<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>

def search(dic, i, j, prefix_sum, stones, user):
    if i+1 == j:
        dic[(i, j)] = [max(stones[i], stones[j]),0]
        return dic[(i, j)]
    else:
        back_val = prefix_sum[i-1] if i>0 else 0 
        option_1 = prefix_sum[j-1] - back_val
        option_2 = prefix_sum[j] - prefix_sum[i]

        new_user = 0 if user == 1 else 1

        track_1 = (i, j-1)
        if track_1 not in dic:
            result = search(dic, i, j-1, prefix_sum, stones, new_user)
        result = dic[track_1]
        option_1 += result[1]
        rem_1 = result[0]

        track_2 = (i+1, j)
        if track_2 not in dic:
            result = search(dic, i+1, j, prefix_sum, stones, new_user)
        result = dic[track_2]
        option_2 += result[1]
        rem_2 = result[0]
        
        
        if (i,j) == (0,3):
            print(option_1, rem_1, option_2, rem_2)
        if abs(option_1 -rem_1) > abs(option_2 - rem_2):
            dic[(i,j)] = [option_1, rem_1]
        else:
            dic[(i,j)] = [option_2, rem_2]





class Solution:
    def stoneGameVII(self, stones: List[int]) -> int:
        
        

        prefix_sum = []
        cur = 0

        for i in range(len(stones)):
            cur += stones[i]
            prefix_sum.append(cur)

        i = 0
        j = len(stones) - 1

        dic = {}
        user =0

        search(dic, i, j, prefix_sum, stones, user)
        return dic[(i,j)][0] - dic[(i,j)][1] 
        return 2
        

            



            
            
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