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 8.8.3) (judge verdict: Accepted)

import Control.Monad
import Data.Maybe
import qualified Data.ByteString.Char8 as BS
import Data.List
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import Data.Array.Unboxed

readInt = fst . fromJust . BS.readInt
readIntList = map readInt . BS.words
getInt = readInt <$> BS.getLine
getIntList = readIntList <$> BS.getLine

main = do
    [n,c] <- getIntList
    ds <- array ((1,1),(c,c)) . zip [(i,j)|i<-[1..c],j<-[1..c]] . concat <$> replicateM c getIntList :: IO(UArray (Int,Int) Int)
    cs <- array ((1,1),(n,n)) . zip [(i,j)|i<-[1..n],j<-[1..n]] . concat <$> replicateM n getIntList :: IO(UArray (Int,Int) Int)

    v0 <- VUM.replicate c (0 :: Int)
    v1 <- VUM.replicate c (0 :: Int)
    v2 <- VUM.replicate c (0 :: Int)
    forM_ [1..n] $ \i -> do
        forM_ [1..n] $ \j -> do
            let a = (i+j) `mod` 3
            let cij = (cs ! (i,j)) - 1
            when (a == 0) $ VUM.modify v0 succ cij
            when (a == 1) $ VUM.modify v1 succ cij
            when (a == 2) $ VUM.modify v2 succ cij
    fv0 <- VU.unsafeFreeze v0
    fv1 <- VU.unsafeFreeze v1
    fv2 <- VU.unsafeFreeze v2

    let colors = [(c0,c1,c2)| c0 <- [0..c-1], c1 <-[0..c-1], c2 <- [0..c-1], c0 /= c1, c1 /= c2, c2 /= c0]
    let ans = minimum $ map f colors
            where f (c0,c1,c2) = s0 + s1 + s2
                      where s0 = sum $ map (\ci -> (fv0 VU.! ci) * (ds ! (ci+1,c0+1))) [0..c-1]
                            s1 = sum $ map (\ci -> (fv1 VU.! ci) * (ds ! (ci+1,c1+1))) [0..c-1]
                            s2 = sum $ map (\ci -> (fv2 VU.! ci) * (ds ! (ci+1,c2+1))) [0..c-1]
    print ans