<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>

from collections import deque
from functools import lru_cache

class Solution:
    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:
        # Precompute knight's move distances on the chessboard using BFS
        def bfs_knight(start_x, start_y):
            directions = [(2, 1), (1, 2), (-1, 2), (-2, 1), (-2, -1), (-1, -2), (1, -2), (2, -1)]
            dist = [[float('inf')] * 50 for _ in range(50)]
            dist[start_x][start_y] = 0
            queue = deque([(start_x, start_y)])
            
            while queue:
                x, y = queue.popleft()
                current_dist = dist[x][y]
                
                for dx, dy in directions:
                    nx, ny = x + dx, y + dy
                    if 0 <= nx < 50 and 0 <= ny < 50 and dist[nx][ny] == float('inf'):
                        dist[nx][ny] = current_dist + 1
                        queue.append((nx, ny))
            return dist

        # Precompute all distances from the knight and between all pawns
        n = len(positions)
        knight_dist = bfs_knight(kx, ky)  # Distance from knight to every cell
        pawn_dist = [[0] * n for _ in range(n)]  # Distance between all pawns
        
        # Precompute distances from each pawn to every other pawn
        for i in range(n):
            px, py = positions[i]
            dist_from_pawn = bfs_knight(px, py)
            for j in range(n):
                if i != j:
                    px2, py2 = positions[j]
                    pawn_dist[i][j] = dist_from_pawn[px2][py2]
        
        # Memoized DP function with bitmask representing which pawns are left
        @lru_cache(None)
        def dp(mask, is_alice_turn, last_pawn):
            if mask == 0:
                return 0  # No pawns left to capture
            
            max_moves = 0 if is_alice_turn else float('inf')
            
            for i in range(n):
                if mask & (1 << i):  # Pawn i is still on the board
                    new_mask = mask ^ (1 << i)  # Remove pawn i
                    if last_pawn == -1:
                        # First move from the knight to a pawn
                        moves_to_pawn = knight_dist[positions[i][0]][positions[i][1]]
                    else:
                        # Move from the last captured pawn to this pawn
                        moves_to_pawn = pawn_dist[last_pawn][i]
                    
                    if is_alice_turn:
                        # Alice's turn: she tries to maximize the total moves
                        max_moves = max(max_moves, moves_to_pawn + dp(new_mask, False, i))
                    else:
                        # Bob's turn: he tries to minimize the total moves
                        max_moves = min(max_moves, moves_to_pawn + dp(new_mask, True, i))
            
            return max_moves
        
        # Start the game with all pawns available (initial_mask = all bits set to 1)
        initial_mask = (1 << n) - 1
        return dp(initial_mask, True, -1)  # -1 indicates the knight starts first

    

        
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