Count Total Number of Colored Cells - Practice Coding Problems

Count Total Number of Colored Cells - Problem

Math Medium

Imagine you're playing a cellular expansion game on an infinite grid! You start by coloring a single cell blue, and each minute, the blue color spreads to all neighboring uncolored cells (up, down, left, right).

Given a positive integer n, you need to determine how many cells will be colored blue after exactly n minutes of this spreading process.

The Process:

Minute 1: Color any arbitrary cell blue (1 cell total)
Minute 2: Color all uncolored cells adjacent to blue cells (5 cells total)
Minute 3: Continue the expansion (13 cells total)

This creates a diamond-shaped pattern that grows outward each minute. Your task is to calculate the total number of colored cells after n minutes.

Goal: Return the total count of blue cells after the expansion completes.

Input & Output

example_1.py — Python

$ Input: n = 1

› Output: 1

💡 Note: At minute 1, we color only the initial cell, so the total is 1 colored cell.

example_2.py — Python

$ Input: n = 2

› Output: 5

💡 Note: Minute 1: 1 cell colored. Minute 2: The 4 adjacent cells get colored (up, down, left, right). Total = 1 + 4 = 5 cells.

example_3.py — Python

$ Input: n = 3

› Output: 13

💡 Note: The diamond pattern continues to expand. At minute 3, we add 8 more cells to the outer ring, giving us 5 + 8 = 13 total colored cells.

Constraints

1 ≤ n ≤ 10⁵
n is a positive integer
The result will fit in a 64-bit signed integer

Visualization

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Understanding the Visualization

Initial Drop

One blue cell appears at the center

First Spread

Ink spreads to 4 adjacent cells, forming a plus shape

Diamond Formation

The pattern forms a clear diamond with 13 total cells

Pattern Recognition

Each ring adds 4k cells where k is the ring number

Key Takeaway

🎯 Key Insight: The diamond pattern grows predictably - each minute n creates a diamond containing all cells within Manhattan distance ≤ (n-1) from the center, leading to the elegant O(1) formula.

Asked in

G Google 15 f Meta 12 a Amazon 8 ⊞ Microsoft 6

The optimal solution recognizes that the colored cells form a diamond pattern where at minute n, all cells within Manhattan distance ≤ (n-1) from the center are colored. This gives us the elegant mathematical formula: 2n² - 2n + 1, providing an O(1) time and space solution.

Common Approaches

Approach	Time	Space	Notes
✓ Mathematical Formula (Optimal)	O(1)	O(1)	Use the mathematical pattern: 2n² - 2n + 1 to directly calculate the result

Mathematical Formula (Optimal) — Algorithm Steps

Recognize that colored cells form a diamond pattern
Observe that at minute n, all cells with Manhattan distance ≤ (n-1) from center are colored
Count cells in diamond: sum of cells at each distance level
Apply the derived formula: 2n² - 2n + 1

Visualization

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Step-by-Step Walkthrough

Pattern Analysis

Minute 1: 1 cell, Minute 2: 5 cells, Minute 3: 13 cells

Formula Derivation

Sequence: 1, 5, 13, 25, 41... follows 2n² - 2n + 1

Verification

Check: n=1→1, n=2→5, n=3→13, n=4→25

Code -

solution.c — C

#include <stdio.h>

long long coloredCells(int n) {
    // Mathematical formula: 2n² - 2n + 1
    // This counts all cells in a diamond with Manhattan distance <= (n-1)
    return 2LL * n * n - 2LL * n + 1;
}

int main() {
    int n;
    scanf("%d", &n);
    printf("%lld\n", coloredCells(n));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(1)

Single arithmetic calculation regardless of input size

✓ Linear Growth

Space Complexity

O(1)

Only uses constant extra space for the calculation

✓ Linear Space

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Medium Frequency

~12 min Avg. Time

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Ln 1, Col 1

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Mathematical Formula (Optimal) — Algorithm Steps

Visualization

Code -

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