Number of Valid Move Combinations On Chessboard
<p>There is an <code>8 x 8</code> chessboard containing <code>n</code> pieces (rooks, queens, or bishops). You are given a string array <code>pieces</code> of length <code>n</code>, where <code>pieces[i]</code> describes the type (rook, queen, or bishop) of the <code>i<sup>th</sup></code> piece. In addition, you are given a 2D integer array <code>positions</code> also of length <code>n</code>, where <code>positions[i] = [r<sub>i</sub>, c<sub>i</sub>]</code> indicates that the <code>i<sup>th</sup></code> piece is currently at the <strong>1-based</strong> coordinate <code>(r<sub>i</sub>, c<sub>i</sub>)</code> on the chessboard.</p>

<p>When making a <strong>move</strong> for a piece, you choose a <strong>destination</strong> square that the piece will travel toward and stop on.</p>

<ul>
	<li>A rook can only travel <strong>horizontally or vertically</strong> from <code>(r, c)</code> to the direction of <code>(r+1, c)</code>, <code>(r-1, c)</code>, <code>(r, c+1)</code>, or <code>(r, c-1)</code>.</li>
	<li>A queen can only travel <strong>horizontally, vertically, or diagonally</strong> from <code>(r, c)</code> to the direction of <code>(r+1, c)</code>, <code>(r-1, c)</code>, <code>(r, c+1)</code>, <code>(r, c-1)</code>, <code>(r+1, c+1)</code>, <code>(r+1, c-1)</code>, <code>(r-1, c+1)</code>, <code>(r-1, c-1)</code>.</li>
	<li>A bishop can only travel <strong>diagonally</strong> from <code>(r, c)</code> to the direction of <code>(r+1, c+1)</code>, <code>(r+1, c-1)</code>, <code>(r-1, c+1)</code>, <code>(r-1, c-1)</code>.</li>
</ul>

<p>You must make a <strong>move</strong> for every piece on the board simultaneously. A <strong>move combination</strong> consists of all the <strong>moves</strong> performed on all the given pieces. Every second, each piece will instantaneously travel <strong>one square</strong> towards their destination if they are not already at it. All pieces start traveling at the <code>0<sup>th</sup></code> second. A move combination is <strong>invalid</strong> if, at a given time, <strong>two or more</strong> pieces occupy the same square.</p>

<p>Return <em>the number of <strong>valid</strong> move combinations</em>​​​​​.</p>

<p><strong>Notes:</strong></p>

<ul>
	<li><strong>No two pieces</strong> will start in the<strong> same</strong> square.</li>
	<li>You may choose the square a piece is already on as its <strong>destination</strong>.</li>
	<li>If two pieces are <strong>directly adjacent</strong> to each other, it is valid for them to <strong>move past each other</strong> and swap positions in one second.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/2000-2099/2056.Number%20of%20Valid%20Move%20Combinations%20On%20Chessboard/images/a1.png" style="width: 215px; height: 215px;" />
<pre>
<strong>Input:</strong> pieces = [&quot;rook&quot;], positions = [[1,1]]
<strong>Output:</strong> 15
<strong>Explanation:</strong> The image above shows the possible squares the piece can move to.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/2000-2099/2056.Number%20of%20Valid%20Move%20Combinations%20On%20Chessboard/images/a2.png" style="width: 215px; height: 215px;" />
<pre>
<strong>Input:</strong> pieces = [&quot;queen&quot;], positions = [[1,1]]
<strong>Output:</strong> 22
<strong>Explanation:</strong> The image above shows the possible squares the piece can move to.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/2000-2099/2056.Number%20of%20Valid%20Move%20Combinations%20On%20Chessboard/images/a3.png" style="width: 214px; height: 215px;" />
<pre>
<strong>Input:</strong> pieces = [&quot;bishop&quot;], positions = [[4,3]]
<strong>Output:</strong> 12
<strong>Explanation:</strong> The image above shows the possible squares the piece can move to.
</pre>

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

<ul>
	<li><code>n == pieces.length </code></li>
	<li><code>n == positions.length</code></li>
	<li><code>1 &lt;= n &lt;= 4</code></li>
	<li><code>pieces</code> only contains the strings <code>&quot;rook&quot;</code>, <code>&quot;queen&quot;</code>, and <code>&quot;bishop&quot;</code>.</li>
	<li>There will be at most one queen on the chessboard.</li>
	<li><code>1 &lt;= r<sub>i</sub>, c<sub>i</sub> &lt;= 8</code></li>
	<li>Each <code>positions[i]</code> is distinct.</li>
</ul>
Array; String; Backtracking; Simulation
rook_dirs = [(1, 0), (-1, 0), (0, 1), (0, -1)]
bishop_dirs = [(1, 1), (1, -1), (-1, 1), (-1, -1)]
queue_dirs = rook_dirs + bishop_dirs


def get_dirs(piece: str) -> List[Tuple[int, int]]:
    match piece[0]:
        case "r":
            return rook_dirs
        case "b":
            return bishop_dirs
        case _:
            return queue_dirs


class Solution:
    def countCombinations(self, pieces: List[str], positions: List[List[int]]) -> int:
        def check_stop(i: int, x: int, y: int, t: int) -> bool:
            return all(dist[j][x][y] < t for j in range(i))

        def check_pass(i: int, x: int, y: int, t: int) -> bool:
            for j in range(i):
                if dist[j][x][y] == t:
                    return False
                if end[j][0] == x and end[j][1] == y and end[j][2] <= t:
                    return False
            return True

        def dfs(i: int) -> None:
            if i >= n:
                nonlocal ans
                ans += 1
                return
            x, y = positions[i]
            dist[i][:] = [[-1] * m for _ in range(m)]
            dist[i][x][y] = 0
            end[i] = (x, y, 0)
            if check_stop(i, x, y, 0):
                dfs(i + 1)
            dirs = get_dirs(pieces[i])
            for dx, dy in dirs:
                dist[i][:] = [[-1] * m for _ in range(m)]
                dist[i][x][y] = 0
                nx, ny, nt = x + dx, y + dy, 1
                while 1 <= nx < m and 1 <= ny < m and check_pass(i, nx, ny, nt):
                    dist[i][nx][ny] = nt
                    end[i] = (nx, ny, nt)
                    if check_stop(i, nx, ny, nt):
                        dfs(i + 1)
                    nx += dx
                    ny += dy
                    nt += 1

        n = len(pieces)
        m = 9
        dist = [[[-1] * m for _ in range(m)] for _ in range(n)]
        end = [(0, 0, 0) for _ in range(n)]
        ans = 0
        dfs(0)
        return ans
