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: Accepted)

# util functions

# debug
$debugFlag = ( ARGV[0] != nil )
p "==== DEBUG MODE ====" if $debugFlag

def pd(str)
  p str if $debugFlag
end

def putsd(str)
  puts str if $debugFlag
end

# input
lc = 0
cost = []
stat = []
while l = $stdin.gets do
  if lc == 0
    tmp = l.chomp.split(" ").map(&:to_i)
    cc = tmp[1]
    3.times do |i|
      stat[i] = [0]*cc
    end
  elsif lc <= cc
    tmp = l.chomp.split(" ").map(&:to_i)
    cost[lc-1] = []
    cc.times do |i|
      cost[lc-1][i] = tmp[i]
    end
  else
    tmp = l.chomp.split(" ").map(&:to_i)
    tmp.length.times do |i|
      pd tmp[i]
      print ":" if $debugFlag
      pd (i+lc-cc)%3
      stat[(i+lc-cc)%3][tmp[i]-1] += 1
    end
  end
  lc += 1
end

stat.length.times do |i|
  print "#{i}: " if $debugFlag
  pd stat[i]
end
pd cost

numArr = [*(0..cc-1)]
perm = numArr.permutation(3).to_a
print "perm:" if $debugFlag
pd perm

cand = []
perm.each do |p|
  tmp_cand = 0
  p.length.times do |i|
    pd "i=#{i}, p[i]=#{p[i]}"
    cc.times do |j|
      tmp_cand += stat[0][j] * cost[j][p[0]] if j != p[0]
      tmp_cand += stat[1][j] * cost[j][p[1]] if j != p[1]
      tmp_cand += stat[2][j] * cost[j][p[2]] if j != p[2]
    end
    cand << tmp_cand
  end
end

puts cand.min