<p>Alice and Bob play a game with piles of stones. There are an <strong>even</strong> number of piles arranged in a row, and each pile has a <strong>positive</strong> integer number of stones <code>piles[i]</code>.</p>

<p>The objective of the game is to end with the most stones. The <strong>total</strong> number of stones across all the piles is <strong>odd</strong>, so there are no ties.</p>

<p>Alice and Bob take turns, with <strong>Alice starting first</strong>. Each turn, a player takes the entire pile of stones either from the <strong>beginning</strong> or from the <strong>end</strong> of the row. This continues until there are no more piles left, at which point the person with the <strong>most stones wins</strong>.</p>

<p>Assuming Alice and Bob play optimally, return <code>true</code><em> if Alice wins the game, or </em><code>false</code><em> if Bob wins</em>.</p>

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

<pre>
<strong>Input:</strong> piles = [5,3,4,5]
<strong>Output:</strong> true
<strong>Explanation:</strong> 
Alice starts first, and can only take the first 5 or the last 5.
Say she takes the first 5, so that the row becomes [3, 4, 5].
If Bob takes 3, then the board is [4, 5], and Alice takes 5 to win with 10 points.
If Bob takes the last 5, then the board is [3, 4], and Alice takes 4 to win with 9 points.
This demonstrated that taking the first 5 was a winning move for Alice, so we return true.
</pre>

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

<pre>
<strong>Input:</strong> piles = [3,7,2,3]
<strong>Output:</strong> true
</pre>

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

<ul>
	<li><code>2 &lt;= piles.length &lt;= 500</code></li>
	<li><code>piles.length</code> is <strong>even</strong>.</li>
	<li><code>1 &lt;= piles[i] &lt;= 500</code></li>
	<li><code>sum(piles[i])</code> is <strong>odd</strong>.</li>
</ul>

class Solution {
public:
    bool stoneGame(vector<int>& piles) {

        std::map<std::pair<int,int>,int> memo;
       
        auto solve=[&](int left,int right,bool is_alice,auto& self)->int{
            if(left>right) return 0;

            if(memo.find({left,right})!=memo.end()) return memo[{left,right}];

            int score_left=0;
            if(is_alice) score_left=piles[left];
            int score_right=0;
            if(is_alice) score_right=piles[right];

            /*
                If Alice pick from the left,
                the remaining array for Bob is [l+1,r]
                If she picks from the right side,
                the remaining array for Bob is [l,r-1]
            */
            return memo[{left,right}]=std::max(self(left+1,right,!is_alice,self)+score_left,self(left,right-1,!is_alice,self)+score_right);
        };

        int n=piles.size();
        int all_stones=std::accumulate(piles.begin(),piles.end(),0);
        int alice_score=solve(0,n-1,true,solve);
        int bob_score=n-alice_score;
        return alice_score>bob_score;
    }
};
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