Sliding Puzzle
<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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/0700-0799/0773.Sliding%20Puzzle/images/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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/0700-0799/0773.Sliding%20Puzzle/images/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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/0700-0799/0773.Sliding%20Puzzle/images/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>
Breadth-First Search; Memoization; Array; Dynamic Programming; Backtracking; Matrix
class Solution:
    def slidingPuzzle(self, board: List[List[int]]) -> int:
        t = [None] * 6

        def gets():
            for i in range(2):
                for j in range(3):
                    t[i * 3 + j] = str(board[i][j])
            return ''.join(t)

        def setb(s):
            for i in range(2):
                for j in range(3):
                    board[i][j] = int(s[i * 3 + j])

        def f():
            res = []
            i, j = next((i, j) for i in range(2) for j in range(3) if board[i][j] == 0)
            for a, b in [[0, -1], [0, 1], [1, 0], [-1, 0]]:
                x, y = i + a, j + b
                if 0 <= x < 2 and 0 <= y < 3:
                    board[i][j], board[x][y] = board[x][y], board[i][j]
                    res.append(gets())
                    board[i][j], board[x][y] = board[x][y], board[i][j]
            return res

        start = gets()
        end = "123450"
        if start == end:
            return 0
        vis = {start}
        q = deque([start])
        ans = 0
        while q:
            ans += 1
            for _ in range(len(q)):
                x = q.popleft()
                setb(x)
                for y in f():
                    if y == end:
                        return ans
                    if y not in vis:
                        vis.add(y)
                        q.append(y)
        return -1
