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
![Example magic squares of even order [6X6] Example magic squares of even order [6X6]](./px/matrix/example-magic-squares-of-even-order-6x6.png)
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
-
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.
-
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.
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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.
-
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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1) Magic squares of even order in java
2) Magic squares of even order in c
3) Magic squares of even order in node js
4) Magic squares of even order in python
5) Magic squares of even order in ruby
6) Magic squares of even order in c++
7) Magic squares of even order in golang
8) Magic squares of even order in c#
9) Magic squares of even order in php
10) Magic squares of even order in scala
11) Magic squares of even order in swift
12) Magic squares of even order in kotlin
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.
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