Magic Squares of Even Order

The Magic Squares of Even Order is a mathematical problem that involves constructing a square matrix of even dimension (order) where the sum of elements in each row, column, and diagonals is the same. In other words, the sum of all the numbers in each row, column, and both diagonals of the square should be equal. This property gives the square a magic quality.

Problem Statement

Given an even integer 'n', the task is to generate a magic square of order 'n'. There are specific methods to construct such magic squares.

Example

Let's take the example of constructing a magic square of order 4 to illustrate the problem.

A standard magic square of order 4 looks like this:

16  2   3  13
5   11  10  8
9   7   6  12
4  14  15  1

In this example, each row, column, and diagonal sums up to 34.

Idea to Solve

There are different algorithms to construct magic squares of even order. One common approach is to use the Strachey method for singly even order and the LUX method for doubly even order magic squares.

Pseudocode

magicSquare(n):
    if n is even:
        if n mod 4 ≠ 0:
            Construct a singly even magic square
        else:
            Construct a doubly even magic square
    else:
        Print "Magic square of even order is not possible"

Algorithm Explanation

1. If 'n' is not even, then it's not possible to create a magic square of even order, so the program should handle this case.

2. If 'n' is even:

   • If 'n' is not divisible by 4, construct a singly even magic square.
   • If 'n' is divisible by 4, construct a doubly even magic square.

3. To construct a singly even magic square:

   • Divide the grid into four quadrants.
   • Fill each quadrant using a counter-clockwise pattern starting from the top-left corner.

4. To construct a doubly even magic square:

   • Fill the grid with consecutive numbers in order.
   • Reverse the numbers in the top-left, top-right, bottom-left, and bottom-right corners.

Code Solution

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Time Complexity

The time complexity of the algorithm to construct a magic square of even order is O(n^2), where 'n' is the order of the magic square. The algorithm fills each cell of the grid exactly once, resulting in a quadratic time complexity.