Problem:
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.

Sample Input 1

3 2
1 2
5 5
-2 8

Sample Output 1

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.

Sample Input 2

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

Sample Output 2

2

Sample Input 3

5 1
1
2
3
4
5

Sample Output 3

10

Solution in OCaml (4.10.0) (judge verdict: Accepted)

let is_integer xs ys =
  let dim = Array.length xs in
  let sq_dist = Array.fold_left ( + ) 0 @@
    Array.init dim @@ fun i -> (xs.(i) - ys.(i)) * (xs.(i) - ys.(i)) in
  let check = Array.exists (fun d -> d * d = sq_dist) (Array.init 160 @@ fun i -> (i + 1)) in
  if check then 1 else 0

let () = Scanf.scanf "%d %d\n" @@ fun n d ->
  let points =
    Array.init n @@ fun _ ->
      Array.init d @@ fun _ -> Scanf.scanf " %d" Fun.id in
  let check_boxes =
    Array.init (n - 1) @@ fun i ->
      Array.init (n - 1 - i) @@ fun j -> is_integer points.(i) points.(i + j + 1) in
  let ans = 
    Array.fold_left (Array.fold_left ( + )) 0 check_boxes in
  Printf.printf "%d\n" ans 


(*
  Array.fold_left: ('a -> 'b -> 'a) -> 'a -> 'b array -> 'a
  1つ目の fold_left は (int -> int array -> int) -> int -> int array array -> int
  2つ目の fold_left は ( + ) を渡しているので ('a -> 'a array -> 'a) になる
  ここで期待するのは int -> int list -> int
*)