README_EN.md

2579. Count Total Number of Colored Cells

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Description

There exists an infinitely large two-dimensional grid of uncolored unit cells. You are given a positive integer n, indicating that you must do the following routine for n minutes:

• At the first minute, color any arbitrary unit cell blue.
• Every minute thereafter, color blue every uncolored cell that touches a blue cell.

Below is a pictorial representation of the state of the grid after minutes 1, 2, and 3.

Return the number of colored cells at the end of n minutes.

 

Example 1:

Input: n = 1
Output: 1
Explanation: After 1 minute, there is only 1 blue cell, so we return 1.

Example 2:

Input: n = 2
Output: 5
Explanation: After 2 minutes, there are 4 colored cells on the boundary and 1 in the center, so we return 5. 

 

Constraints:

• 1 <= n <= 105

Solutions

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 $2\times n\times (n-1)+1$.

Time complexity $O(1)$, space complexity $O(1)$.

Python3

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

Java

class Solution {
    public long coloredCells(int n) {
        return 2L * n * (n - 1) + 1;
    }
}

C++

class Solution {
public:
    long long coloredCells(int n) {
        return 2LL * n * (n - 1) + 1;
    }
};

Go

func coloredCells(n int) int64 {
	return int64(2*n*(n-1) + 1)
}

TypeScript

function coloredCells(n: number): number {
    return 2 * n * (n - 1) + 1;
}

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