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 Scala (2.13.1) (judge verdict: Accepted)

object Main extends App {
  val in = new java.util.Scanner(System.in)
  val N, C = in.nextInt
  val Dxy = Array.fill(C)(Array.fill(C)(in.nextInt))
  val grid: Array[List[Int]] = Array.fill(3)(Nil)

  for {
    r <- 0 until N
    c <- 0 until N
  } {
    grid((r + c) % 3) = in.nextInt :: grid((r + c) % 3)
  }

  val gridColorChanges: Array[Array[Int]] = {
    grid.map { colors =>
      val colorSizes: List[(Int, Int)] = colors.groupBy(i => i - 1).view.mapValues(_.size).toList
      (0 until C).map(to => colorSizes.map{case (from, size) => Dxy(from)(to) * size}.sum).toArray
    }
  }

  val min = (for {
    p0 <- 0 until C
    p1 <- 0 until C
    if p0 != p1
    p2 <- 0 until C
    if p0 != p2 && p1 != p2
  } yield {
    gridColorChanges(0)(p0) + gridColorChanges(1)(p1) + gridColorChanges(2)(p2)
  }).min
  println(min)
}