'

A Latin square is an n ×n matrix in which every row and every column contains a rearrangement of all the elements of the set {1, 2, . . . , n}.

For example, below is a 6 ×6 Latin square:

6 2 3 4 5 1

2 4 5 6 1 3

3 5 6 1 2 4

4 6 1 2 3 5

5 1 2 3 4 6

1 3 4 5 6 2

Similar to some other famous games, the Latin square puzzle is a partially filled n × n square which needs to be completed to the full square.

For example, the puzzle on the left has a unique completion to the square on the right:

0 2 3 0 0 0

6 2 3 4 5 1

0 0 0 6 1 0

2 4 5 6 1 3

3 0 0 0 0 4

3 5 6 1 2 4

4 0 0 0 0 5

---->

4 6 1 2 3 5

0 1 2 0 0 0

5 1 2 3 4 6

0 0 0 5 6 0

1 3 4 5 6 2 (0s denote the missing elements.)

In this challenge, you are asked to devise a recursive algorithm which given a partial Latin square completes it to a full one. You can write your own implementation (it must be recursive) or follow the suggestions in the provided starter file. Your program should take its input from a text file and produce the output on stdout. Examples of input and output format are provided in the attached files.