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| 1 | +# 486. Predict the Winner |
| 2 | +You are given an integer array `nums`. Two players are playing a game with this array: player 1 and player 2. |
| 3 | + |
| 4 | +Player 1 and player 2 take turns, with player 1 starting first. Both players start the game with a score of `0`. At each turn, the player takes one of the numbers from either end of the array (i.e., `nums[0]` or `nums[nums.length - 1]`) which reduces the size of the array by `1`. The player adds the chosen number to their score. The game ends when there are no more elements in the array. |
| 5 | + |
| 6 | +Return `true` if Player 1 can win the game. If the scores of both players are equal, then player 1 is still the winner, and you should also return `true`. You may assume that both players are playing optimally. |
| 7 | + |
| 8 | +#### Example 1: |
| 9 | +<pre> |
| 10 | +<strong>Input:</strong> nums = [1,5,2] |
| 11 | +<strong>Output:</strong> false |
| 12 | +<strong>Explanation:</strong> Initially, player 1 can choose between 1 and 2. |
| 13 | +If he chooses 2 (or 1), then player 2 can choose from 1 (or 2) and 5. If player 2 chooses 5, then player 1 will be left with 1 (or 2). |
| 14 | +So, final score of player 1 is 1 + 2 = 3, and player 2 is 5. |
| 15 | +Hence, player 1 will never be the winner and you need to return false. |
| 16 | +</pre> |
| 17 | + |
| 18 | +#### Example 2: |
| 19 | +<pre> |
| 20 | +<strong>Input:</strong> nums = [1,5,233,7] |
| 21 | +<strong>Output:</strong> true |
| 22 | +<strong>Explanation:</strong> Player 1 first chooses 1. Then player 2 has to choose between 5 and 7. No matter which number player 2 choose, player 1 can choose 233. |
| 23 | +Finally, player 1 has more score (234) than player 2 (12), so you need to return True representing player1 can win. |
| 24 | +</pre> |
| 25 | + |
| 26 | +#### Constraints: |
| 27 | +* `1 <= nums.length <= 20` |
| 28 | +* <code>0 <= nums[i] <= 10<sup>7</sup></code> |
| 29 | + |
| 30 | +## Solutions (Python) |
| 31 | + |
| 32 | +### 1. Solution |
| 33 | +```Python |
| 34 | +from functools import cache |
| 35 | + |
| 36 | + |
| 37 | +class Solution: |
| 38 | + def predictTheWinner(self, nums: List[int]) -> bool: |
| 39 | + @cache |
| 40 | + def subArrayMaxDiff(i: int, j: int) -> int: |
| 41 | + if i == j: |
| 42 | + return nums[i] |
| 43 | + |
| 44 | + return max(nums[i] - subArrayMaxDiff(i + 1, j), nums[j] - subArrayMaxDiff(i, j - 1)) |
| 45 | + |
| 46 | + return subArrayMaxDiff(0, len(nums) - 1) >= 0 |
| 47 | +``` |
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