Magical String

Magical String

What is this problem about?

The Magical String problem revolves around a specific string composed only of '1's and '2's. The string is "magical" because if you count the number of contiguous '1's and '2's, it generates the exact same string itself! For example, the string starts "1221121221221121122...". If you group it: "1", "22", "11", "2", "1", "22", the lengths of these groups are 1, 2, 2, 1, 1, 2, which exactly matches the string. You are asked to return the number of '1's in the first n characters of this string.

Why is this asked in interviews?

This is a pattern-generation and simulation problem. Interviewers ask it to test a candidate's ability to implement self-referential logic. It evaluates whether you can use Two Pointers to manage two different states simultaneously: one pointer that dictates what to build, and another pointer that represents the end of the string you are actively building.

Algorithmic pattern used

This problem leverages the Two Pointers / Simulation pattern. You seed an array with the initial sequence [1, 2, 2]. You place a read pointer at index 2 (pointing to the value that tells you how many times to add the next number) and a write pointer at the end of the array. You toggle the number to append (if the last number was 1, append 2; if 2, append 1) and append it array[read] times. Then, advance the read pointer.

Example explanation

Let's generate the string up to $n=6$. Seed: arr = [1, 2, 2]. read = 2 (points to 2). The last number written was 2, so the next number to write is 1.

• read says we need two 1s. Append 1, 1.
• arr is now [1, 2, 2, 1, 1]. read moves to index 3 (which holds a 1).
• The last number was 1, so next is 2.
• read says we need one 2. Append 2.
• arr is now [1, 2, 2, 1, 1, 2]. We have 6 elements. We loop through the array and count the '1's. The '1's are at index 0, 3, and 4. Total = 3.

Common mistakes candidates make

A major pitfall is trying to figure out a purely mathematical formula to count the '1's without generating the string. The sequence is highly irregular, and no simple $O(1)$ formula exists; you must simulate the generation. Another common mistake is going out of bounds when n is very small (like n=1), so make sure to handle base cases before entering the simulation loop.

Interview preparation tip

For the Magical String interview pattern, mastering the concept of a "generator queue" or "self-referential array" is key. When an array's future values are dictated by its own previous values, using a read pointer lagging behind a write pointer is the standard, foolproof way to orchestrate the simulation cleanly.