<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
import copy

class Solution:
    def minimumMoves(self, grid: List[List[int]]) -> int:
        # maintain a queue of states
        # for each state, find all cell > 1
        # add 1 to surrounding cells, and append to queue
        # if the current state has no 0s, return count

        count = 0 

        queue = deque()
        queue.append(grid)

        tuples = [(-1,0),(1,0),(0,-1),(0,1)]

        def convTuple(li):
            res = tuple(tuple(x) for x in li)
            return res

        ms = set()

        def isValid(x,y):
            return 0 <= x < 3 and 0 <= y < 3

        while queue:
            n = len(queue)
            for i in range(n):
                curr = queue.popleft()
                tups = convTuple(curr)
                if tups in ms:
                    continue
                ms.add(tups)
                zero_count = 0
                for i in range(3):
                    for j in range(3):
                        if curr[i][j] == 0:
                            zero_count += 1
                if not zero_count:
                    return count
            
                for i in range(3):
                    for j in range(3):
                        if curr[i][j] > 1:
                            for dx, dy in tuples:
                                nx, ny = i+dx, j+dy
                                if isValid(nx,ny):
                                    curr[i][j] -= 1
                                    curr[nx][ny] += 1
                                    queue.append(copy.deepcopy(curr))
                                    curr[i][j] += 1
                                    curr[nx][ny] -= 1

            count += 1
21200