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 PHP (7.4.4) (judge verdict: Accepted)

<?php
list($N, $C) = ints();
for ($i = 0; $i < $C; ++$i) $D[] = ints();
for ($i = 0; $i < $N; ++$i) {
    $c[] = ints();
    for ($j = 0; $j < $N; ++$j) --$c[$i][$j];
}
echo dfs();
function dfs($d = 0, $selected = [])
{
    global $C;
    if ($d === 3) return calcD(array_keys($selected));
    for ($i = 0; $i < $C; ++$i) {
        if (isset($selected[$i])) continue;
        $_selected = $selected;
        $_selected[$i] = true;
        $res[] = dfs($d + 1, $_selected);
    }
    return min($res);
}
function calcD($a)
{
    global $memo;
    $res = 0;
    for ($i = 0; $i < 3; ++$i) {
        if (!isset($memo[$i][$a[$i]])) $memo[$i][$a[$i]] = subCalcD($i, $a[$i]);
        $res += $memo[$i][$a[$i]];
    }
    return $res;
}
function subCalcD($x, $to)
{
    global $N, $D, $c;
    $res = 0;
    for ($i = 0; $i < $N; ++$i) {
        for ($j = ($x + 2 * $i) % 3; $j < $N; $j += 3) {
            $from = $c[$i][$j];
            $res += $D[$from][$to];
        }
    }
    return $res;
}
function ints()
{
    return array_map('intval', explode(' ', trim(fgets(STDIN))));
}