<p>Alice and Bob take turns playing a game, with <strong>Alice starting first</strong>.</p>

<p>There are <code>n</code> stones arranged in a row. On each player&#39;s turn, they can <strong>remove</strong> either the leftmost stone or the rightmost stone from the row and receive points equal to the <strong>sum</strong> of the remaining stones&#39; values in the row. The winner is the one with the higher score when there are no stones left to remove.</p>

<p>Bob found that he will always lose this game (poor Bob, he always loses), so he decided to <strong>minimize the score&#39;s difference</strong>. Alice&#39;s goal is to <strong>maximize the difference</strong> in the score.</p>

<p>Given an array of integers <code>stones</code> where <code>stones[i]</code> represents the value of the <code>i<sup>th</sup></code> stone <strong>from the left</strong>, return <em>the <strong>difference</strong> in Alice and Bob&#39;s score if they both play <strong>optimally</strong>.</em></p>

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

<pre>
<strong>Input:</strong> stones = [5,3,1,4,2]
<strong>Output:</strong> 6
<strong>Explanation:</strong> 
- Alice removes 2 and gets 5 + 3 + 1 + 4 = 13 points. Alice = 13, Bob = 0, stones = [5,3,1,4].
- Bob removes 5 and gets 3 + 1 + 4 = 8 points. Alice = 13, Bob = 8, stones = [3,1,4].
- Alice removes 3 and gets 1 + 4 = 5 points. Alice = 18, Bob = 8, stones = [1,4].
- Bob removes 1 and gets 4 points. Alice = 18, Bob = 12, stones = [4].
- Alice removes 4 and gets 0 points. Alice = 18, Bob = 12, stones = [].
The score difference is 18 - 12 = 6.
</pre>

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

<pre>
<strong>Input:</strong> stones = [7,90,5,1,100,10,10,2]
<strong>Output:</strong> 122</pre>

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

<ul>
	<li><code>n == stones.length</code></li>
	<li><code>2 &lt;= n &lt;= 1000</code></li>
	<li><code>1 &lt;= stones[i] &lt;= 1000</code></li>
</ul>

import numpy as np
class Solution:
    def stoneGameVII(self, stones: List[int]) -> int:
        seen = dict()
        n = len(stones)
        pref = dict({-1:0})
        for i in range(len(stones)): pref[i] = pref[i-1] + stones[i]
        print(pref)
        dp = np.zeros((n, n))

        for r in range(1, n): 
            for l in range(r-1, -1, -1): 
                dp[l][r] = max(pref[r] - pref[l] - dp[l+1][r], pref[r-1] - pref[l-1] - dp[l][r-1])
        return(int(dp[0][n-1]))        

        """
        def game(l, r): 
            #ret first and second optimal (diff f - s maximized) 
            if r - l == 1: return (max(stones[l], stones[r]), 0)
            if (l, r) in seen.keys(): return seen[(l,r)]
            #pick left
            f1, s1 = (pref[r] - pref[l] + game(l+1, r)[1]), game(l+1, r)[0]
            f2, s2 = (pref[r-1] - pref[l-1] + game(l, r-1)[1]), game(l, r-1)[0]
            ans = 0 
            if f1 - s1 >= f2 - s2: ans = (f1, s1)
            else: ans = (f2, s2)
            seen[(l, r)] = ans
            return ans
        a, b = game(0, len(stones) - 1)
        print(a,b)
        """
                

            




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