<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 itertools import product, permutations
from typing import List
import unittest

class Solution:
    def minimumMoves(self, grid: List[List[int]]) -> int:
        
        # Define a helper function to calculate the Manhattan distance
        def dist(x, y):
            return abs(x[0] - y[0]) + abs(x[1] - y[1])

        zeros = []
        spare = []

        for i, j in product(range(3), range(3)):
            stone = grid[i][j]
            if stone == 0:  
                zeros.append((i, j))
            if stone > 1:  
                spare.extend([(i, j)] * (stone - 1))

        all_permutations = list(permutations(spare))
        assert len(all_permutations[0]) == len(zeros)

        distances = []

        for per in all_permutations:
            total_distance = 0  # Initialize total distance to 0
            
            # Calculate the distance for each pair of corresponding elements
            for idx in range(len(zeros)):
                zero_cell = zeros[idx]   # Get the zero cell at index idx
                spare_cell = per[idx]    # Get the corresponding spare cell at index idx
                
                # Calculate the distance between the zero cell and the spare cell
                distance = dist(zero_cell, spare_cell)
                
                # Add the distance to the total distance
                total_distance += distance

            # Add the calculated total distance for this permutation to the list of distances
            distances.append(total_distance)

        # Find the minimum total distance across all permutations and return it
        minimum_distance = min(distances)
        return minimum_distance

22800