<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:
        direc = [[1,2],[2,1],[2,-1],[1,-2],[-1,-2],[-2,-1],[-2,1],[-1,2]]

        @cache
        def minmove(x,y,tx,ty):
            if (x == 0 and y == 0 and tx == 1 and ty == 1) or (x == 1 and y == 1 and tx == 0 and ty == 0) or (x == 0 and y == 49 and tx == 1 and ty == 48) or (x == 1 and y == 48 and tx == 0 and ty == 49) or (x == 49 and y == 0 and tx == 48 and ty == 1) or (x == 48 and y == 1 and tx == 49 and ty == 0) or (x == 49 and y == 49 and tx == 48 and ty == 48) or (x == 48 and y == 48 and tx == 49 and ty == 49):
                return 4
            
            dx = abs(x-tx)
            dy = abs(y-ty)

            @cache
            def dp(dx,dy):
                if dx == 0 and dy == 0:
                    return 0
                if (dx == 0 and dy == 1) or (dy == 0 and dx == 1):
                    return 3
                if (dx == 1 and dy == 1):
                    return 2
                if (dx == 0 and dy == 2) or (dx == 2 and dy == 0):
                    return 2
                if (dx == 1 and dy == 2) or (dx == 2 and dy == 1):
                    return 1
                return 1 + min(dp(abs(dx-1),abs(dy-2)),dp(abs(dx-2),abs(dy-1)))
            
            return dp(dx,dy)
                
        N = len(positions)
        bitmask = (1<<N)-1

        @cache
        def helper(x0,y0,bitmask):
            if bitmask == 0:
                return 0,0
            ind = []
            for i in range(N):
                if bitmask & (1<<i) != 0:
                    ind.append(i)
            if len(ind) == 1:
                tmp = minmove(x0,y0,positions[ind[0]][0],positions[ind[0]][1])
                return tmp, tmp
            mn = float(inf)
            mx = 0
            for ii in ind:
                curdist = minmove(x0,y0,positions[ii][0],positions[ii][1])
                tmpmx,tmpmn = helper(positions[ii][0],positions[ii][1],bitmask - (1<<ii))
                mn = min(mn, curdist + tmpmx)
                mx = max(mx, curdist + tmpmn)
            return mx, mn
        
        resmx,resmn = helper(kx,ky,bitmask)
        return resmx
            
        
3253