Magic Square Calculator - 3x3, 4x4, and Magic Constant

A magic square calculator fills a grid of numbers so every row, column, and main diagonal sums to the same total, called the magic constant. Pick an order, start, and step, and the calculator returns the solved grid, the magic constant, and every line sum in one read.

• • Puzzle solving: Solve a missing cell in a partially filled 3x3 or 4x4 square by reading the solved grid and magic constant.
• • Math class demos: Show the Lo Shu 3x3 and Dürer 4x4 grids next to the magic constant formula.
• • Recreational math: Try unusual start and step values and watch the line-sum property hold.
• • Verification of own work: Confirm a hand-built square against the eight or ten line sums.

A magic square is one of the oldest objects in recreational mathematics. The 3x3 Lo Shu square has been known in China for at least two thousand years, and Albrecht Dürer included a 4x4 magic square in his engraving Melencolia I in 1514. A normal magic square of order n contains each integer from 1 to n^2 once, and every line sums to the magic constant.

A magic square of order n has a built-in cross-check: the arithmetic mean of its n^2 cells equals the magic constant divided by n (5 for a normal 3x3 Lo Shu, 8.5 for a normal 4x4 Dürer). To confirm that mean on a hand-built grid, Average Calculator returns the arithmetic mean, sum, and count of any list of cell values in one pass.

The calculator combines the closed-form magic constant M = n(n^2 + 1) / 2 with the Lo Shu pattern (n = 3) and the Dürer pattern (n = 4), and applies a linear transform so the same shape works for any start and step. Every line sum is recomputed from the final grid so the displayed sums always match the displayed cells.

• n: The order of the square (3 for a 3x3 grid, 4 for a 4x4 grid).
• M: The magic constant, the value every row, column, and main diagonal sums to.
• start: The number placed in the cell that corresponds to a 1 in the base pattern. Default 1.
• step: The difference between consecutive integers in the square. Default 1.

The calculator re-sums each row, each column, and both main diagonals from the final cells, so the displayed sums always match the displayed grid. With step > 1 the cells become a stretched version of the Lo Shu or Dürer pattern, and the magic constant scales with step and shifts with start.

According to Wolfram MathWorld, a normal magic square of order n contains each integer from 1 to n^2 once, and every row, column, and main diagonal sums to the magic constant n(n^2+1)/2.

When the grid uses many cells with multi-digit values, Long Addition Calculator lays out each row sum in stacked long form so the addition can be checked column by column.

Four small ideas explain why the Lo Shu and Dürer squares work and how the line-sum property extends to other start and step values.

These four ideas are enough to build and verify a magic square by hand for n = 3 and n = 4. For larger orders, the Siamese, Strachey, and LUX methods start from the same magic constant and use the same line-sum property.

If you want to treat a magic square as a 2D array and explore its linear-algebra properties, Adjoint Matrix Calculator returns the adjugate of the corresponding matrix in one step.

Pick the order, set the start and step, and read the solved grid, the magic constant, and every line sum on the right. The result panel updates as you change any input.

1. 1 Choose the order: 3x3 (Lo Shu, M = 15) or 4x4 (Dürer, M = 34).
2. 2 Set the start value: The smallest number in the grid. Use 1 for a normal square, 0 for 0 to n^2 - 1.
3. 3 Set the step between cells: The difference between consecutive integers. Use 1 for normal, 2 for even numbers only.
4. 4 Read the magic constant: The highlighted Magic Constant row shows the value that every row, column, and main diagonal must add up to.
5. 5 Compare each line sum: All row, column, and diagonal rows show the recomputed line sum from the displayed grid, and they should all equal the magic constant.
6. 6 Reset for a new puzzle: Press Reset to restore the default 3x3, start 1, step 1 inputs.

Example: a student is given the partial 3x3 grid [8, 1, 6 / 3, ?, 7 / 4, 9, 2] and asked for the missing cell. They set Order 3, Start 1, Step 1. The result panel shows magic constant 15 and the full Lo Shu grid, with the missing cell as 5.

When the puzzle asks for the total of 1 to n^2 in a normal magic square, Arithmetic Sequence Calculator returns the closed-form triangular sum n^2(n^2+1)/2 in one step, which is the same total used to derive the magic constant n(n^2+1)/2.

Pattern recognition is enough to build a 3x3 or 4x4 magic square, but writing one by hand invites arithmetic mistakes, and the magic constant is easy to forget. The calculator removes both risks.

• • Two orders in one tool: Switch between the 3x3 Lo Shu and 4x4 Dürer pattern with a single dropdown.
• • Magic constant built in: M = n(n^2 + 1) / 2 is applied to the start and step, so the highlighted Magic Constant row is always the correct target.
• • All line sums at once: Eight sums for 3x3 and ten for 4x4 appear next to the magic constant.
• • Reusable for offset squares: Build squares that use 0 to 8, even numbers only, or any equally spaced sequence, and the line sums stay equal.
• • Verifies hand-built squares: Read the recomputed line sums against the displayed magic constant to confirm a square you built on paper.

The biggest benefit is the immediate cross-check: if any line sum disagrees with the magic constant, the grid or formula is wrong, and the cause is usually a misplaced cell. With the Lo Shu and Dürer patterns built in, swapping two cells in row 2 is caught at a glance.

To check a hand-built square at a glance, copy the eight or ten line sums into a list and Mean Median Mode Range Calculator reports the range, median, and mean; for a true magic square the range is 0 and the mean equals the magic constant.

A magic square is a discrete object, so there is no continuous parameter to estimate. The factors that change the result are the integer inputs you choose, and the limits are the supported orders.

• • The calculator returns a clear error for any order other than 3 or 4. The 2x2 case has no solution, and orders 5 and above need a separate method.
• • For each supported order the calculator returns one canonical pattern. The full family of order-4 normal magic squares (7040 total) is not enumerated, but any one of them has the same magic constant and the same line-sum property.
• • The cells must form an arithmetic sequence defined by start and step. General magic squares with non-equally-spaced values are outside the scope of this tool.

The integer constraints above are the only real limits. Within them, the same line-sum property holds for every combination of start and step.

According to Wikipedia (Magic square), the order-n normal magic square uses the integers 1 through n^2 and has the magic constant n(n^2+1)/2; for n=3 the classical pattern is the Lo Shu square, and for n=4 Albrecht Dürer published a well-known example in 1514.

When you want to confirm a hand-built square by computing the difference between the largest and smallest line sums, Long Subtraction Calculator sets up the subtraction in stacked long form.