/* sqtest.cpp - Program for testing magic squares.

This program takes n^2 numbers as input and reports
whether the input forms a magic square of size n. If so,
it reports the smallest (start) value, step size, and the total
of each row, column and diagonal (which should be the same).
For example, if the input is:

 100 240 230 130
 210 150 160 180
 170 190 200 140
 220 120 110 250

Then the program reports n=4, sum=700, start=100, step=10.

The program does not prompt for any input. There should be no
other input except the list of numbers separated by white space.
*/

#include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>
#include <cstdlib>
using namespace std;

int main()
{

  // Read the square
  vector<int> sq;
  int x;
  while (cin >> x)
    sq.push_back(x);

  // Test whether the input size is a perfect square
  int n = int(sqrt(int(sq.size())) + 0.5);  // round
  if (n < 1) {
    cout << "No input\n";
    exit(1);
  }
  if (n*n == int(sq.size()))
    cout << "n = " << n << endl;
  else
  {
    cout << int(sq.size()) << " elements is not a perfect square\n";
    exit(1);
  }

  // Add up the top row
  int sum=0;
  for (int i=0; i<n; ++i)
    sum += sq[i];
  cout << "sum = " << sum << endl;

  // Check sum of diagonal from upper left to lower right
  int t = 0;
  for (int i=0; i<n; ++i)
    t += sq[i*(n+1)];
  if (t != sum)
  {
    cout << "Diagonal from upper left has incorrect sum " << t << endl;
    exit(1);
  }

  // Check sum of diagonal from upper right to lower left
  t = 0;
  for (int i=0; i<n; ++i)
    t += sq[(i+1)*(n-1)];
  if (t != sum)
  {
    cout << "Diagonal from upper right has incorrect sum " << t << endl;
    exit(1);
  }

  // Check sums over all the rows
  for (int i=0; i<n; ++i)
  {
    t = 0;
    for (int j=0; j<n; ++j)
      t += sq[i*n+j];
    if (t != sum)
    {
      cout << "Row " << i << " has incorrect sum " << t << endl;
      exit(1);
    }
  }

  // Check sums over all the columns
  for (int i=0; i<n; ++i)
  {
    t = 0;
    for (int j=0; j<n; ++j)
      t += sq[i+j*n];
    if (t != sum)
    {
      cout << "Column " << i << " has incorrect sum " << t << endl;
      exit(1);
    }
  }

  // Find start and step
  sort(sq.begin(), sq.end());
  int start = sq[0];
  int step = 0;
  if (n > 1)
    step = (sq[n*n-1] - start) / (n*n - 1);
  cout << "start = " << start << endl;
  cout << "step = " << step << endl;

  // Verify elements form a sequence
  for (int i=0; i<n*n; ++i)
  {
    if (sq[i] != start + i * step)
    {
      cout << sq[i] << " is not in sequence\n";
      exit(1);
    }
  }

  // All is OK
  cout << "The square is magic\n";

  return 0;
}