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 nth minute, there are a total of 2 × n - 1 columns in the grid, and the numbers on each column are respectively 1, 3, 5, …, 2 × n - 1, 2 × n - 3, …, 3, 1. The left and right parts are both arithmetic progressions, and the sum can be obtained by 2 × n × (n - 1) + 1.
The time complexity is O(1), and the space complexity is O(1).
PythonJavaC++GoTypeScriptRust
class Solution: def coloredCells(self, n: int) -> int: return 2 * n * (n - 1) + 1(code-box)
