Count Magic squares in a grid

Given an Grid of integers. The task is to find total numbers of 3 x 3 (contiguous) Magic Square subgrids in the given grid. A Magic square is a 3 x 3 grid filled with all distinct numbers from 1 to 9 such that each row, column, and both diagonals have equal sum.

Examples:

Input: G = { { 4, 3, 8, 4 }, { 9, 5, 1, 9 }, { 2, 7, 6, 2 } }
Output: 1
Explanation: The following subgrid is a 3 x 3 magic square: [ 4 3 8, 9 5 1, 2 7 6 ]

Input : G = { { 1, 2, 3, 4, 5 }, { 6, 7, 8, 9, 10 }, { 10, 11, 12, 13, 14 }, { 15, 16, 17, 18, 19 } }
Output : 0



Approach: Let us check every 3 x 3 subgrid individually. For each grid, all numbers must be unique and between (1 and 9) also every rows, columns, and both diagonals must have the equal sum.

Also notice the fact that a subgrid is a Magic Square if its middle element is 5. Because adding the 12 values from the four lines that crosses the center, add up to 60, but they also add up to the entire grid (45), plus 3 times the middle value. This implies the middle value is 5. Hence we can check this condition which help us skip over various subgrids.

You can learn more about Magic_square here or here.

The procedure to check for a subgrid to be a Magic Square is as follows:

  • The middle element must be 5.
  • The sum of the grid must be 45, and contains all distinct values from 1 to 9.
  • Each horizontal(row) and vertical(column) must add up to 15.
  • Both of the diagonal lines must also sum to 15.
  • Below is the implementation of above approach:

    C++

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    // CPP program to count magic squares
    #include <bits/stdc++.h>
    using namespace std;
      
    const int R = 3;
    const int C = 4;
      
    // function to check is subgrid is Magic Square
    int magic(int a, int b, int c, int d, int e, 
                    int f, int g, int h, int i)
    {
        set<int> s1 = { a, b, c, d, e, f, g, h, i }, 
                 s2 = { 1, 2, 3, 4, 5, 6, 7, 8, 9 };
      
        // Elements of grid must contain all numbers from 1 to
        // 9, sum of all rows, columns and diagonals must be
        // same, i.e., 15.
        if (s1 == s2 && (a + b + c) == 15 && (d + e + f) == 15 && 
           (g + h + i) == 15 && (a + d + g) == 15 &&                     
           (b + e + h) == 15 && (c + f + i) == 15 && 
           (a + e + i) == 15 && (c + e + g) == 15)
           return true;
        return false;    
    }
      
    // Function to cound total Magic square subgrids
    int CountMagicSquare(int Grid[R][C])
    {
        int ans = 0;
      
        for (int i = 0; i < R - 2; i++)
            for (int j = 0; j < C - 2; j++) {
      
                // if condition true skip check
                if (Grid[i + 1][j + 1] != 5)
                    continue;
      
                // check for magic square subgrid
                if (magic(Grid[i][j], Grid[i][j + 1], 
                    Grid[i][j + 2], Grid[i + 1][j],
                    Grid[i + 1][j + 1], Grid[i + 1][j + 2],
                    Grid[i + 2][j], Grid[i + 2][j + 1],
                    Grid[i + 2][j + 2]))
      
                    ans += 1;
            }
      
        // return total magic square
        return ans;
    }
      
    // Driver program
    int main()
    {
        int G[R][C] = { { 4, 3, 8, 4 },
                        { 9, 5, 1, 9 },
                        { 2, 7, 6, 2 } };
      
        // function call to print required answer
        cout << CountMagicSquare(G);
      
        return 0;
    }
      
    // This code is written by Sanjit_Prasad

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    Python3

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    # Python3 program to count magic squares 
    R = 3
    C = 4
      
    # function to check is subgrid is Magic Square 
    def magic(a, b, c, d, e, f, g, h, i): 
      
        s1 = set([a, b, c, d, e, f, g, h, i]) 
        s2 = set([1, 2, 3, 4, 5, 6, 7, 8, 9]) 
      
        # Elements of grid must contain all numbers 
        # from 1 to 9, sum of all rows, columns and 
        # diagonals must be same, i.e., 15. 
        if (s1 == s2 and (a + b + c) == 15 and 
           (d + e + f) == 15 and (g + h + i) == 15 and
           (a + d + g) == 15 and (b + e + h) == 15 and 
           (c + f + i) == 15 and (a + e + i) == 15 and 
           (c + e + g) == 15): 
            return True
              
        return false     
      
    # Function to cound total Magic square subgrids 
    def CountMagicSquare(Grid): 
      
        ans = 0
      
        for i in range(0, R - 2): 
            for j in range(0, C - 2): 
      
                # if condition true skip check 
                if Grid[i + 1][j + 1] != 5
                    continue
      
                # check for magic square subgrid 
                if (magic(Grid[i][j], Grid[i][j + 1], 
                      Grid[i][j + 2], Grid[i + 1][j], 
                      Grid[i + 1][j + 1], Grid[i + 1][j + 2], 
                      Grid[i + 2][j], Grid[i + 2][j + 1], 
                      Grid[i + 2][j + 2]) == True): 
      
                    ans += 1
      
        # return total magic square 
        return ans 
      
    # Driver Code
    if __name__ == "__main__"
      
        G = [[4, 3, 8, 4], 
             [9, 5, 1, 9], 
             [2, 7, 6, 2]] 
      
        # Function call to print required answer 
        print(CountMagicSquare(G))
          
    # This code is contributed by Rituraj Jain

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    Output:

    1
    

    Time Complexity: O(R * C)



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