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 Ruby (2.3.3) (judge verdict: Runtime Error)

N, C = gets.split.map(&:to_i)
dboard = C.times.map { gets.split.map(&:to_i) }
board = N.times.map { gets.split.map(&:to_i) }
group = {}
(0...N).each do |i|
  (0...N).each do |j|
    c = board[i][j] - 1
    n = (i + 1 + j + 1) % 3
    group[n] ||= []
    group[n].push(c)
  end
end
group_sums = {}
group.each do |g, l|
  sums = []
  (0...C).each do |c|
    sum = 0
    l.each do |prev_c|
      sum += dboard[prev_c][c]
    end
    sums.push([c, sum])
  end
  sorted = sums.sort_by { |_, sum| sum }
  group_sums[g] = sorted
end
ans = 1 << 60
group_sums[0].take(3).each do |(zero_c, zero_sum)|
  group_sums[1].take(3).each do |(one_c, one_sum)|
    next if zero_c == one_c
    group_sums[2].take(3).each do |(two_c, two_sum)|
      next if zero_c == two_c || one_c == two_c
      sum = zero_sum + one_sum + two_sum
      ans = sum if sum < ans
    end
  end
end
puts ans