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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using static System.Console;
using static System.Convert;
using static System.Math;

class Program
{
    private static int res = int.MaxValue;
    static void Main(string[] args)
    {
        var nc = Array.ConvertAll(ReadLine().Split(' '), int.Parse);
        var ds = new int[nc[1]][];
        for (var i = 0; i < nc[1]; i++)
            ds[i] = Array.ConvertAll(ReadLine().Split(' '), int.Parse);
        
        var dic = new Dictionary<int, Dictionary<int, int>>();
        for(var i = 1; i <= nc[0]; i++)
        {
            var ar = Array.ConvertAll(ReadLine().Split(' '), int.Parse);
            for (var j = 0; j < nc[0]; j++)
            {
                var mod = (i + j + 1) % 3;
                if (!dic.ContainsKey(mod))
                    dic[mod] = new Dictionary<int, int>();
                if (dic[mod].ContainsKey(ar[j]-1))
                    dic[mod][ar[j]-1]++;
                else
                    dic[mod][ar[j]-1] = 1;
            }
        }
        for (var i = 0; i < nc[1]; i++)
            dfs(Enumerable.Repeat(-1,3).ToArray(), 0,  i, ds, dic, nc[1]);
        WriteLine(res);
    }
    private static void dfs(int[] sel,int index,int num,int[][] ds,Dictionary<int,Dictionary<int,int>> dic,int c)
    {
        sel[index] = num;
        if (index == 2)
        {
            var sum = 0;
            for(var i = 0; i < 3; i++)
            {
                for (var j = 0; j < c; j++)
                    if (dic.ContainsKey(i)&&dic[i].ContainsKey(j))
                        sum += ds[j][sel[i]] * dic[i][j];
            }
            res = Min(res, sum);
            sel[index] = -1;
            return;
        }
        for (var i = 0; i < c; i++)
            if (!sel.Contains(i))
                dfs(sel, index + 1, i, ds, dic, c);
        sel[index] = -1;
    }
    private const string ALFA = "abcdefghijklmnopqrstuvwxyz";
    private const int MOD = 1000000007;
}