<p>There is a <code>50 x 50</code> chessboard with <strong>one</strong> knight and some pawns on it. You are given two integers <code>kx</code> and <code>ky</code> where <code>(kx, ky)</code> denotes the position of the knight, and a 2D array <code>positions</code> where <code>positions[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> denotes the position of the pawns on the chessboard.</p>

<p>Alice and Bob play a <em>turn-based</em> game, where Alice goes first. In each player&#39;s turn:</p>

<ul>
	<li>The player <em>selects </em>a pawn that still exists on the board and captures it with the knight in the <strong>fewest</strong> possible <strong>moves</strong>. <strong>Note</strong> that the player can select <strong>any</strong> pawn, it <strong>might not</strong> be one that can be captured in the <strong>least</strong> number of moves.</li>
	<li><span>In the process of capturing the <em>selected</em> pawn, the knight <strong>may</strong> pass other pawns <strong>without</strong> capturing them</span>. <strong>Only</strong> the <em>selected</em> pawn can be captured in <em>this</em> turn.</li>
</ul>

<p>Alice is trying to <strong>maximize</strong> the <strong>sum</strong> of the number of moves made by <em>both</em> players until there are no more pawns on the board, whereas Bob tries to <strong>minimize</strong> them.</p>

<p>Return the <strong>maximum</strong> <em>total</em> number of moves made during the game that Alice can achieve, assuming both players play <strong>optimally</strong>.</p>

<p>Note that in one <strong>move, </strong>a chess knight has eight possible positions it can move to, as illustrated below. Each move is two cells in a cardinal direction, then one cell in an orthogonal direction.</p>

<p><img src="https://assets.leetcode.com/uploads/2024/08/01/chess_knight.jpg" style="width: 275px; height: 273px;" /></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">kx = 1, ky = 1, positions = [[0,0]]</span></p>

<p><strong>Output:</strong> <span class="example-io">4</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/08/16/gif3.gif" style="width: 275px; height: 275px;" /></p>

<p>The knight takes 4 moves to reach the pawn at <code>(0, 0)</code>.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">kx = 0, ky = 2, positions = [[1,1],[2,2],[3,3]]</span></p>

<p><strong>Output:</strong> <span class="example-io">8</span></p>

<p><strong>Explanation:</strong></p>

<p><strong><img alt="" src="https://assets.leetcode.com/uploads/2024/08/16/gif4.gif" style="width: 320px; height: 320px;" /></strong></p>

<ul>
	<li>Alice picks the pawn at <code>(2, 2)</code> and captures it in two moves: <code>(0, 2) -&gt; (1, 4) -&gt; (2, 2)</code>.</li>
	<li>Bob picks the pawn at <code>(3, 3)</code> and captures it in two moves: <code>(2, 2) -&gt; (4, 1) -&gt; (3, 3)</code>.</li>
	<li>Alice picks the pawn at <code>(1, 1)</code> and captures it in four moves: <code>(3, 3) -&gt; (4, 1) -&gt; (2, 2) -&gt; (0, 3) -&gt; (1, 1)</code>.</li>
</ul>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">kx = 0, ky = 0, positions = [[1,2],[2,4]]</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<ul>
	<li>Alice picks the pawn at <code>(2, 4)</code> and captures it in two moves: <code>(0, 0) -&gt; (1, 2) -&gt; (2, 4)</code>. Note that the pawn at <code>(1, 2)</code> is not captured.</li>
	<li>Bob picks the pawn at <code>(1, 2)</code> and captures it in one move: <code>(2, 4) -&gt; (1, 2)</code>.</li>
</ul>
</div>

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

<ul>
	<li><code>0 &lt;= kx, ky &lt;= 49</code></li>
	<li><code>1 &lt;= positions.length &lt;= 15</code></li>
	<li><code>positions[i].length == 2</code></li>
	<li><code>0 &lt;= positions[i][0], positions[i][1] &lt;= 49</code></li>
	<li>All <code>positions[i]</code> are unique.</li>
	<li>The input is generated such that <code>positions[i] != [kx, ky]</code> for all <code>0 &lt;= i &lt; positions.length</code>.</li>
</ul>

INF = 10 ** 20
MX = 50
        
directions = [
    (2, 1),
    (-2, 1),
    (2, -1),
    (-2, -1),
    (1, 2),
    (-1, 2),
    (1, -2),
    (-1, -2),
]

@cache
def go2(sx, sy):
    distances = [[INF] * MX for _ in range(MX)]
    q = collections.deque()

    distances[sx][sy] = 0
    q.append((sx, sy))

    while len(q) > 0:
        x, y = q.popleft()

        for dx, dy in directions:
            nx, ny = x + dx, y + dy

            if 0 <= nx < MX and 0 <= ny < MX and distances[nx][ny] > distances[x][y] + 1:
                distances[nx][ny] = distances[x][y] + 1
                q.append((nx, ny))
    return distances


class Solution:
    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:
        N = len(positions)
        TOTAL = (1 << N) - 1
        
        def go(sx, sy):
            distances = [[INF] * MX for _ in range(MX)]
            q = collections.deque()
            
            distances[sx][sy] = 0
            q.append((sx, sy))
            
            while len(q) > 0:
                x, y = q.popleft()
                
                for dx, dy in directions:
                    nx, ny = x + dx, y + dy
                    
                    if 0 <= nx < MX and 0 <= ny < MX and distances[nx][ny] > distances[x][y] + 1:
                        distances[nx][ny] = distances[x][y] + 1
                        q.append((nx, ny))
            return distances

        def go(sx, sy):
            distances = go2(sx, sy)
            d = []
            for index, (px, py) in enumerate(positions):
                d.append(distances[px][py])
            return d

        distances = []
        for i in range(N):
            distances.append(go(positions[i][0], positions[i][1]))
            
        kd = go(kx, ky)

        alice_cache = [[None] * (TOTAL + 1) for _ in range(N)]
        def go_alice(index, mask):
            if mask == TOTAL:
                return 0
            
            if alice_cache[index][mask] is not None:
                return alice_cache[index][mask]
            
            best = -INF
            for i in range(N):
                if (mask & (1 << i)) == 0:
                    score = go_bob(i, mask | (1 << i)) + distances[index][i]
                    if score > best:
                        best = score
            
            alice_cache[index][mask] = best
            return best
        
        bob_cache = [[None] * (TOTAL + 1) for _ in range(N)]
        def go_bob(index, mask):
            if mask == TOTAL:
                return 0

            if bob_cache[index][mask] is not None:
                return bob_cache[index][mask]
            
            best = INF
            for i in range(N):
                if (mask & (1 << i)) == 0:
                    score = go_alice(i, mask | (1 << i)) + distances[index][i]
                    if score < best:
                        best = score
            bob_cache[index][mask] = best
            return best
        
        best = 0
        for i in range(N):
            best = max(best, go_bob(i, (1 << i)) + kd[i])
        return best

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