Magic Square Calculator

Use the Magic Square Calculator to explore normal magic squares of orders 3 through 9. For a selected order, it gives the magic constant, number of cells, total of all entries, and complement pair sum. An editable sample grid lets you investigate how individual entries contribute to the square. These calculations assume that a normal square uses every integer from 1 to n² exactly once, where n is the number of rows and columns.

Magic constant calculator

Use each integer from 1 to 9 once. Every row, column and both diagonals must sum to 15.

Edit the sample board to test the normal-square rules. Changing order builds a new sample.

Magic constant: 15
Formula: n(n² + 1) / 2 = 3(3² + 1) / 2 = 15
Total of all numbers: 45
Number count: 9
Complement pair sum: 10

Number board

Blank cells are omitted from these running sums.

Row sums: 15 · 15 · 15
Column sums: 15 · 15 · 15
Main diagonal sum: 15
Other diagonal sum: 15

What the four calculated values mean

An order n square contains n² cells. Adding the integers from 1 to n² gives the total n²(n² + 1)/2. Because the n rows must have equal sums, dividing that total by n gives the magic constant M = n(n² + 1)/2. Every column and both main diagonals must also sum to M. The cell count describes the grid size, not the number of different magic squares that can be formed.

Compare orders from 3 to 9

For orders 3, 4, 5, 6, 7, 8, and 9, the magic constants are respectively 15, 34, 65, 111, 175, 260, and 369. Their cell counts are 9, 16, 25, 36, 49, 64, and 81, while their total sums are 45, 136, 325, 666, 1225, 2080, and 3321. Use the relationship total = n × M as a quick arithmetic check. For example, a sixth-order square has six rows summing to 111, so its entries total 666.

Understand complementary entries

The complement of an entry x is n² + 1 − x, so complementary entries sum to n² + 1. These pair sums range from 10 for order 3 to 82 for order 9. Replacing every entry in a normal magic square with its complement preserves the magic constant. This does not mean that complementary entries must occupy opposite positions in every magic square. For odd orders, the middle value is its own complement.

Experiment with the editable sample grid

Select an order and use the sample as a starting point for changing entries. Compare row, column, and main diagonal sums with the calculated constant. Also check the allowed range and repeated or missing numbers: equal line sums alone do not establish a normal magic square. Editing the sample does not change the formula for the selected order. If you use different numbers, the displayed normal-square target may no longer apply.