Elimination Game

Elimination Game

1. What is this problem about?

The Elimination Game coding problem is a fascinating mathematical puzzle that involves repeatedly narrowing down a list of numbers until only one remains. Starting with a list from 1 to $n$, you perform "sweeps." In the first sweep, you remove every other number starting from the left. In the second sweep, you remove every other number starting from the right. This alternating process continues until a single number is left. It is a test of identifying patterns in sequences and understanding how the "head" of a list moves as elements are removed.

2. Why is this asked in interviews?

This question is a favorite for companies like Microsoft and Amazon because it tests mathematical intuition and the ability to optimize. While a brute-force approach using a list or array is easy to conceive, it is highly inefficient for large values of $n$. Interviewers want to see if you can find a constant-time or logarithmic-time solution by tracking the state of the list (the starting element, the step size, and the remaining count) rather than actually simulating the removal of every element.

3. Algorithmic pattern used

The Elimination Game interview pattern typically involves Recursion or Iterative State Tracking. The key is to realize that after each sweep, the remaining numbers still form an arithmetic progression. You only need to track the first element (the "head"), the "step" (which doubles every round), and whether the head moves during a specific sweep. The head moves if the sweep is from the left, or if it's from the right and the number of elements is odd.

4. Example explanation

Let's take $n = 6$:

• Initial: [1, 2, 3, 4, 5, 6], Head = 1, Step = 1, Left = True
• Sweep 1 (Left): Remove 1, 3, 5. Remaining: [2, 4, 6]. New Head = 2, Step = 2, Left = False.
• Sweep 2 (Right): Remove 6, 2 (since it's every other from the right). Remaining: [4]. New Head = 4, Step = 4, Left = True. The result is 4. Notice how the step doubled each time and the head updated based on the direction and count.

5. Common mistakes candidates make

• Brute Force Simulation: Trying to use a List or LinkedList to literally remove elements. This will lead to Memory Limit Exceeded (MLE) or Time Limit Exceeded (TLE) for large $n$.
• Incorrect Head Update Logic: Failing to identify the specific condition under which the head element changes during a right-to-left sweep (it only changes if the number of elements is odd).
• Ignoring the Pattern: Not noticing that the remaining numbers always maintain a consistent gap (the "step").

6. Interview preparation tip

When you encounter "Josephus-style" or "Elimination" problems, don't jump into coding a simulation. Draw out the first few rounds for a small $n$ and observe how the first element of the set changes. Often, these problems have a recursive structure where the problem for $n$ can be defined by the problem for $n/2$. Mastery of these patterns shows interviewers you can think abstractly and mathematically.