<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>

MAX_VAL = 10000
moves = [(-2, -1), (-2, 1),(2, -1), (2, 1), (-1, 2), (1, 2),(-1, -2), (1, -2)]

dist = [[[[0]* 50 for _ in range(50)] for _ in range(50)] for _ in range(50)]

def calc_dist(kx, ky, pawns):
    
    grid = [[MAX_VAL] * 50 for _ in range(50)]

    queue = [[kx, ky]]
    grid[kx][ky] = 0

    visited = 0

    while queue and visited < len(pawns):
        curr_x, curr_y = queue.pop(0)
        
        if [curr_x, curr_y] in pawns:
            visited += 1
            dist[kx][ky][curr_x][curr_y] = grid[curr_x][curr_y]
            
        for move_x, move_y in moves:
            new_x, new_y = curr_x + move_x, curr_y + move_y

            if 0 <= new_x < 50 and 0 <= new_y < 50 and grid[new_x][new_y] == MAX_VAL:
                grid[new_x][new_y] = 1 + grid[curr_x][curr_y]
                queue.append([new_x, new_y])
                


def solve(curr, mask, turn, pos, n, dp):
    if mask == (1<<n) - 1:
        return 0
    
    if dp[curr][mask] is not None:
        return dp[curr][mask]
    
    kx, ky = pos[curr]

    steps = 0 if turn else float('inf')

    for i in range(n):
        if mask & (1 << i):
            continue
        
        px, py = pos[i]
        
        new_dist = dist[kx][ky][px][py]

        if turn:
            steps = max(steps, new_dist + solve(i, mask | (1 << i), False, pos, n, dp))
        else:
            steps = min(steps, new_dist + solve(i, mask | (1 << i), True, pos, n, dp))

    dp[curr][mask] = steps
    return steps


class Solution:
    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:
        # find distances between pawns
        positions.append([kx, ky])
        for px, py in positions:
            calc_dist(px, py, positions)
        
        n = len(positions)

        dp = [[None] * (1 << n) for _ in range(n)]

        return solve(n-1, 1<<(n-1), True, positions, n, dp)

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