5x5 Magic Square

A normal 5x5 magic square places the integers 1 through 25 into twenty-five cells, using each number exactly once. Every row, every column, and both main diagonals must total 65. Magic Square Archive presents this order as partially filled puzzles: the given numbers stay fixed while you work out the missing entries. Understanding the construction helps, but completing a puzzle also requires checking how its clues intersect.

Magic square practice

Use each integer from 1 to 25 once. Every row, column and both diagonals must sum to 65.

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

Magic constant: 65

Number board

Why the magic sum is 65

The numbers from 1 to 25 total 25 × 26 ÷ 2 = 325. Since the five rows partition those numbers and have equal sums, each row must total 325 ÷ 5 = 65. This is the five-by-five case of the normal magic-square formula n(n² + 1) ÷ 2.

Equal row sums alone are not enough. The five columns and the two corner-to-corner diagonals must also reach 65, and repeated numbers are not allowed. Shorter diagonals and arbitrary groups of five cells are not part of the standard requirement.

Constructing an odd-order square

Five is odd, so the Siamese method gives a direct construction. Put 1 in the middle cell of the top row. For each following number, move one row upward and one column rightward, wrapping around to the opposite edge whenever you leave the grid. If the destination is already occupied, place the next number immediately below the previous number instead.

Continue until 25 is placed. This describes the base construction used by the archive, not an instruction to overwrite fixed puzzle clues. The displayed puzzle may use a rotated or reflected version of that square.

A complete example to check

Read these five rows from left to right: 17, 24, 1, 8, 15; 23, 5, 7, 14, 16; 4, 6, 13, 20, 22; 10, 12, 19, 21, 3; 11, 18, 25, 2, 9.

The first column gives 17 + 23 + 4 + 10 + 11 = 65. The main diagonal gives 17 + 5 + 13 + 21 + 9 = 65. Rotating or reflecting the entire arrangement preserves the required sums, although the locations of individual numbers change.

Reasoning with the archive's clues

Start with a row or column containing only one blank. Subtract its known entries from 65, then check whether the result is unused and agrees with the crossing line. For instance, a row containing 17, 24, 1, and 8 needs 15. With two blanks, the remainder specifies their combined sum, not their individual order.

Enter your deductions in editable cells and use the board's verification button to check number range, distinctness, and all required sums. Resetting restores the same clues. A completed row animation confirms a local sum; it does not replace validation of the whole square.