<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, defaultdict
from math import inf

def is_valid(x, y):
    return 0 <= x < 50 and 0 <= y < 50

def min_knight_moves(kx, ky, tx, ty):
    if (kx, ky) == (tx, ty):
        return 0

    directions = [(2, 1), (2, -1), (-2, 1), (-2, -1),
                  (1, 2), (1, -2), (-1, 2), (-1, -2)]

    queue = deque([(kx, ky, 0)])
    visited = set([(kx, ky)])  # Correct initialization
    
    while queue:
        x, y, moves = queue.popleft()
        
        for dx, dy in directions:
            nx, ny = x + dx, y + dy
            
            if (nx, ny) == (tx, ty):
                return moves + 1
            
            if is_valid(nx, ny) and (nx, ny) not in visited:
                visited.add((nx, ny))
                queue.append((nx, ny, moves + 1))
    
    return -1

class Solution:
    def panda(self, i, mask, n, positions, mp, f, dp):
        if mask == (1 << n) - 1:
            return 0
        
        # Check if this state has already been computed
        if dp[i][mask][f] != -1:
            return dp[i][mask][f]
        
        ans = 0
        xi, yi = positions[i]
        
        if f:  # Bob's turn (minimizing moves)
            ans = float(inf)
            for j in range(n):
                if not ((1 << j) & mask):
                    xj, yj = positions[j]
                    ans = min(ans, mp[(xi, yi, xj, yj)] + self.panda(j, mask | (1 << j), n, positions, mp, 0, dp))
        else:  # Alice's turn (maximizing moves)
            ans = -float(inf)
            for j in range(n):
                if not ((1 << j) & mask):
                    xj, yj = positions[j]
                    ans = max(ans, mp[(xi, yi, xj, yj)] + self.panda(j, mask | (1 << j), n, positions, mp, 1, dp))
        
        dp[i][mask][f] = ans  # Memoize the result
        return ans
    
    def maxMoves(self, kx: int, ky: int, positions: list[list[int]]) -> int:
        mp = defaultdict(lambda: -1)
        n = len(positions)
        
        # Initialize the dp array
        dp = [[[-1 for _ in range(2)] for _ in range(1 << n)] for _ in range(n)]
        
        # Calculate minimum moves between all pairs of pawn positions
        for i in range(n):
            xk, yk = positions[i]
            for j in range(i + 1, n):
                xt, yt = positions[j]
                d = min_knight_moves(xk, yk, xt, yt)
                mp[(xk, yk, xt, yt)] = d
                mp[(xt, yt, xk, yk)] = d
        
        ans = -float(inf)
        
        # Try starting from every pawn position
        for i in range(n):
            x, y = positions[i]
            # Knight's initial move to the first pawn
            initial_moves = min_knight_moves(kx, ky, x, y)
            # Calculate the rest of the moves for Alice and Bob
            res = self.panda(i, 1 << i, n, positions, mp, 1, dp)
            ans = max(ans, initial_moves + res)
        
        return ans

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