The Two Sneaky Numbers of Digitville

The Two Sneaky Numbers of Digitville

What is this problem about?

In the world of Digitville, numbers usually appear in a very organized fashion. You are given an array of size n + 2 that contains all integers from 0 to n - 1 at least once. However, there's a catch: two numbers in the array are "sneaky" and appear twice. Your task is to identify these two duplicate numbers.

Why is this asked in interviews?

This the Two Sneaky Numbers of Digitville interview question is a variation of the classic "find duplicates" problem. It is frequently asked by Microsoft, Meta, and Amazon to test a candidate's ability to manipulate arrays and utilize mathematical properties for optimization. While a hash table provides an easy solution, interviewers often look for more space-efficient methods, such as using bit manipulation (XOR) or mathematical equations, to demonstrate a deeper understanding of computer science fundamentals.

Algorithmic pattern used

This problem follows the Array, Math, Hash Table interview pattern.

1. The simplest approach is using a Hash Table or a frequency array to count occurrences. When you encounter a number already in the set, it's one of the sneaky ones.
2. A more space-efficient (O(1)) approach involves Math:
   • Calculate the sum of all elements in the array and compare it to the sum of numbers 0 to n-1. The difference is x + y.
   • Calculate the sum of squares and compare it to the sum of squares of numbers 0 to n-1. The difference is x^2 + y^2.
   • Solve the resulting system of equations to find x and y.
3. Another O(1) space approach is XOR:
   • XOR all elements in the array and all numbers from 0 to n-1. The result is x XOR y.
   • Use the rightmost set bit in the XOR result to partition the numbers into two groups and find x and y separately.

Example explanation

n = 3, Array = [0, 1, 2, 1, 0].

1. Target numbers: 0, 1, 2.
2. Frequency count:
   • 0: appears twice.
   • 1: appears twice.
   • 2: appears once. The sneaky numbers are 0 and 1. Using XOR:

• XOR of array = 0 ^ 1 ^ 2 ^ 1 ^ 0 = 2.
• XOR of 0..2 = 0 ^ 1 ^ 2 = 3.
• Combined XOR = 2 ^ 3 = 1.
• Partition based on the 1st bit... (and so on) to find 0 and 1.

Common mistakes candidates make

In "The Two Sneaky Numbers of Digitville coding problem," a common mistake is using a sorting approach (O(n log n)) when a linear time solution is possible. Another error is not correctly handling the range of the numbers (0 to n-1), leading to off-by-one errors in mathematical formulas. Finally, forgetting to return exactly two numbers or returning them in the wrong format can be an issue.

Interview preparation tip

Finding duplicates is a "bread and butter" interview topic. Master the XOR trick and the mathematical sum/square-sum method, as they are applicable to many variations of this problem. Always consider the trade-off between time complexity and space complexity (e.g., using a hash table vs. O(1) space).