Java Program to Check Given Matrix is Magic Square or Not

A magic square is a square matrix (an (n times n) grid) of unique positive integers organized in such a way that the total of the numbers in each row, column, and both main diagonals is equal. This constant sum is known as the magical constant. Magic squares have captivated mathematicians and enthusiasts for generations, combining parts of number theory and symmetry in novel ways.

Magic Square

A standard magic square of order (n) employs numbers between (1) and (n^2).

• The matrix should be square, with an equal number of rows and columns.
• Every row, column, and diagonal must add up to the same amount.
• Each integer between 1 and (n^2) should only appear once in the square.

The magic constant (or magic sum) for an (n times n) magic square can be calculated using the formula:

Magic Constant = n(n^2 + 1)/2

For example, in a (3 times 3) magic square, the magic constant is:

Magic Constant = 3(3^2 + 1)/2 = 3 * 10/2 = 15

Properties of a Magic Square

Magic squares exhibit various interesting properties:

1. The median value, 5, is always found in the center of a 3x3 magic square.
2. A magic square is symmetrical, meaning the opposite rows and columns will yield the same sums.
3. Magic squares of order 3 are unique up to rotation and reflection. Higher-order magic squares (like (4 times 4), (5 times5)) allow for many more possible configurations.

Types of Magic Squares

There are several types of magic squares based on properties and configurations:

1. Odd-Order Magic Squares: These have an odd number of rows and columns (like (3 times 3), (5 times 5)).
2. Doubly Even-Order Magic Squares: These squares have an order that is a multiple of 4 (like (4 times 4), (8 times 8)).
3. Singly Even-Order Magic Squares: These have an order that is an even number but not a multiple of 4 (like (6 times 6)).

Method for Determining Whether a Matrix is a Magic Square

In order to identify if a given (n x n) matrix is a magic square, we must:

• Find the sum of prime diagonal and secondary diagonal.
• Calculate the sum of each row and column.
• If the prime diagonal and secondary diagonal sums are equal to every row's sum and every column's sum, then it is the magic matrix.

Matrix Square Java Program

Let's implement the above steps in a Java program.

File Name: MagicSquareChecker.java

Output:

 
The given matrix is a magic square.   

Explanation

The Java code above defines a colass, 'MagicSquareChecker', which has a method isMagicSquare() that receives a matrix (2D array) as input and assesses if it meets the criteria for a magic square. The algorithm first determines if the matrix is square, and then calculates the magic constant based on its size. It then iterates through each row and column to see if their sums equal the magic constant.

It also checks the sums of both main diagonals. Finally, the function ensures that all numbers in the matrix are distinct and range from (1) to ( n^2 ). If all these checks pass, the matrix is identified as a magic square. In the main() method, a sample ( 3 times 3 ) matrix is tested, and a message is displayed indicating whether it is a magic square.

This approach effectively identifies magic squares by verifying both structural properties and numerical conditions, ensuring the output is accurate and efficient.

Complexity Analysis

The time complexity of this algorithm is (O(n^2)), where (n) is the dimension of the matrix. This is because we must iterate through all elements to check row sums, column sums, and both diagonals, as well as to confirm the presence of distinct values.

Each check requires scanning all elements, resulting in a combined complexity of (O(n^2) ). The space complexity is (O(n^2)) due to the input matrix itself and an additional (O(n^2) ) for the boolean array used to track uniqueness, making the overall space complexity (O(n^2)) as well. This is efficient for small to medium-sized matrices, making the algorithm suitable for general-purpose magic square validation.

Conclusion

Magic squares are fascinating mathematical constructs characterized by unique properties of symmetry and numerical patterns. The Java code provided efficiently checks if a given matrix qualifies as a magic square by verifying the sum consistency of rows, columns, and diagonals, and ensuring the matrix contains unique elements within a specified range.

This approach ensures the matrix is validated against the strict requirements for magic squares, providing a robust solution suitable for square matrices of any size. By implementing this algorithm, we gain deeper insight into the computational requirements and structure of magic squares, further appreciating their complexity and elegance.