<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 typing import List
from collections import deque

class Solution:
    def slidingPuzzle(self, board: List[List[int]]) -> int:
        # Directions map: key is the index of the blank space (0), and values are the indices it can swap with.
        dirs = {
            0: [1, 3], 
            1: [0, 2, 4], 
            2: [1, 5], 
            3: [0, 4], 
            4: [1, 3, 5], 
            5: [2, 4]
        }
        # Flatten the board to a single list and start the BFS queue with this initial state and move count of 0.
        state = list((*board[0], *board[1]))
        q = deque([(state, 0)])  # Queue for BFS, containing tuples of (current state, move count)
        visit = set()  # Set to keep track of visited states

        # Begin BFS
        while q:
            state, count = q.popleft()  # Current state and its move count
            # If the current state is the goal state, return the move count.
            if state == [1, 2, 3, 4, 5, 0]:
                return count

            idx = state.index(0)  # Find the index of the blank space (0)

            # Iterate over all possible directions the blank space can move
            for new_i in dirs[idx]:
                cur = state[:]  # Create a copy of the current state
                cur[idx], cur[new_i] = cur[new_i], 0  # Swap the blank space with the adjacent number
                
                # If we haven't visited this state before, add it to the queue and the visited set
                cur_tuple = tuple(cur)  # Convert list to tuple for hashability
                if cur_tuple not in visit:
                    visit.add(cur_tuple)
                    q.append((cur, count + 1))  # Add new state with incremented move count to the queue
        
        # If the loop ends without returning, the puzzle is unsolvable.
        return -1
16600