<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:
        target = 1
        n = len(grid)
        queue = deque([(grid, 0)])
        visited = set()
        visited.add(tuple(tuple(row) for row in grid))
        
        def is_solved(current_grid: List[List[int]]) -> bool:
            for i in range(n):
                for j in range(n):
                    if current_grid[i][j] != 1:
                        return False
            return True
        
        def move_stones(current_grid: List[List[int]], x: int, y: int) -> List[Tuple[List[List[int]], int]]:
            directions = [(0, 1), (0, -1), (1, 0), (-1, 0)]
            moves = []
            for dx, dy in directions:
                nx, ny = x + dx, y + dy
                if 0 <= nx < n and 0 <= ny < n and current_grid[x][y] > 1:
                    new_grid = [row[:] for row in current_grid]
                    new_grid[x][y] -= 1
                    new_grid[nx][ny] += 1
                    moves.append((new_grid, 1))
            return moves
        
        while queue:
            current_grid, steps = queue.popleft()
            if is_solved(current_grid):
                return steps
            
            for i in range(n):
                for j in range(n):
                    if current_grid[i][j] > 1:
                        for new_grid, move_cost in move_stones(current_grid, i, j):
                            grid_tuple = tuple(tuple(row) for row in new_grid)
                            if grid_tuple not in visited:
                                visited.add(grid_tuple)
                                queue.append((new_grid, steps + move_cost))
        
        return -1
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