<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:
        
        pools = []
        zero_position = [] 
        for i in range(3):
            for j in range(3):
                if grid[i][j]>1:
                    for _ in range(grid[i][j]-1):
                        pools.append((i,j))
                if grid[i][j]==0:
                    zero_position.append((i,j))

        n = len(zero_position)
        if n == 0:
            return 0


        res = float('inf')
        for p in permutations(pools):
            dist=0
            for (x1,y1),(x2,y2) in zip(zero_position, p):
                dist += abs(x2-x1)+abs(y1-y2)
            res = min(res, dist)


        return res

16400