B - Good Distance

Score : $200$ points

Problem Statement

There are $N$ points in a $D$-dimensional space.

The coordinates of the $i$-th point are $(X_{i1}, X_{i2}, ..., X_{iD})$.

The distance between two points with coordinates $(y_1, y_2, ..., y_D)$ and $(z_1, z_2, ..., z_D)$ is $\sqrt{(y_1 - z_1)^2 + (y_2 - z_2)^2 + ... + (y_D - z_D)^2}$.

How many pairs $(i, j)$ $(i < j)$ are there such that the distance between the $i$-th point and the $j$-th point is an integer?

Constraints

• All values in input are integers.
• $2 \leq N \leq 10$
• $1 \leq D \leq 10$
• $-20 \leq X_{ij} \leq 20$
• No two given points have the same coordinates. That is, if $i \neq j$, there exists $k$ such that $X_{ik} \neq X_{jk}$.

Input

Input is given from Standard Input in the following format:

$N$ $D$

$X_{11}$ $X_{12}$ $...$ $X_{1D}$

$X_{21}$ $X_{22}$ $...$ $X_{2D}$

$\vdots$

$X_{N1}$ $X_{N2}$ $...$ $X_{ND}$

Output

Print the number of pairs $(i, j)$ $(i < j)$ such that the distance between the $i$-th point and the $j$-th point is an integer.

3 2
1 2
5 5
-2 8

1

The number of pairs with an integer distance is one, as follows:

• The distance between the first point and the second point is $\sqrt{|1-5|^2 + |2-5|^2} = 5$, which is an integer.
• The distance between the second point and the third point is $\sqrt{|5-(-2)|^2 + |5-8|^2} = \sqrt{58}$, which is not an integer.
• The distance between the third point and the first point is $\sqrt{|-2-1|^2+|8-2|^2} = 3\sqrt{5}$, which is not an integer.

3 4
-3 7 8 2
-12 1 10 2
-2 8 9 3

2

5 1
1
2
3
4
5

10