<p>You are given a <strong>0-indexed</strong> 2D integer matrix <code>grid</code> of size <code>3 * 3</code>, representing the number of stones in each cell. The grid contains exactly <code>9</code> stones, and there can be <strong>multiple</strong> stones in a single cell.</p>

<p>In one move, you can move a single stone from its current cell to any other cell if the two cells share a side.</p>

<p>Return <em>the <strong>minimum number of moves</strong> required to place one stone in each cell</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/08/23/example1-3.svg" style="width: 401px; height: 281px;" />
<pre>
<strong>Input:</strong> grid = [[1,1,0],[1,1,1],[1,2,1]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> One possible sequence of moves to place one stone in each cell is: 
1- Move one stone from cell (2,1) to cell (2,2).
2- Move one stone from cell (2,2) to cell (1,2).
3- Move one stone from cell (1,2) to cell (0,2).
In total, it takes 3 moves to place one stone in each cell of the grid.
It can be shown that 3 is the minimum number of moves required to place one stone in each cell.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2023/08/23/example2-2.svg" style="width: 401px; height: 281px;" />
<pre>
<strong>Input:</strong> grid = [[1,3,0],[1,0,0],[1,0,3]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> One possible sequence of moves to place one stone in each cell is:
1- Move one stone from cell (0,1) to cell (0,2).
2- Move one stone from cell (0,1) to cell (1,1).
3- Move one stone from cell (2,2) to cell (1,2).
4- Move one stone from cell (2,2) to cell (2,1).
In total, it takes 4 moves to place one stone in each cell of the grid.
It can be shown that 4 is the minimum number of moves required to place one stone in each cell.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>grid.length == grid[i].length == 3</code></li>
	<li><code>0 &lt;= grid[i][j] &lt;= 9</code></li>
	<li>Sum of <code>grid</code> is equal to <code>9</code>.</li>
</ul>

from collections import deque

class Solution:
    def minimumMoves(self, grid: List[List[int]]) -> int:        
        target = tuple([(1, 1, 1) for _ in range(3)])
        if target == grid:
            return 0
        initial = tuple(tuple(row) for row in grid)
        q = deque([initial])
        visited = {initial}
        directions = {
            (1, 0),
            (0, 1),
            (-1, 0),
            (0, -1),
        }
        step = 0
        while q:
            size = len(q)
            step += 1
            for _ in range(size):
                curr = q.popleft()
                for i in range(3):
                    for j in range(3):
                        if curr[i][j] <= 1:
                            continue
                        for di, dj in directions:
                            curr_grid = list(list(row) for row in curr)
                            new_i, new_j = i + di, j + dj
                            if not 0 <= new_i < 3 or not 0 <= new_j < 3:
                                continue
                            curr_grid[i][j] -= 1
                            curr_grid[new_i][new_j] += 1
                            next_state = tuple(tuple(row) for row in curr_grid)
                            if next_state in visited:
                                continue
                            if next_state == target:
                                return step
                            visited.add(next_state)
                            q.append(next_state)
        return -1

            

            


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