<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>

"""
Alice starts
Alice wins: True
False ow

[]

A 3 B 2 A 1 => A wins
A 1 B 3 A 3 => A wins
A 3 B 3 A 1 => A wins

[1, 3, 3]
[1,4,4]
[1, 1, 1, 15, 3, 2]

[5,3,4,5]

[3,7,2,3]

[False, True, True, False],
[-1, True, False, True],
[-1, -1, True, False]
[-1, -1, -1, True]]

"""
class Solution:
    def stoneGame(self, piles: List[int]) -> bool:
        n = len(piles)
        total = sum(piles)
        dp = [[-1 for _ in range(n)] for _ in range(n)]
        player = 1 # 1 if Alice, 0 if Bob
        res = self.helper(piles, 0, n - 1, 1, dp)
        return res > total - res

    def helper(self, piles, i_left, i_right, player, dp):
        if i_left > i_right:
            return 0

        if dp[i_left][i_right] != -1:
            return dp[i_left][i_right]
        
        if player:
            take_left = piles[i_left] + self.helper(piles, i_left + 1, i_right, abs(player - 1), dp)
            take_right =  piles[i_right] + self.helper(piles, i_left, i_right - 1, abs(player - 1), dp)
            res = max(take_left, take_right)
            dp[i_left][i_right] = res
            return res
        else:
            take_left = self.helper(piles, i_left + 1, i_right, abs(player - 1), dp)
            take_right = self.helper(piles, i_left, i_right - 1, abs(player - 1), dp)
            res = max(take_left, take_right)
            dp[i_left][i_right] = res
            return res
        
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