9x9 Magic Square

A 9x9 normal magic square places the integers 1 through 81 in an eighty-one-cell grid, using each number exactly once. Every row, every column, and both main diagonals must total 369. Magic Square Archive offers partially filled nine-order puzzles: preserve the fixed clues and use the missing totals to reason about the empty cells.

Magic square practice

Use each integer from 1 to 81 once. Every row, column and both diagonals must sum to 369.

Given clues are locked. Fill the empty cells, check your work, or reveal one hint.

Magic constant: 369

Number board

Why the magic sum is 369, not a Sudoku rule

The integers from 1 to 81 total 81 × 82 ÷ 2 = 3321. Nine rows divide that total equally, so each row must sum to 3321 ÷ 9 = 369. Equivalently, the normal-square formula is n(n² + 1) ÷ 2, with n = 9.

The shared grid size does not make this Sudoku. Standard Sudoku repeats the digits 1 through 9 and imposes rules on nine 3x3 boxes. Here, all eighty-one numbers are different, the two main diagonals matter, and the small boxes have no separate constraint.

Turn intersecting lines into deductions

Start with a row, column, or main diagonal containing only one blank. Subtract its known entries from 369. If eight entries sum to 327, the missing value is 42. Then check the crossing line and confirm that 42 has not already appeared elsewhere.

Two blanks require more care: their combined value is known, but their individual values may not be. Compare unused numbers with both crossing columns rather than guessing from the row total alone. A correct sum cannot rescue a repeated number or an entry outside 1–81.

An odd-order construction behind the puzzles

Nine is odd, so the Siamese method provides a construction. Place 1 at the middle of the top row, then move up and right for each successive integer, wrapping across edges. When that destination is occupied, move down one cell from the previous position instead.

The archive builds its nine-order boards from this method and rotated or reflected versions. In this family, the center is 41 and opposite cells across the center sum to 82. These are construction-specific observations, not extra rules defining every normal magic square.

Check the completed board, not just its rows

Enter numbers in the editable cells and use the board’s validation control when finished. It checks the integer range, rejects repeats, and compares all nine rows, nine columns, and two main diagonals with 369. Keep the fixed clues unchanged.

Use the archive as a clue-based puzzle, not an exhaustive catalog of every possible nine-order square. Its generated reference arrangement supplies the clues; successful validation tests the mathematical conditions rather than requiring an exact match to that arrangement.