Stone Game IV

Stone Game IV

What is this problem about?

Stone Game IV is a game where Alice and Bob take turns removing a non-zero square number of stones (1, 4, 9, 16, ...) from a single pile of $n$ stones. The player who takes the last stone wins. You need to determine if Alice (the first player) can win given $n$ stones.

Why is this asked in interviews?

This problem is a pure test of Game Theory and Dynamic Programming. It assesses whether you can recognize a "winning state" vs a "losing state." It's common in Microsoft interviews to see if a candidate can optimize a recursive solution with memoization or implement an iterative DP approach efficiently.

Algorithmic pattern used

The pattern is Dynamic Programming.

• Winning State: A state $n$ is a winning state if there exists some square number $k^2 \le n$ such that $n - k^2$ is a losing state.
• Losing State: A state $n$ is a losing state if for all possible square numbers $k^2 \le n$, $n - k^2$ is a winning state. We can build a boolean DP array canWin[n+1] where canWin[i] is true if the current player can win starting with $i$ stones.

Example explanation (use your own example)

Let $n = 7$.

• $i=0$: False (Losing state).
• $i=1$: Can take $1^2=1$. $1-1=0$ (Losing). So 1 is Winning (True).
• $i=2$: Can only take 1. $2-1=1$ (Winning). So 2 is Losing (False).
• $i=3$: Can only take 1. $3-1=2$ (Losing). So 3 is Winning (True).
• $i=4$: Can take $2^2=4$. $4-4=0$ (Losing). So 4 is Winning (True).
• $i=5$: Can take 1 or 4. $5-1=4$ (W), $5-4=1$ (W). All lead to Winning states, so 5 is Losing (False).
• $i=6$: Can take 1 or 4. $6-1=5$ (L), so 6 is Winning (True).
• $i=7$: Can take 1 or 4. $7-4=3$ (W), $7-1=6$ (W). Wait, if $7-4=3$ and 3 is Winning, and $7-1=6$ and 6 is Winning, then 7 is a losing state? No, we check if any move leads to a false. $7-1=6(T)$, $7-4=3(T)$. Both true, so 7 is False.

Common mistakes candidates make

• Slow iteration: Not optimizing the inner loop that checks square numbers ($O(N \sqrt{N})$ is the goal).
• Base case: Not realizing that 0 stones is a losing state for the person whose turn it is.
• Confusing Win/Loss: Forgetting the logic that "I win if I can force you to lose."
• Greedy logic: Thinking that taking the largest possible square is always the best move (it's not; any square that leads to a losing state is fine).

Interview preparation tip

Game Theory problems often boil down to: "Can I make a move such that the remaining game state is a loss for my opponent?" This simple boolean logic is the heart of many Hard coding problems.