzle problem is the sliding puzzle.