Triangles (Editorial)

Consider a triangle whose each side is divided into $n$ equal parts. The largest number of identical shapes will be among the smallest equal triangles. Let us count them.

Let us divide the triangle into $n$ horizontal strips, numbered from top to bottom.

• In the last, $n$-th (bottom) strip, there are $n$ triangles pointing upwards and n - 1 triangles pointing downwards.

• In the penultimate, (n - 1)-th strip, there are n - 1 triangles pointing upwards and n - 2 triangles pointing downwards.

Continuing this reasoning, we obtain:

• the number of triangles pointing upwards is n + (n - 1) + (n - 2) + ... + 2 + 1;

• the number of triangles pointing downwards is (n - 1) + (n - 2) + ... + 2 + 1.

Thus, the total number of the smallest equal triangles is the sum of these two quantities:

Example

For n = 1, 2, and $3$, the number of equal triangles is $1,4$, and $9$, respectively. For example, when n = 3, the answer is (1 + 2 + 3) + (1 + 2) = 9.

Algorithm implementation

Read the input value $n$.

scanf("%lld",&n);

Compute and print the answer.

res = n * n;
printf("%lld\n",res);

Java implementation

import java.util.*;

class Main 
{
  public static void main(String[] args) 
  {
    Scanner con = new Scanner(System.in);
    long n = con.nextLong();
    long res = n * n;
    System.out.println(res);
    con.close();
  }
}

Python implementation

Read the input value $n$.

n = int(input())

Compute and print the answer.

res = n * n
print(res)