<p>On an <code>2 x 3</code> board, there are five tiles labeled from <code>1</code> to <code>5</code>, and an empty square represented by <code>0</code>. A <strong>move</strong> consists of choosing <code>0</code> and a 4-directionally adjacent number and swapping it.</p>

<p>The state of the board is solved if and only if the board is <code>[[1,2,3],[4,5,0]]</code>.</p>

<p>Given the puzzle board <code>board</code>, return <em>the least number of moves required so that the state of the board is solved</em>. If it is impossible for the state of the board to be solved, return <code>-1</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/29/slide1-grid.jpg" style="width: 244px; height: 165px;" />
<pre>
<strong>Input:</strong> board = [[1,2,3],[4,0,5]]
<strong>Output:</strong> 1
<strong>Explanation:</strong> Swap the 0 and the 5 in one move.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/29/slide2-grid.jpg" style="width: 244px; height: 165px;" />
<pre>
<strong>Input:</strong> board = [[1,2,3],[5,4,0]]
<strong>Output:</strong> -1
<strong>Explanation:</strong> No number of moves will make the board solved.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/06/29/slide3-grid.jpg" style="width: 244px; height: 165px;" />
<pre>
<strong>Input:</strong> board = [[4,1,2],[5,0,3]]
<strong>Output:</strong> 5
<strong>Explanation:</strong> 5 is the smallest number of moves that solves the board.
An example path:
After move 0: [[4,1,2],[5,0,3]]
After move 1: [[4,1,2],[0,5,3]]
After move 2: [[0,1,2],[4,5,3]]
After move 3: [[1,0,2],[4,5,3]]
After move 4: [[1,2,0],[4,5,3]]
After move 5: [[1,2,3],[4,5,0]]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>board.length == 2</code></li>
	<li><code>board[i].length == 3</code></li>
	<li><code>0 &lt;= board[i][j] &lt;= 5</code></li>
	<li>Each value <code>board[i][j]</code> is <strong>unique</strong>.</li>
</ul>

from collections import deque

class Solution:
    def slidingPuzzle(self, board: List[List[int]]) -> int:
        # Flatten the board and find the index of 0
        start = ''.join(str(num) for row in board for num in row)
        target = '123450'
        zero_index = start.index('0')

        # BFS setup
        q = deque([(start, zero_index, 0)])  # board state, zero_index, steps
        visited = set([start])
        
        # Possible movements for the empty space (0)
        neighbors = {
            0: [1, 3], 1: [0, 2, 4], 2: [1, 5],
            3: [0, 4], 4: [1, 3, 5], 5: [2, 4]
        }

        while q:
            state, zero_index, steps = q.popleft()
            
            # If it's the target state, return the steps
            if state == target:
                return steps

            # Explore neighbors
            for neighbor in neighbors[zero_index]:
                # Swap 0 with the neighbor
                new_state = list(state)
                new_state[zero_index], new_state[neighbor] = new_state[neighbor], new_state[zero_index]
                new_state_str = ''.join(new_state)
                
                if new_state_str not in visited:
                    visited.add(new_state_str)
                    q.append((new_state_str, neighbor, steps + 1))

        return -1

16800