Marble Bucket Game

This article is part of the Scaler Topics Fortnightly Contest - 27

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Solution Approach

Winning Move for First Bucket with More Than 1 Marble:
- Player who encounters this bucket controls the outcome.
- They can either remove all marbles or leave only 1, forcing the opponent into a losing position.
Counting Consecutive 1-Marble Buckets:
- If the first non-empty bucket has only 1 marble, count subsequent 1-marble buckets.
- This count determines the winning condition based on parity (odd or even).
Winning Condition Based on Consecutive 1s:
- Case 1: Not All Buckets Have 1 Marble
  - Even count of consecutive 1-marble buckets: First player (Jack) wins.
  - Odd count of consecutive 1-marble buckets: Second player (friend) wins.
- Case 2: All Buckets Have 1 Marble
  - Odd number of total buckets: First player (Jack) wins.
  - Even number of total buckets: Second player (friend) wins.