Forming a Magic Square
We define a magic square to be an matrix of distinct positive integers from to where the sum of any row, column, or diagonal of length is always equal to the same number: the magic constant.
You will be given a matrix of integers in the inclusive range . We can convert any digit to any other digit in the range at cost of . Given , convert it into a magic square at minimal cost. Print this cost on a new line.
Note: The resulting magic square must contain distinct integers in the inclusive range .
For example, we start with the following matrix :
5 3 4
1 5 8
6 4 2
We can convert it to the following magic square:
8 3 4
1 5 9
6 7 2
This took three replacements at a cost of .
Input Format
Each of the lines contains three space-separated integers of row .
Constraints
Output Format
Print an integer denoting the minimum cost of turning matrix into a magic square.
Sample Input 0
4 9 2
3 5 7
8 1 5
Sample Output 0
1
Explanation 0
If we change the bottom right value, , from to at a cost of , becomes a magic square at the minimum possible cost.
Sample Input 1
4 8 2
4 5 7
6 1 6
Sample Output 1
4
Explanation 1
Using 0-based indexing, if we make
- -> at a cost of
- -> at a cost of
- -> at a cost of ,
then the total cost will be .