Chalkboard XOR Game

Chalkboard XOR Game

What is this problem about?

The "Chalkboard XOR Game interview question" is a mathematical game played with an array of numbers. Two players take turns removing one number from the chalkboard. A player loses if the XOR sum of all remaining numbers on the board is 0 after their move. However, if the XOR sum is already 0 at the start of a player's turn, that player wins immediately. You need to determine if the first player will win, assuming both play optimally.

Why is this asked in interviews?

This "Chalkboard XOR Game coding problem" is a high-level Brainteaser asked by companies like Google to test a candidate's mathematical intuition and ability to find invariants in a system. It evaluates whether you can move beyond simulation and find a logical "short-circuit" answer based on the properties of the XOR operation and the game's rules.

Algorithmic pattern used

This problem follows the Game Theory and Bit Manipulation interview pattern. The solution relies on two key observations:

1. Immediate Win: If the initial XOR sum of the array is 0, the first player wins immediately.
2. Parity of Array Length: If the initial XOR sum is NOT 0, the game depends on the number of elements. If the number of elements is even, the first player can always find a move that doesn't lead to an XOR sum of 0, eventually forcing the second player into a losing position.

Example explanation

Array: [1, 1, 2]

1. XOR sum: $1 \oplus 1 \oplus 2 = 2$. (Not 0).
2. Length: 3 (Odd). In this case, the first player removes a number. If they remove 2, the board becomes [1, 1], XOR sum is 0. Player 1 loses. If they remove 1, board is [1, 2], XOR sum is 3. Player 2 then removes either 1 or 2, making XOR sum non-zero for themselves? No, this requires deep optimal play analysis, but the pattern boils down to: "If XOR sum != 0 and length is odd, Player 1 loses."

Common mistakes candidates make

• Simulating the Game: Trying to use recursion or minimax to simulate all possible moves. The constraints on the array size make this impossible ($O(2^N)$).
• Misunderstanding XOR: Failing to realize that removing an element $x$ from a set with XOR sum $S$ results in a new XOR sum of $S \oplus x$.
• Ignoring the "Win on Start" rule: Forgetting to check if the XOR sum is already 0 before any moves are made.

Interview preparation tip

When you see XOR combined with a game, think about the properties of XOR. It is commutative, associative, and $x \oplus x = 0$. Many "Game Theory interview pattern" problems involving XOR have simple solutions based on parity or initial sums.