<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>

class Solution:
    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:
        import functools

        num_rows = 50
        num_cols = 50

        def get_next_coords(src_row, src_col):
            for row in [src_row - 2, src_row + 2]:
                if row < 0 or row >= num_rows:
                    continue
                for col in [src_col - 1, src_col + 1]:
                    if col < 0 or col >= num_cols:
                        continue
                    yield row, col
            for col in [src_col - 2, src_col + 2]:
                if col < 0 or col >= num_cols:
                    continue
                for row in [src_row - 1, src_row + 1]:
                    if row < 0 or row >= num_rows:
                        continue
                    yield row, col

        @functools.cache
        def get_num_moves(src_row, src_col, tgt_row, tgt_col):
            if src_row == tgt_row and src_col == tgt_col:
                return 0
            enqueued = [
                [False for _ in range(num_cols)] for _ in range(num_rows)
            ]
            enqueued[src_row][src_col] = True
            nodes_curr = [(src_row, src_col)]
            dist = 0
            while nodes_curr:
                dist += 1
                nodes_next = []
                for src in nodes_curr:
                    for row, col in get_next_coords(*src):
                        if enqueued[row][col]:
                            continue
                        enqueued[row][col] = True
                        if row == tgt_row and col == tgt_col:
                            return dist
                        nodes_next.append((row, col))
                nodes_curr = nodes_next           
            return dist

        full_pos_mask = (1 << len(positions)) - 1
        def get_pos(pos_mask):
            for i, pos in enumerate(positions):
                if (1 << i) & pos_mask:
                    continue
                yield i, pos

        @functools.cache
        def find_min_moves(src_row, src_col, pos_mask):
            if full_pos_mask == pos_mask:                
                return 0            
            min_moves = None
            def _update_min_moves(curr_moves):
                nonlocal min_moves
                if min_moves is None:
                    min_moves = curr_moves
                else:
                    min_moves = min(curr_moves, min_moves)

            for i, (tgt_row, tgt_col) in get_pos(pos_mask):
                sub = get_num_moves(src_row, src_col, tgt_row, tgt_col) 
                if sub is None:
                    continue
                _update_min_moves(sub + find_max_moves(tgt_row, tgt_col, pos_mask | (1 << i)))
            
            return min_moves

        @functools.cache
        def find_max_moves(src_row, src_col, pos_mask):
            if full_pos_mask == pos_mask:                
                return 0            
            max_moves = None
            def _update_max_moves(curr_moves):
                nonlocal max_moves
                if max_moves is None:
                    max_moves = curr_moves
                else:
                    max_moves = max(curr_moves, max_moves)                

            for i, (tgt_row, tgt_col) in get_pos(pos_mask):
                sub = get_num_moves(src_row, src_col, tgt_row, tgt_col) 
                if sub is None:
                    continue
                _update_max_moves(sub + find_min_moves(tgt_row, tgt_col, pos_mask | (1 << i)))
            
            return max_moves


        return find_max_moves(kx, ky, 0)


                
                
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