Problem:
Score : 400 points

Problem Statement

There is a grid with N rows and N columns of squares. Let (i,j) be the square at the i-th row from the top and the j-th column from the left.

These squares have to be painted in one of the C colors from Color 1 to Color C. Initially, (i,j) is painted in Color c_{i,j}.

We say the grid is a good grid when the following condition is met for all i,j,x,y satisfying 1 \leq i,j,x,y \leq N:

If (i+j) \% 3=(x+y) \% 3, the color of (i,j) and the color of (x,y) are the same.

If (i+j) \% 3 \neq (x+y) \% 3, the color of (i,j) and the color of (x,y) are different.

Here, X \% Y represents X modulo Y.

We will repaint zero or more squares so that the grid will be a good grid.

For a square, the wrongness when the color of the square is X before repainting and Y after repainting, is D_{X,Y}.

Find the minimum possible sum of the wrongness of all the squares.

Constraints

1 \leq N \leq 500

3 \leq C \leq 30

1 \leq D_{i,j} \leq 1000 (i \neq j),D_{i,j}=0 (i=j)

1 \leq c_{i,j} \leq C

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N C
D_{1,1} ... D_{1,C}
:
D_{C,1} ... D_{C,C}
c_{1,1} ... c_{1,N}
:
c_{N,1} ... c_{N,N}

Output

If the minimum possible sum of the wrongness of all the squares is x, print x.

Sample Input 1

2 3
0 1 1
1 0 1
1 4 0
1 2
3 3

Sample Output 1

3

Repaint (1,1) to Color 2. The wrongness of (1,1) becomes D_{1,2}=1.

Repaint (1,2) to Color 3. The wrongness of (1,2) becomes D_{2,3}=1.

Repaint (2,2) to Color 1. The wrongness of (2,2) becomes D_{3,1}=1.

In this case, the sum of the wrongness of all the squares is 3.

Note that D_{i,j} \neq D_{j,i} is possible.

Sample Input 2

4 3
0 12 71
81 0 53
14 92 0
1 1 2 1
2 1 1 2
2 2 1 3
1 1 2 2

Sample Output 2

428

Solution in OCaml (4.02.3) (judge verdict: Accepted)

(* 与えられたリストからk要素を選ぶ順列 *)
let perm k xs =
  let rec perm_aux trace n k i xs ys acc =
    match k, i, xs with
    | 0, _, _ -> trace :: acc
    | _, 0, _ -> acc
    | _, _, [] -> perm_aux trace n k i (List.rev ys) [] acc
    | _, _, x :: xs ->
        perm_aux trace n k (i - 1) xs (x :: ys) @@
        perm_aux (x :: trace) (n - 1) (k - 1) (n - 1) xs ys acc in
  let n = List.length xs in
  perm_aux [] n k n xs [] []

let () = Scanf.scanf "%d %d\n" @@ fun n c ->
  let dss =
    Array.init c @@ fun _ ->
      Array.init c @@ fun _ ->
        Scanf.scanf "%d " @@ fun d -> d in
  let css =
    Array.init n @@ fun _ ->
      Array.init n @@ fun _ ->
        Scanf.scanf "%d " @@ fun c -> c in
  let colours = Array.make_matrix 3 c 0 in
  Array.iteri (fun i ->
    Array.iteri (fun j c ->
      colours.((i + j + 2) mod 3).(c - 1) <- colours.((i + j + 2) mod 3).(c - 1) + 1)) css;
  Array.init c (fun c -> c)
  |> Array.to_list
  |> perm 3
  |> List.map (List.mapi (fun m y ->
      Array.mapi (fun x n -> n * dss.(x).(y)) colours.(m)
      |> Array.fold_left ( + ) 0))
  |> List.map (List.fold_left ( + ) 0)
  |> List.fold_left min max_int
  |> Printf.printf "%d\n"