<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

# Knight movement directions (8 possible moves for knight)
KNIGHT_MOVES = [
    (2, 1), (2, -1), (-2, 1), (-2, -1), 
    (1, 2), (-1, 2), (1, -2), (-1, -2)
]

class Solution:
    def bfs_knight_moves(self, start_x, start_y):
        # BFS to compute minimum moves from (start_x, start_y) to any cell on the board
        dist = [[float('inf')] * 50 for _ in range(50)]
        dist[start_x][start_y] = 0
        q = deque([(start_x, start_y)])
        
        while q:
            x, y = q.popleft()
            current_dist = dist[x][y]
            
            for dx, dy in KNIGHT_MOVES:
                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
                    q.append((nx, ny))
                    
        # Return the distance array which holds the min steps to each position.
        return dist

    def maxMoves(self, kx: int, ky: int, positions: list[list[int]]) -> int:
        # Number of pawns
        n = len(positions)
        
        # BFS from each pawn and the initial knight position
        knight_dist = self.bfs_knight_moves(kx, ky)
        pawn_dist = []
        
        for x, y in positions:
            # BFS for each pawn position -> distance to any other place
            pawn_dist.append(self.bfs_knight_moves(x, y))
        
        # Memoized recursive helper
        @lru_cache(None)
        def dp(mask, curr_pawn_idx, turn):
            # mask -> bitmask of remaining pawns (1: remaining, 0: captured)
            # curr_pawn_idx -> where the knight currently is
            # turn -> which player's turn (0 for Alice, 1 for Bob)
            
            if mask == 0:
                return 0  # no pawns left to capture

            # Get the current knight's position (or initial knight position)
            if curr_pawn_idx == -1:
                curr_x, curr_y = kx, ky  # initial knight position
                from_dist = knight_dist
            else:
                curr_x, curr_y = positions[curr_pawn_idx]
                from_dist = pawn_dist[curr_pawn_idx]
            
            if turn == 0:  # Alice's turn - maximize the moves
                result = -float('inf')
                for i in range(n):
                    if mask & (1 << i):  # if pawn i is still on the board
                        new_mask = mask & ~(1 << i)
                        result = max(result, from_dist[positions[i][0]][positions[i][1]] + dp(new_mask, i, 1))
                return result
            else:  # Bob's turn - minimize the moves
                result = float('inf')
                for i in range(n):
                    if mask & (1 << i):  # if pawn i is still on the board
                        new_mask = mask & ~(1 << i)
                        result = min(result, from_dist[positions[i][0]][positions[i][1]] + dp(new_mask, i, 0))
                return result
        
        # Start from initial knight position (-1 indicates starting position, no previous pawn captured)
        return dp((1 << n) - 1, -1, 0)


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