Magic Square Magic Square
Moderate
English

In mathematics a magic square is an arrangement of numbers (integers in our case) in a square grid, where the numbers in each row, each column and the numbers in the forward and backward main diagonals, all sum up to the same number.

example example

You have been given an incomplete magic square (with a size from 3 to 5). With your coding skills, you must finish the square. It must be a normal magic square and contain integers from 1 to n2 without repeating. The square is presented as a list of lists with integers. Zero is used to mark an empty cell. You should return a completed magic square. The task may have multiple solutions.

You can read more about the about Magic Squares on Wikipedia.

Input: A partially filled magic square as a list of lists with integers.

Output: The completed magic square as a list of lists with integers.

Example:

checkio([
    [2, 7, 6],
    [9, 5, 1],
    [4, 3, 0]
])
#must return [[2, 7, 6], [9, 5, 1], [4, 3, 8]]

checkio([
    [0, 0, 0],
    [0, 5, 0],
    [0, 0, 0]
])
#can return [[2, 7, 6], [9, 5, 1], [4, 3, 8]] or
# [[4, 9, 2], [3, 5, 7], [8, 1, 6]

checkio([[1, 15, 14, 4],
         [12, 0, 0, 9],
         [8, 0, 0, 5],
         [13, 3, 2, 16]])
# answer [[1, 15, 14, 4], [12, 6, 7, 9], [8, 10, 11, 5], [13, 3, 2, 16]]

How it is used: This is a constraint satisfaction problem. It's used not only for solving puzzles, but also for planning and resource allocation in city planning, construction and just about everywhere else.

Precondition: 3 ≤ |square| ≤ 5
square_width == square_height