Buggy python submission (verdict: Time Limit Exceeded):
# divide this matrix into 3 groups according to the remainder
# then consider the cost

from itertools import permutations

N, C = [int(elem) for elem in input().split(' ')]
D_matrix = [[int(elem) for elem in input().split(' ')] for _ in range(C)]
assert len(D_matrix) == C
assert len(D_matrix[0]) == C
c_matrix = [[int(elem) for elem in input().split(' ')] for _ in range(N)]
assert len(c_matrix) == N
assert len(c_matrix[0]) == N

# grouping
# O(N^2)
mod_0, mod_1, mod_2 = [], [], []
for i in range(N):
    for j in range(N):
        if (i + j) % 3 == 0:
            mod_0.append((i, j))
        elif (i + j) % 3 == 1:
            mod_1.append((i, j))
        else:
            mod_2.append((i, j))

# O(C^3 * N^2)
# muda
# only considering top 3 transitions is enough?
min_cost = float('inf')
for colors in permutations(range(C), 3):
    total_cost = 0
    for idx_color, mod in enumerate((mod_0, mod_1, mod_2)):
        for i, j in mod:
            color = c_matrix[i][j] - 1
            total_cost += D_matrix[color][colors[idx_color] - 1]
    min_cost = min(min_cost, total_cost)

print(min_cost)

Fixed submission:
from itertools import permutations

N, C = [int(elem) for elem in input().split(' ')]
D_matrix = [[int(elem) for elem in input().split(' ')] for _ in range(C)]
assert len(D_matrix) == C
assert len(D_matrix[0]) == C
c_matrix = [[int(elem) for elem in input().split(' ')] for _ in range(N)]
assert len(c_matrix) == N
assert len(c_matrix[0]) == N

# grouping
# O(N^2)
mod_list = [[] for _ in range(3)]
for i in range(N):
    for j in range(N):
        mod_list[(i + j) % 3].append((i, j))

# O(N^2)
mod_color_list = [[0] * C for _ in range(3)]
for k, mod in enumerate(mod_list):
    for i, j in mod:
        mod_color_list[k][c_matrix[i][j] - 1] += 1

# O(C^3 * N^2) -> O(C^4)
min_cost = float('inf')
for colors in permutations(range(C), 3):
    total_cost = 0
    for color, mod_color in zip(colors, mod_color_list):
        total_cost += sum([D_matrix[X][color] * mod_color[X] for X in range(C)])
    min_cost = min(min_cost, total_cost)

print(min_cost)

Buggy python submission (verdict: Time Limit Exceeded):
# divide this matrix into 3 groups according to the remainder
# then consider the cost

from itertools import permutations

N, C = [int(elem) for elem in input().split(' ')]
D_matrix = [[int(elem) for elem in input().split(' ')] for _ in range(C)]
assert len(D_matrix) == C
assert len(D_matrix[0]) == C
c_matrix = [[int(elem) for elem in input().split(' ')] for _ in range(N)]
assert len(c_matrix) == N
assert len(c_matrix[0]) == N

# grouping
# O(N^2)
mod_0, mod_1, mod_2 = [], [], []
for i in range(N):
    for j in range(N):
        if (i + j) % 3 == 0:
            mod_0.append((i, j))
        elif (i + j) % 3 == 1:
            mod_1.append((i, j))
        else:
            mod_2.append((i, j))

# O(C^3 * N^2)
min_cost = float('inf')
for colors in permutations(range(C), 3):
    total_cost = 0
    for idx_color, mod in enumerate((mod_0, mod_1, mod_2)):
        for i, j in mod:
            color = c_matrix[i][j] - 1
            total_cost += D_matrix[color][colors[idx_color] - 1]
    min_cost = min(min_cost, total_cost)

print(min_cost)

Fixed submission:
from itertools import permutations

N, C = [int(elem) for elem in input().split(' ')]
D_matrix = [[int(elem) for elem in input().split(' ')] for _ in range(C)]
assert len(D_matrix) == C
assert len(D_matrix[0]) == C
c_matrix = [[int(elem) for elem in input().split(' ')] for _ in range(N)]
assert len(c_matrix) == N
assert len(c_matrix[0]) == N

# grouping
# O(N^2)
mod_list = [[] for _ in range(3)]
for i in range(N):
    for j in range(N):
        mod_list[(i + j) % 3].append((i, j))

# O(N^2)
mod_color_list = [[0] * C for _ in range(3)]
for k, mod in enumerate(mod_list):
    for i, j in mod:
        mod_color_list[k][c_matrix[i][j] - 1] += 1

# O(C^3 * N^2) -> O(C^4)
min_cost = float('inf')
for colors in permutations(range(C), 3):
    total_cost = 0
    for color, mod_color in zip(colors, mod_color_list):
        total_cost += sum([D_matrix[X][color] * mod_color[X] for X in range(C)])
    min_cost = min(min_cost, total_cost)

print(min_cost)