<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:
        sx = 50
        sy = 50
        offsets = [(1,2),(1,-2),(2,1),(2,-1),(-1,2),(-1,-2),(-2,1),(-2,-1)]
        def minknightmove(startx,starty):
            # Finds minimum knights distance to all points on the square
            # Returns minimum distance only for x,y in positions
            pq = [(0,startx,starty)]
            dist = [[float('inf')]*sy for i in range(sx)]
            dist[startx][starty] = 0
            while pq:
                d,x,y = heapq.heappop(pq)
                if d==dist[x][y]:
                    for i,j in offsets:
                        newx = x+i
                        newy = y+j
                        if 0<=newx and newx<sx and 0<=newy and newy<sy:
                            if d+1<dist[newx][newy]:
                                dist[newx][newy] = d+1
                                heapq.heappush(pq,(d+1,newx,newy))
            out = []
            for x,y in positions:
                out.append(dist[x][y])
            return out
        
        dists = []
        initial = minknightmove(kx,ky)
        for x,y in positions:
            dists.append(minknightmove(x,y))
        
        n = len(positions)
        visited = [0]*n
        d = {}
        def helper(i,turn):
            # Returns sum of moves in optimal play
            # turn 0 corresponds to Alice
            # turn 1 corresponds to Bob
            key = tuple(visited+[i])
            if key in d:
                return d[key]
            if turn:
                # Need to minimze total
                mintot = float('inf')
                for j in range(n):
                    if visited[j]==0:
                        visited[j]=1
                        mintot = min(dists[i][j]+helper(j,1-turn),mintot)
                        visited[j]=0
                if mintot==float('inf'):
                    # Nothing to visit
                    d[key] = 0
                    return 0
                d[key] = mintot
                return mintot
            else:
                # Need to maximize total
                maxtot = 0
                for j in range(n):
                    if visited[j]==0:
                        visited[j]=1
                        maxtot = max(dists[i][j]+helper(j,1-turn),maxtot)
                        visited[j]=0
                d[key] = maxtot
                return maxtot
        
        maxout = 0
        for i in range(n):
            visited[i]=1
            maxout = max(initial[i]+helper(i,1),maxout)
            visited[i]=0
        return maxout


        


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