<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>

class Solution:
    def minimumMoves(self, grid: List[List[int]]) -> int:
        
        def get_state(g):
            return ''.join(str(cell) for row in g for cell in row)
        
        def move_stone(g, i, j, di, dj):
            # Create a new grid with the stone moved
            new_grid = [row[:] for row in g]  # Shallow copy of each row
            new_grid[i][j] -= 1
            new_grid[i + di][j + dj] += 1
            return new_grid

        target = "111111111"  # Target grid state as a string
        queue = deque([(grid, 0)])  # Tuple of (grid, move)
        visited = set([get_state(grid)])
        
        while queue:
            cur_grid, move = queue.popleft()
            cur_string = get_state(cur_grid)
            if cur_string == target: 
                return move
            
            for i in range(3):
                for j in range(3):
                    if cur_grid[i][j] > 1:
                        for di, dj in [(-1, 0), (1, 0), (0, -1), (0, 1)]:
                            if 0 <= i + di < 3 and 0 <= j + dj < 3:
                                next_grid = move_stone(cur_grid, i, j, di, dj)
                                next_state = get_state(next_grid)
                                if next_state not in visited:
                                    visited.add(next_state)
                                    queue.append((next_grid, move + 1))
        
        return -1  # Return -1 if no solution found
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