<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 typing import List, Tuple

class Solution:
    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:
        directions = [(-2, -1), (-2, 1), (2, -1), (2, 1), (-1, -2), (-1, 2), (1, -2), (1, 2)]
        
        # BFS to calculate the shortest distance (in moves) from knight to every position
        def bfs(start_x, start_y):
            dist = [[float('inf')] * 50 for _ in range(50)]
            queue = deque([(start_x, start_y, 0)])
            dist[start_x][start_y] = 0
            while queue:
                x, y, d = queue.popleft()
                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] = d + 1
                        queue.append((nx, ny, d + 1))
            return dist
        
        # Get the distance matrix for knight starting at (kx, ky)
        knight_dist = bfs(kx, ky)
        
        # Precompute distances from each pawn to every other position
        n = len(positions)
        pawn_dists = []
        for px, py in positions:
            pawn_dists.append(bfs(px, py))
        
        # Memoization dictionary to avoid recalculating states
        memo = {}
        
        # Minimax to simulate the game between Alice and Bob
        def minimax(mask: int, isAliceTurn: bool, knight_pos: Tuple[int, int]) -> int:
            if mask == (1 << n) - 1:  # All pawns captured
                return 0
            
            # Memoization key
            state_key = (mask, isAliceTurn, knight_pos)
            if state_key in memo:
                return memo[state_key]
            
            if isAliceTurn:
                max_moves = -float('inf')
                for i in range(n):
                    if not mask & (1 << i):  # If pawn i has not been captured
                        px, py = positions[i]
                        # Calculate distance from the current knight position
                        if knight_pos == (-1, -1):  # Knight starts from initial position
                            dist = knight_dist[px][py]
                        else:
                            dist = pawn_dists[knight_pos][px][py]
                        max_moves = max(max_moves, dist + minimax(mask | (1 << i), False, i))
                memo[state_key] = max_moves
                return max_moves
            else:  # Bob's turn
                min_moves = float('inf')
                for i in range(n):
                    if not mask & (1 << i):  # If pawn i has not been captured
                        px, py = positions[i]
                        # Calculate distance from the current knight position
                        if knight_pos == (-1, -1):  # Knight starts from initial position
                            dist = knight_dist[px][py]
                        else:
                            dist = pawn_dists[knight_pos][px][py]
                        min_moves = min(min_moves, dist + minimax(mask | (1 << i), True, i))
                memo[state_key] = min_moves
                return min_moves
        
        # Start minimax from Alice's turn with no pawns captured (mask = 0)
        return minimax(0, True, (-1, -1))  # (-1, -1) represents the starting knight position


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