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**Euclidean distance heuristic**?

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The 8-puzzle is a sliding puzzle that consists of a frame of numbered square tiles in random order with one tile missing. The goal is to slide the tiles until the whole puzzle is in order.

It turns out that A* can be used to solve this puzzle. If we represent a certain configuration of the board as a node, we can say its neighboring nodes are the 2-4 possible states achievable by moving the empty space. Thus the problem is reduced to a path finding problem. Our goal state is of-course the solved 8-puzzle.(The puzzle on the right in the image above)

Suppose we use the following heuristic for finding \(h\). The heuristic is a simple function \(h = \sum d(x_i)\) where \(d(x)\) is the Manhattan distance of each square \(x_i\) from its goal state.

For example for the start state in the image above , \(h = 1 + 1 + 1 + 3 + 1 + 3 + 3 + 2 = 15 \)

Suppose we solve the same start state using this heuristic coupled with A* search. How many moves would it take to solve the puzzle?

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Consider the given maze below. You want to get from the red dot to the yellow dot. What is the length of the path found by an A* search that uses a Manhattan distance heuristic?

**Details and assumptions**

- The length is simply the number of cells that constitute the path. This includes the start and ending cells.

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