<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>

from itertools import count

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

        n = len(stones)

        stone_sums = [0] + stones.copy()
        for i in range(2, n + 1):
            stone_sums[i] += stone_sums[i - 1]

        def get_stone_sum(i, j): # inclusive
            return stone_sums[j + 1] - stone_sums[i]

        # memo = {}
        # def get_score_diff(i, j):
        #     if i == j:
        #         return 0
        #     elif i == j - 1:
        #         return max(stones[i], stones[j])
        #     elif (i, j) in memo:
        #         return memo[(i, j)]
        #     else:
        #         memo[(i, j)] = max(get_stone_sum(i + 1, j) - get_score_diff(i + 1, j),
        #                            get_stone_sum(i, j - 1) - get_score_diff(i, j - 1))
        #         return memo[(i, j)]
                
        # return get_score_diff(0, n - 1)

        table = [[0] * n for _ in range(n)]

        for i in range(n - 1):
            table[i][i + 1] = max(stones[i], stones[i + 1])

        for diff in range(2, n):
            for i in range(n - diff):
                j = i + diff
                table[i][j] = max(get_stone_sum(i + 1, j) - table[i + 1][j],
                                  get_stone_sum(i, j - 1) - table[i][j - 1])

        return table[0][n - 1]


        
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