An 8-puzzle is a 3×3 grid of tiles, numbered 1-8, with the last square in the grid being empty. A tile can be slid into the blank spot, thus changing the configuration of the puzzle. For example,

2 4 71 3 5 6 8

Can become any of these:

2 4 7 2 4 2 4 7 1 3 1 3 7 1 3 85 6 8 5 6 8 5 6

The goal is to arrange the tiles in this order:

1 2 34 5 67 8

Given an arbitrary arrangement of tiles, can this goal state be reached? Not necessarily; half of all permutations are such that reaching the goal state is impossible. (All states have either odd or even parity; no state with odd parity is reachable from any state with even parity, and vice-versa. The proof relies on a bit of number theory, not relevant here, and not essential for this program.) Your task is to write a program that determines if the goal state can be reached, and if so, the series of moves (path) needed to reach it.

Your input is a text file with a series of puzzles. Each is a 3×3 grid of integers, with 0 indicating the blank square. For each, output is either a statement that no path exists, or a listing of the tiles (by number) that must be moved, in order, to reach a solution.

Determining a path exists: You will be searching through possible puzzle states, tracking already-visited states so you don’t return to a state you’ve already examined. This will eventually search all states reachable from your starting state. If none of them is the goal–it’s not reachable! (This isn’t as awful as it sounds; from any state, there are only about 180,000 reachable states, and moving from one state to another is a fast operation. For larger problems–a 15-puzzle or 24-puzzle–you’d definitely want a more efficient determination.) Of course, if you want to read up on parity and check that directly, that’s OK too.

Finding the path: Use the A* algorithm. You will need a data structure that stores puzzle states you have already checked (so you don’t visit them again), and a ‘pending’ queue which can keep unvisited states ordered by estimated distance from the goal state. Use the Manhattan (city-block) distance from the goal as your heuristic. That is, for each tile, how far away is this tile from its goal position?

Submission: Submit your source code (C++) and text file showing solutions.

* You have until 11:00 pm central time US, 2/09/20.

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