Stone Game IX

Problem Description

Alice and Bob take turns removing stones from a row of stones, each with an associated value. Alice starts first, and the player who removes a stone loses if the sum of the values of all removed stones is divisible by 3. Bob wins automatically if there are no remaining stones. Determine if Alice wins assuming both play optimally.

Key Insights

• The outcome of the game depends on the sum of the values of the removed stones modulo 3.
• The game can be analyzed based on how many stones have values that contribute to the total being divisible by 3.
• The optimal strategy involves understanding the counts of stones with values modulo 3 (i.e., how many have a value of 0, 1, or 2 modulo 3).
• The player who is forced into a position where they have to remove a stone that makes the sum divisible by 3 will lose.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve the problem, we can count the number of stones with values that are congruent to 0, 1, and 2 modulo 3. Based on these counts, we can determine if Alice can force Bob into a losing position. The key steps involve:

1. Counting how many stones have values equivalent to 0, 1, and 2 when taken modulo 3.
2. Analyzing the counts to determine the winning strategy.

The approach uses a greedy algorithm to ensure that players make optimal choices based on the current state of the game.

Code Solutions