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 (2.11.7) (judge verdict: Time Limit Exceeded)

import scala.io.{StdIn => in}
import scala.collection.{mutable => mu}

object Main extends App {
  val Array(n, c) = in.readLine.split(" ").map(_.toInt)
  val dcf = Iterator.continually(in.readLine)
    .take(c).map(_.split(" ").map(_.toInt)).toArray
  val pallet = Iterator.continually(in.readLine)
    .take(n).map(_.split(" ").map(_.toInt)).toArray

  val table = Array.fill(3)(mu.ArrayBuffer.empty[Int])
  for {
    i <- 0 until n
    j <- 0 until n
  } {
    val rem = (i + j) % 3
    table(rem) += pallet(i)(j)
  }
  table.foreach(s => debug(s.mkString(" ")))

  def dfsEnum(d:Int, n:Int):Int = {
    // d進 n個列挙
    val enum = Array.ofDim[Int](n)
    var ret = Int.MaxValue
  
    def evalute():Unit = {
      if(enum.toSet.size == 3) {
        val res = 
          enum.indices.map(i => table(i)./:(0)((acc, j) => {
            acc + dcf(j-1)(enum(i))
          })).sum
        ret = math.min(ret, res)
        // debug(enum.mkString(" "), " ", res)
      }
    }
  
    def dfs(cnt:Int):Unit = {
      if(cnt == n) evalute
      else {
        for(i <- 0 until c) {
          enum(cnt) = i
          dfs(cnt+1)
        }
      }
    }
  
    dfs(0)
    ret
  }

  val ans = dfsEnum(c,3)
  println(ans)

  def debug(any:Any*):Unit = {
    if(this.getClass.toString != "class Main$")
      println(any.mkString(" "))
  }
}