- Practice
- Algorithms
- Implementation
- Forming a Magic Square

# Forming a Magic Square

# 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 .