Manhattan distance is not an admissible heuristic for the problem of moving a rook from square A to square B in the smallest number of moves, given the rook's movement restrictions.

Here's why:

Admissible Heuristic:

A heuristic is considered admissible if it never overestimates the true cost $($number of moves in this case$)$ to reach the goal state from any given state.

It always underestimates or equals the actual cost.

Manhattan Distance:

Manhattan distance calculates the distance between two points on a grid as the sum of the absolute differences of their horizontal and vertical coordinates.

Rook's Movement:

A rook can move any number of squares vertically or horizontally, but it cannot jump over other pieces.

The Problem:

Manhattan distance can overestimate the number of moves required for a rook when other pieces are present.

It assumes unobstructed movement, ignoring potential blockages.

Example:

Consider a rook at square A $(1,1)$ and the goal at square B $(5,5).$

Manhattan distance $=4+4=8.$

However, if a piece blocks the direct path, the rook might require more moves to navigate around it$,$ making the true cost greater than $8.$

Admissible Alternative:

A more appropriate heuristic would account for potential obstacles and consider the rook's movement constraints:

Count the minimum number of horizontal and vertical moves required to reach the goal, ignoring obstacles.

This heuristic never overestimates the true cost, making it admissible.