<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 typing import List
from functools import cache
from collections import deque

dist = [[1000000] * 2500 for _ in range(2500)]

for x in range(50):
    for y in range(50):
        dist[x * 50 + y][x * 50 + y] = 0
        # for i, j in [
        #     (x + 1, y + 2),
        #     (x + 1, y - 2),
        #     (x + 2, y + 1),
        #     (x + 2, y - 1),
        #     (x - 1, y + 2),
        #     (x - 1, y - 2),
        #     (x - 2, y + 1),
        #     (x - 2, y - 1),
        # ]:
        #     if 0 <= i < 50 and 0 <= j < 50:
        #         dist[x * 50 + y][i * 50 + j] = 1


for i in range(2500):
    I = i
    # print("i", i)
    seen = set([i])
    queue = deque([(i, 0)])
    while queue:
        i, d = queue.popleft()
        xi = i // 50
        yi = i % 50
        # print(xi, yi)
        for xj, yj in [
            (xi + 1, yi + 2),
            (xi + 1, yi - 2),
            (xi + 2, yi + 1),
            (xi + 2, yi - 1),
            (xi - 1, yi + 2),
            (xi - 1, yi - 2),
            (xi - 2, yi + 1),
            (xi - 2, yi - 1),
        ]:
            if 0 <= xj < 50 and 0 <= yj < 50:
                j = xj * 50 + yj
                if j not in seen:
                    seen.add(j)
                    dist[I][j] = d + 1
                    # print(i, j, d + 1)
                    queue.append((j, d + 1))

# print(dist[0][0])
# print(dist[0][1])  # WHY it's 1000000??? it should be a number small
# print(dist[500][501])  # WHY it's 1000000??? it should be a number small
# print(dist[1000][2000])


class Solution:
    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:
        n = len(positions)

        @cache
        def dfs(mask: int, x: int, y: int, alice: bool) -> int:
            if not mask:
                return 0

            moves = []

            for i in range(n):
                if mask & (1 << i):
                    a, b = positions[i]
                    move = dist[x * 50 + y][a * 50 + b]
                    moves.append(dfs(mask ^ (1 << i), a, b, not alice) + move)

            return max(moves) if alice else min(moves)

        return dfs((1 << n) - 1, kx, ky, True)

 
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