<p>You are given an array of integers <code>stones</code> where <code>stones[i]</code> is the weight of the <code>i<sup>th</sup></code> stone.</p>

<p>We are playing a game with the stones. On each turn, we choose any two stones and smash them together. Suppose the stones have weights <code>x</code> and <code>y</code> with <code>x &lt;= y</code>. The result of this smash is:</p>

<ul>
	<li>If <code>x == y</code>, both stones are destroyed, and</li>
	<li>If <code>x != y</code>, the stone of weight <code>x</code> is destroyed, and the stone of weight <code>y</code> has new weight <code>y - x</code>.</li>
</ul>

<p>At the end of the game, there is <strong>at most one</strong> stone left.</p>

<p>Return <em>the smallest possible weight of the left stone</em>. If there are no stones left, return <code>0</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> stones = [2,7,4,1,8,1]
<strong>Output:</strong> 1
<strong>Explanation:</strong>
We can combine 2 and 4 to get 2, so the array converts to [2,7,1,8,1] then,
we can combine 7 and 8 to get 1, so the array converts to [2,1,1,1] then,
we can combine 2 and 1 to get 1, so the array converts to [1,1,1] then,
we can combine 1 and 1 to get 0, so the array converts to [1], then that&#39;s the optimal value.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> stones = [31,26,33,21,40]
<strong>Output:</strong> 5
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= stones.length &lt;= 30</code></li>
	<li><code>1 &lt;= stones[i] &lt;= 100</code></li>
</ul>

class Solution:
    def lastStoneWeightII(self, stones: List[int]) -> int:
        # [2,7]
        # - 2 + 7
        # [2,7,4]
        # -2 + 7 - 4
        # [2,7, 4, 8]
        # - 2 + 7  + 4 - 8
        # [2,5]
        # dp = [0]*stones()
        #           2
        #     -2 + 7  (5)          +2 -7
        # -2+7+4. -2+7-4        +2-7+4 +2-7-4
        # (9, 1)      
        # (8, 0)
        # (0, 8):
        # (1, 7)

        mp = {}
        def fn(i, v): 
            """Return minimum weight of stones[i:] given existing weight."""
            if i == len(stones): 
                return abs(v)
            if (i,v) in mp:
                return mp[(i,v)]
            res =  min(fn(i+1, v - stones[i]), fn(i+1, v + stones[i]))
            mp[(i, v)] = res
            return res
        
        return fn(0, 0)

        
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