Interview Problem Explained: Find the Maximum Achievable Number (LeetCode 2769)

Find the Maximum Achievable Number: Problem Statement

You are given two integers num and t. An integer x is called achievable if, after performing at most t operations, you can make x equal to num. Each operation is one of the following:

1. Increment both x and num by 1

2. Decrement both x and num by 1

3. Increment x by 1 and decrement num by 1

4. Decrement x by 1 and increment num by 1

Your task is to find the maximum possible value of x that can be achieved under these conditions.

Example (num = 5, t = 3):

• Starting with x = 5 → Already matches num.

• x = 6 → Impossible to match num after 3 operations.

• x = 7 → Possible via:

  1. Decrement x (7→6), increment num (5→6) → Match in 1 operation.

The maximum achievable x here is 11.

Tip: This is LeetCode Problem #2769. Codetree provides detailed explanations and examples to help you grasp concepts that might be unclear in LeetCode’s concise problem statements.

Simplifying the Problem with Perspective Shifts

Idea

While the four operations seem complex, focusing on the difference between x and num simplifies the problem. The goal is to make this difference zero.

• Operations that preserve the difference: Incrementing/decrementing both x and num (cases 1 and 2) → No change.

• Operations that reduce the difference by 2: Decrement x and increment num (case 3) → Difference decreases by 2.

• Operations that increase the difference by 2: Increment x and decrement num (case 4) → Difference increases by 2.

To maximize x, choose operations that reduce the difference by 2 each time. After t operations, the initial difference can be reduced by 2 * t. Thus, the maximum initial x is num + 2 * t.

This approach exemplifies perspective-shifting in problem-solving, commonly associated with Ad-hoc problems.

Logic Breakdown

Formula: Maximum x = num + 2 * t

Code Implementation

class Solution:
    def theMaximumAchievableX(self, num, t):
        """
        :type num: int
        :type t: int
        :rtype: int
        """
        # By choosing to increment x and decrement num in each operation,
        # the difference increases by 2 each time.
        # Thus, the maximum achievable x is num + 2 * t.
        return num + 2 * t

Time & Space Complexity

• Time Complexity: O(1)

• Space Complexity: O(1)

The solution uses a single formula, making it extremely efficient.

Frequently Asked Questions

Q. Why does num + 2 * t guarantee the maximum achievable x?

A. Choosing the operation that reduces the difference by 2 each time allows x to start as high as num + 2 * t and still match num in exactly t steps.

Q. Does the initial value of x affect the strategy?

A. No. The formula num + 2 * t works universally because it maximizes x while ensuring the difference can be eliminated in t steps.

How to Prepare for Interviews with the "Find the Maximum Achievable Number" Problem

The "Find the Maximum Achievable Number" problem is a classic example of an Ad-hoc problem that tests your ability to shift perspectives and derive simple formulas from seemingly complex operations.

This problem helps in developing:

• Ad-hoc thinking: Simplifying multiple cases into a single perspective.

• Mathematical intuition: Expressing iterative operations as a concise formula (num + 2 * t).

Similar Problems to Practice

• Number of Steps to Reduce a Number to Zero

• Minimum Moves to Equal Array Elements

• Moving Blocks

These problems also involve Ad-hoc reasoning, formula derivation, or simple operation rules.

How to Review After Solving

1. Solve other Ad-hoc problems: Look for problems with keywords like "difference" or "sum" and try to simplify naive approaches into formulas.

2. Validate the formula's correctness: Understand why num + 2 * t always yields the maximum achievable x.