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
func countCombinations(pieces []string, positions [][]int) (ans int) {
	n := len(pieces)
	m := 9
	dist := make([][][]int, n)
	for i := range dist {
		dist[i] = make([][]int, m)
		for j := range dist[i] {
			dist[i][j] = make([]int, m)
		}
	}

	end := make([][3]int, n)

	rookDirs := [][2]int{{1, 0}, {-1, 0}, {0, 1}, {0, -1}}
	bishopDirs := [][2]int{{1, 1}, {1, -1}, {-1, 1}, {-1, -1}}
	queenDirs := [][2]int{{1, 0}, {-1, 0}, {0, 1}, {0, -1}, {1, 1}, {1, -1}, {-1, 1}, {-1, -1}}

	resetDist := func(i int) {
		for j := 0; j < m; j++ {
			for k := 0; k < m; k++ {
				dist[i][j][k] = -1
			}
		}
	}

	checkStop := func(i, x, y, t int) bool {
		for j := 0; j < i; j++ {
			if dist[j][x][y] >= t {
				return false
			}
		}
		return true
	}

	checkPass := func(i, x, y, t int) bool {
		for j := 0; j < i; j++ {
			if dist[j][x][y] == t {
				return false
			}
			if end[j][0] == x && end[j][1] == y && end[j][2] <= t {
				return false
			}
		}
		return true
	}

	isValid := func(x, y int) bool {
		return x >= 1 && x < m && y >= 1 && y < m
	}

	getDirs := func(piece string) [][2]int {
		switch piece[0] {
		case 'r':
			return rookDirs
		case 'b':
			return bishopDirs
		default:
			return queenDirs
		}
	}

	var dfs func(i int)
	dfs = func(i int) {
		if i >= n {
			ans++
			return
		}

		x, y := positions[i][0], positions[i][1]
		resetDist(i)
		dist[i][x][y] = 0
		end[i] = [3]int{x, y, 0}

		if checkStop(i, x, y, 0) {
			dfs(i + 1)
		}

		dirs := getDirs(pieces[i])
		for _, dir := range dirs {
			resetDist(i)
			dist[i][x][y] = 0
			nx, ny, nt := x+dir[0], y+dir[1], 1

			for isValid(nx, ny) && checkPass(i, nx, ny, nt) {
				dist[i][nx][ny] = nt
				end[i] = [3]int{nx, ny, nt}
				if checkStop(i, nx, ny, nt) {
					dfs(i + 1)
				}
				nx += dir[0]
				ny += dir[1]
				nt++
			}
		}
	}

	dfs(0)
	return
}
