<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>

inline const auto optimize = []() {
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);
    std::cout.tie(nullptr);
    return 0;
}();

class Solution {
    static constexpr int solve(const std::vector<int>& stones, const int N,
                     std::vector<int>& cache, const int l, const int r) {
        // Maybe adda special case for SN == 3?

        const int key = l * N + r;
        if (cache[key] != -1) {
            return cache[key];
        }

        const int solve1 = solve(stones, N, cache, l + 2, r);
        const int solve2 = solve(stones, N, cache, l + 1, r - 1);
        const int solve3 = solve(stones, N, cache, l, r-2);

        
        const int leftChoice =
            std::min(stones[l + 1] + solve1,
                     stones[r] + solve2);
        const int rightChoice =
            std::min(stones[r - 1] + solve3,
                     stones[l] +solve2);

        const int ans = std::max(leftChoice, rightChoice);
        cache[key] = ans;
        return ans;
    }

public:
    static constexpr int stoneGameVII(const std::vector<int>& stones) {
        const int N = stones.size();

        std::vector<int> cache(N * N, -1);
        for (int i = 0; i < N; i++) {
            cache[i * N + i] = 0;
        }
        for (int i = 0; i < N - 1; i++) {
            cache[i * N + i + 1] = std::max(stones[i], stones[i + 1]);
        }

        return solve(stones, N, cache, 0, N - 1);
    }
};
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