End-Picking Number Game: Can the First Player Win Under Optimal Play?

Two players play a game on a row of non-negative integers nums. They alternate turns, and Player 1 moves first. On each turn, the current player removes either the leftmost or the rightmost remaining number and adds it to their own score. The game ends when no numbers remain.

Both players play optimally. Return whether Player 1 can finish with a score at least as large as Player 2's.

Function Signature

def first_player_can_win(nums: list[int]) -> bool:

Rules

• Both scores start at 0.
• "Optimally" means each player always chooses the move that maximizes their own final score minus the opponent's final score, assuming the opponent does the same.
• Return True if Player 1's final score is greater than or equal to Player 2's under optimal play (a tie counts as a win for Player 1), and False otherwise.

Constraints

• 1 <= len(nums) <= 20
• 0 <= nums[i] <= 10^7

Examples

Example 1

• Input: nums = [2, 9, 4]
• Output: False
• Explanation: Whichever end Player 1 takes, Player 2 can then take 9. Player 1's best total is 6 against 9.

Example 2

• Input: nums = [3, 7, 2, 2]
• Output: True
• Explanation: Player 1 takes the rightmost 2, leaving [3, 7, 2]. Whichever end Player 2 takes, Player 1 can then take 7. Player 1 finishes with 9 against Player 2's 5.

Example 3

• Input: nums = [5]
• Output: True