#2579

Medium

auto_awesomeMath-driven solution strategy

LeetCode Problem Workspace

Count Total Number of Colored Cells

The "Count Total Number of Colored Cells" problem requires deriving a formula to compute colored cells based on elapsed time.

math

Problem Statement

You are given an infinitely large 2D grid of uncolored cells. After n minutes, cells on the boundary of the grid will be colored, with each minute expanding the colored area by one unit outward. The task is to calculate how many cells are colored after n minutes.

For example, after 1 minute, 1 cell is colored; after 2 minutes, 5 cells are colored, and so on. The goal is to derive a formula to efficiently compute the total number of colored cells after any given number of minutes, n.

Examples

Example 1

Input: n = 1

Output: 1

After 1 minute, there is only 1 blue cell, so we return 1.

Example 2

Input: n = 2

Output: 5

After 2 minutes, there are 4 colored cells on the boundary and 1 in the center, so we return 5.

Constraints

1 <= n <= 105

Solution Approach

Identify the Pattern

Each minute increases the number of colored cells in a predictable pattern. The formula for the total number of colored cells after n minutes is derived by recognizing the boundary layers added at each step.

Derive the Formula

The number of colored cells after n minutes follows the formula: total = n^2 + (n-1)^2. This formula represents the expansion of colored cells with each minute adding an additional layer of cells on the boundary.

Constant Time Calculation

Using the derived formula, the total number of colored cells can be computed in constant time O(1), which allows for a very efficient solution even for large values of n.

Complexity Analysis

Metric	Value
Time	O(1)
Space	O(1)

Both the time and space complexity are O(1) because the solution involves a constant-time mathematical calculation with no iterative steps or space-dependent computations.

What Interviewers Usually Probe

Candidate should demonstrate an ability to recognize mathematical patterns and derive efficient formulas.
Look for clarity in explaining the relation between n and the total colored cells.
Check if the candidate is able to explain how constant time complexity is achieved in this solution.

Common Pitfalls or Variants

Common pitfalls

Candidates might confuse the total number of colored cells with a simpler linear growth pattern, missing the quadratic nature.
Over-complicating the problem by trying to simulate the grid's growth instead of using a direct formula.
Failure to recognize that the problem can be solved in constant time may lead to inefficient solutions.

Follow-up variants

Modify the problem to include a constraint on the grid size, limiting the total number of colored cells after n minutes.
Ask the candidate to calculate the total number of colored cells after k minutes given multiple values of n.
Extend the problem to handle multiple grids and calculate the total number of colored cells across several test cases.

FAQ

How do I calculate the total number of colored cells in the "Count Total Number of Colored Cells" problem?

The formula to calculate the total number of colored cells after n minutes is total = n^2 + (n-1)^2.

What is the time complexity of the solution for this problem?

The solution has a time complexity of O(1) because the formula can be computed directly without iteration.

What is the relationship between n and the total number of colored cells?

The total number of colored cells grows quadratically as n increases, with the formula total = n^2 + (n-1)^2.

How can I optimize the solution for larger values of n?

The formula provides a constant-time solution, O(1), so no further optimization is needed, even for large values of n.

What are common mistakes to avoid in this problem?

Avoid confusing the problem as a linear growth scenario and ensure you understand the quadratic pattern in the number of colored cells.

terminal

Solution

Solution 1: Mathematics

We find that after the $n$th minute, there are a total of $2 \times n - 1$ columns in the grid, and the numbers on each column are respectively $1, 3, 5, \cdots, 2 \times n - 1, 2 \times n - 3, \cdots, 3, 1$. The left and right parts are both arithmetic progressions, and the sum can be obtained by $2 \times n \times (n - 1) + 1$.

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class Solution:
    def coloredCells(self, n: int) -> int:
        return 2 * n * (n - 1) + 1

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