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"title": "Stone Game VIII",
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"content": "<p>Alice and Bob take turns playing a game, with <strong>Alice starting first</strong>.</p>\r\n\r\n<p>There are <code>n</code> stones arranged in a row. On each player's turn, while the number of stones is <strong>more than one</strong>, they will do the following:</p>\r\n\r\n<ol>\r\n\t<li>Choose an integer <code>x > 1</code>, and <strong>remove</strong> the leftmost <code>x</code> stones from the row.</li>\r\n\t<li>Add the <strong>sum</strong> of the <strong>removed</strong> stones' values to the player's score.</li>\r\n\t<li>Place a <strong>new stone</strong>, whose value is equal to that sum, on the left side of the row.</li>\r\n</ol>\r\n\r\n<p>The game stops when <strong>only</strong> <strong>one</strong> stone is left in the row.</p>\r\n\r\n<p>The <strong>score difference</strong> between Alice and Bob is <code>(Alice's score - Bob's score)</code>. Alice's goal is to <strong>maximize</strong> the score difference, and Bob's goal is the <strong>minimize</strong> the score difference.</p>\r\n\r\n<p>Given an integer array <code>stones</code> of length <code>n</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>score difference</strong> between Alice and Bob if they both play <strong>optimally</strong>.</em></p>\r\n\r\n<p> </p>\r\n<p><strong>Example 1:</strong></p>\r\n\r\n<pre>\r\n<strong>Input:</strong> stones = [-1,2,-3,4,-5]\r\n<strong>Output:</strong> 5\r\n<strong>Explanation:</strong>\r\n- Alice removes the first 4 stones, adds (-1) + 2 + (-3) + 4 = 2 to her score, and places a stone of\r\n value 2 on the left. stones = [2,-5].\r\n- Bob removes the first 2 stones, adds 2 + (-5) = -3 to his score, and places a stone of value -3 on\r\n the left. stones = [-3].\r\nThe difference between their scores is 2 - (-3) = 5.\r\n</pre>\r\n\r\n<p><strong>Example 2:</strong></p>\r\n\r\n<pre>\r\n<strong>Input:</strong> stones = [7,-6,5,10,5,-2,-6]\r\n<strong>Output:</strong> 13\r\n<strong>Explanation:</strong>\r\n- Alice removes all stones, adds 7 + (-6) + 5 + 10 + 5 + (-2) + (-6) = 13 to her score, and places a\r\n stone of value 13 on the left. stones = [13].\r\nThe difference between their scores is 13 - 0 = 13.\r\n</pre>\r\n\r\n<p><strong>Example 3:</strong></p>\r\n\r\n<pre>\r\n<strong>Input:</strong> stones = [-10,-12]\r\n<strong>Output:</strong> -22\r\n<strong>Explanation:</strong>\r\n- Alice can only make one move, which is to remove both stones. She adds (-10) + (-12) = -22 to her\r\n score and places a stone of value -22 on the left. stones = [-22].\r\nThe difference between their scores is (-22) - 0 = -22.\r\n</pre>\r\n\r\n<p> </p>\r\n<p><strong>Constraints:</strong></p>\r\n\r\n<ul>\r\n\t<li><code>n == stones.length</code></li>\r\n\t<li><code>2 <= n <= 10<sup>5</sup></code></li>\r\n\t<li><code>-10<sup>4</sup> <= stones[i] <= 10<sup>4</sup></code></li>\r\n</ul>",
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"hints": [
"Let's note that the only thing that matters is how many stones were removed so we can maintain dp[numberOfRemovedStones]",
"dp[x] = max(sum of all elements up to y - dp[y]) for all y > x"
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