## USACO 2020 February Contest, Platinum

## Problem 2. Equilateral Triangles

Contest has ended.

**Log in to allow submissions in analysis mode**

FJ believes that the beauty of his pasture is directly proportional to the number of triples of cows such that their positions are equidistant from each other. In other words, they form an equilateral triangle. Unfortunately, it was only quite recently that FJ realized that since all of his cows are located at integer coordinates, no beautiful triples can possibly exist if Euclidean distance is used! Thus, FJ has decided to switch to use "Manhattan" distance instead. Formally, the Manhattan distance between two positions $(x_0,y_0)$ and $(x_1,y_1)$ is equal to $|x_0-x_1|+|y_0-y_1|$.

Given the grid representing the positions of the cows, compute the number of equilateral triples.

#### SCORING:

There will be fourteen test cases aside from the sample, one for each of $N\in \{50,75,100,125,150,175,200,225,250,275,300,300,300,300\}.$#### INPUT FORMAT (file triangles.in):

The first line contains a single integer $N.$For each $1\le i\le N,$ line $i+1$ of the input contains a string of length $N$ consisting solely of the characters '*' and '.'. The $j$th character describes whether there exists a cow at position $(i,j)$ or not.

#### OUTPUT FORMAT (file triangles.out):

Output a single integer containing the answer. It can be shown that it fits into a signed 32-bit integer.#### SAMPLE INPUT:

3 *.. .*. *..

#### SAMPLE OUTPUT:

1

There are three cows, and they form an equilateral triple because the Manhattan distance between each pair of cows is equal to two.

Problem credits: Benjamin Qi