3x3 Magic Square

A normal 3x3 magic square arranges the integers 1 through 9 in nine cells, using each number once. Every row, column, and main diagonal must have the same total: 15. This small grid is a practical starting point for exploring magic squares because its structure can be checked directly, without a lengthy calculation.

Magic square practice

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

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

Magic constant: 15

Number board

What makes the square normal, and why is the sum 15?

The word normal specifies the entries: exactly the consecutive integers from 1 to 9. Their total is 45. The three rows partition those nine entries, so equal row totals must each be 45 ÷ 3 = 15. That calculation determines the required magic sum, but it does not prove an arrangement works. Columns and both corner-to-corner diagonals must also total 15; checking only the rows can miss an invalid square.

The center is 5, and opposite cells sum to 10

Consider the middle row, middle column, and two main diagonals together. Their totals add to 60. Every outer cell appears once, while the center appears four times. Since all nine entries total 45, the extra three copies of the center contribute 15, making the center 5. Each line through that center therefore leaves 10 for its two opposite cells. The pairs are 1 and 9, 2 and 8, 3 and 7, and 4 and 6.

Build the square with the Siamese method

Place 1 in the top middle cell. For each next number, try moving one row upward and one column rightward, wrapping around whenever you cross an edge. If the destination is already occupied, move one row down from the previous number instead. Continue through 9. This produces the rows 8, 1, 6; 3, 5, 7; and 4, 9, 2. For example, the diagonal move after 3 would return to the occupied cell containing 1, so 4 goes below 3. The same collision rule places 7 below 6.

Read and verify the displayed grid

Use the grid as a visual record of the construction. First confirm that every integer from 1 to 9 appears exactly once. Then check all eight required lines: three rows, three columns, and two main diagonals. Center 5 and opposite totals of 10 are useful additional checks, not substitutes for those line sums. Rotating or reflecting a valid square preserves its sums, so a different orientation can still be correct.