This magic square calculator, provided by Hesapstan, generates normal magic squares for supported orders using the numbers from 1 to n² and shows the magic constant with row, column and diagonal sum checks.
What does this magic square calculator generate?
This calculator generates normal magic squares for the supported orders 3, 4, 5, 7, 8 and 9. The output includes the n×n grid, the magic constant and verification sums for rows, columns and the two main diagonals.
The calculator does not generate singly-even orders such as 6, 10 or 14. These require a different algorithm, so the page should not be understood as an all-order magic square generator.
What is a magic square?
A magic square is a square arrangement of numbers where every row, every column and the two main diagonals have the same sum. A normal magic square uses each number from 1 to n² exactly once.
For example, a 3×3 normal magic square uses the numbers from 1 to 9. It is magic only when the horizontal, vertical and diagonal sums all match the same value.
How is the magic constant calculated?
The magic constant of a normal n×n magic square is calculated with M = n(n² + 1) / 2. This is the target sum for every row, every column and both main diagonals.
- For n = 3, M = 3(9 + 1) / 2 = 15
- For n = 4, M = 4(16 + 1) / 2 = 34
- For n = 5, M = 5(25 + 1) / 2 = 65
This formula applies to normal magic squares using the numbers from 1 to n². Custom starting values or custom step sizes are outside this calculator's scope.
Which magic square orders are supported?
The calculator supports odd orders 3, 5, 7 and 9, plus doubly-even orders 4 and 8. It does not support singly-even orders such as 6 or 10.
- Supported odd orders: 3, 5, 7, 9
- Supported doubly-even orders: 4, 8
- Unsupported singly-even orders: 6, 10, 14 and similar values
How are odd-order magic squares created?
Odd-order magic squares in this calculator are created using the classic Siamese method, also known as the De la Loubère method. This method works for 3, 5, 7 and 9.
Users do not need to follow the construction manually. The calculator applies the method and then displays the grid with the sum checks.
How are 4x4 and 8x8 magic squares created?
For 4 and 8, the calculator uses a doubly-even construction method. This is different from the odd-order method, which is why 4×4 and 8×8 can be supported while 6×6 is not supported here.
Orders such as 6, 10 and 14 are singly-even. They require a separate construction method that is not implemented in this calculator.
Example: 3x3 magic square
For n = 3, the magic constant is 15. One generated grid can be read as 2 7 6 / 9 5 1 / 4 3 8.
The first row gives 2+7+6 = 15, the second row gives 9+5+1 = 15, and the third row gives 4+3+8 = 15. The same target sum also holds for the columns and the main diagonals.
Example: 4x4 magic square
For n = 4, the magic constant is 34. A normal 4×4 magic square uses all numbers from 1 to 16, and every row, column and main diagonal must sum to 34.
The calculator shows the grid and the verification sums together, so the result can be checked as a magic square rather than just viewed as a number table.
What are the limitations of this calculator?
This calculator generates normal magic squares only. It does not generate custom ranges, custom step values, all possible solutions for the same order, or higher-dimensional magic cubes.
- Only 3, 4, 5, 7, 8 and 9 are supported.
- Singly-even orders such as 6 and 10 are not supported.
- Only one magic square is shown for a selected order.
- The numbers follow the normal magic square range from 1 to n².
Frequently Asked Questions
What is a magic square?
A magic square is a square grid where every row, every column and the two main diagonals have the same sum. A normal magic square uses the numbers from 1 to n².
What is the magic constant?
The magic constant is the common sum that each row, column and main diagonal must reach. For a normal n×n magic square, it is M = n(n² + 1) / 2.
Which orders does this calculator support?
It supports 3, 4, 5, 7, 8 and 9. It does not support 6, 10 or other singly-even orders.
Why is 6x6 not supported?
A 6×6 magic square is singly-even and requires a different construction method. This runtime supports odd orders and doubly-even orders only.
Are there multiple magic squares for the same order?
Yes, several orders can have many possible magic squares. This calculator shows one generated square for the selected supported order, not every possible solution.
Can I choose a custom starting number?
No. This tool generates normal magic squares using the numbers from 1 to n². Custom starting values and custom step sizes are not supported.