Chalkboard XOR Game

Problem Statement

In the given problem, an array of integers nums represents the set of numbers written on a chalkboard. Alice and Bob are in a competition where they alternately erase one number from the chalkboard, starting with Alice. The challenge lies in the effect of the bitwise XOR operation applied to the numbers left on the board after each move. A player loses if their move results in the XOR of all remaining numbers to be zero. Additionally, a player immediately wins if the XOR result of all the chalkboard numbers at the start of their turn is zero. The objective is to determine if Alice can win the game, given that both players are using their best strategies throughout the game.

Examples

Example 1

Input:

nums = [1,1,2]

Output:

false

Explanation:

Alice has two choices: erase 1 or erase 2.
If she erases 1, the nums array becomes [1, 2]. The bitwise XOR of all the elements of the chalkboard is 1 XOR 2 = 3. Now Bob can remove any element he wants, because Alice will be the one to erase the last element and she will lose.
If Alice erases 2 first, now nums become [1, 1]. The bitwise XOR of all the elements of the chalkboard is 1 XOR 1 = 0. Alice will lose.

Example 2

Input:

nums = [0,1]

Output:

true

Example 3

Input:

nums = [1,2,3]

Output:

true

Constraints

• 1 <= nums.length <= 1000
• 0 <= nums[i] < 216

Approach and Intuition

Understanding Bitwise XOR for Gameplay:

The XOR (exclusive OR) operation is a fundamental concept utilized here:

• XOR of a number with itself is zero.
• XOR of a number with zero is the number itself.

Thus, the game's outcome can be dictated by the initial condition of the chalkboard numbers.

Strategy Analysis Using Examples:

1. Example 1 Analysis (nums = [1,1,2]):

   • Alice’s options are:
     • If she removes a 1, the remaining numbers are [1, 2]. Their XOR equals 3, which isn't zero. However, Alice eventually loses after more moves, since removing any number would leave a situation for Bob to respond optimally and force Alice into a losing position.
     • If she removes the 2, the remaining numbers are [1, 1]. Their XOR is 0, thus Alice immediately loses.
   • Conclusion: Regardless of the approach, Alice ends up losing.

2. Example 2 Analysis (nums = [0, 1]):

   • Initially, the XOR of [0, 1] is 1, which isn’t zero. Alice can remove 0, leading to a XOR remaining of [1] which is 1. On Bob's turn, any move leads to zero, causing Bob to lose.
   • Conclusion: With optimal play, Alice wins.

3. Example 3 Analysis (nums = [1, 2, 3]):

   • Initial XOR is 0 when computed for all integers, meaning if Alice beginnings with such a scenario, she wins immediately with no move required.
   • In any nuanced plays, strategic removals keep the scenario mostly favorable to the starter, which in every game case, is Alice.

Strategy for Winning:

• Alice will win if the XOR of all numbers in nums at the start of her turn isn’t zero. She then should play strategically to either maintain non-zero XOR conditions or manipulate the board such that Bob faces a zero-XOR scenario on his turn.
• If the game starts with a XOR sum of zero naturally, Alice wins automatically without making any move.

Considering these examples and the application of bitwise operations, the game heavily relies on the cumulative XOR of the starting array and the ongoing strategic interaction between Alice and Bob. Understanding the XOR conditions after each potential move is key to determining the optimal strategy and predicting the game's outcome, fundamentally benefiting Alice if she begins optimally and adapts as the board evolves.

Solutions

class Solution {
    public boolean playXORGame(int[] elements) {
        int result = 0;
        for (int element : elements) result ^= element;
        return result == 0 || elements.length % 2 == 0;
    }
}