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 Python (3.8.2) (judge verdict: Accepted)

# D - Good Grid

from collections import defaultdict
import itertools

N, C = map(int, input().split())
D = [[0] * C for _ in range(C)]
for i in range(C):
    D[i] = list(int(d) for d in input().split())

# MOD3の世界で0,1,2ごとに、最初の値の数をカウントしておく
dict_c = [defaultdict(int) for _ in range(3)]
for i in range(N):
    c = list(int(x) for x in input().split())
    for j in range(N):
        dict_c[(i+j)%3][c[j]-1] += 1

# ここからは全探索。
# 0, 1, 2の各配列に対して、全ての色を試す(C**3)
list_C = list(range(C))
ans = 10**18
for (p, q, r) in itertools.permutations(list_C, 3):
    tmp = 0
    for i, j in enumerate((p, q, r)):
        for m, n in dict_c[i].items():
            tmp += D[m][j] * n
    ans = min(ans, tmp)

print(ans)