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 Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

import Data.Array.IO
import Data.Array.Unboxed
import Data.List
import Control.Monad
import Control.Applicative

main = do
  (n:c:_) <- getLine >>= return . map read . words :: IO [Int]
  darr <- newArray ((1,1),(c,c)) 0 :: IO (IOUArray (Int, Int) Int)
  carr <- newArray ((1,1),(n,n)) 0 :: IO (IOUArray (Int, Int) Int)

  forM_ [1..c] $ \i -> do
    temp <- getLine >>= return . map read . words :: IO [Int]
    forM_ (zip [1..c] temp) $ \(j, d) -> do
      writeArray darr (i,j) d

  forM_ [1..n] $ \i -> do
    temp <- getLine >>= return . map read . words :: IO [Int]
    forM_ (zip [1..n] temp) $ \(j, c) -> do
      writeArray carr (i,j) c

  let
    solve :: Int -> IO [Int]
    solve m = do
        forM [1..c] $ \color -> do
          res <- forM [(j,i) | j<-[1..n], i<-(takeWhile (<=n) $ iterate (+3) (mod (m-j) 3 + 1))] $ \(j,i) -> do
            current <- readArray carr (j,i)
            readArray darr (current, color)
          return $ sum res
            
  m0 <- zip [1..] <$> solve 0
  m1 <- zip [1..] <$> solve 1
  m2 <- zip [1..] <$> solve 2

  print $ minimum [d0+d1+d2 | (i,d0)<-m0, (j,d1)<-m1, (k,d2)<-m2, i/=j, j/=k, i/=k]