5x5 Magic Square: Sum 65 and How to Build One

Last updated 9 November 2026 · By Perfect Magic Squares

A 5x5 magic square is a grid of the numbers 1 to 25, each used once, in which every row, column and both long diagonals add up to 65. That total comes from M = 5(25 + 1)/2, and the Siamese method below builds a valid square in 25 simple moves.

The 3x3 and 4x4 squares are small enough to study completely. The 5x5 is the first size where that stops being possible, because the number of different squares jumps into the hundreds of millions. So this page does two things: it walks through one construction you can repeat on any odd size, and it explains why nobody hands you a list of 5x5 squares.

Why every line adds to 65

For a normal magic square of order n, the magic constant is M = n(n² + 1)/2 (MathWorld), and Wikipedia lists 65 as the constant for order 5. With n = 5 that is 5 × 26 ÷ 2 = 65.

The reasoning is short. The numbers 1 to 25 add up to 25 × 26 ÷ 2 = 325. Five rows share that total equally, so each row must be 325 ÷ 5 = 65, and the columns and diagonals have to match. The magic constant guide proves the general case and tabulates other sizes, so this page does not repeat that table.

Build a 5x5 magic square with the Siamese method

The Siamese method (also called de la Loubère's method) works for every odd order. Wikipedia dates its description in Europe to 1691, when Simon de la Loubère described the Indian continuous method. The rules fit in three lines:

1. Write 1 in the middle cell of the top row.
2. Write each next number one cell up and one cell to the right. If that move leaves the grid, wrap around: off the top, reappear at the bottom of the same column; off the right edge, reappear on the left of the same row. If the move leaves at the top-right corner, it wraps to the bottom-left corner. That cell is always already filled, so rule 3 applies.
3. If the cell you would move to is already taken, write the next number directly below the one you just placed.

The walk-through below uses eight stages instead of 25 separate clicks, grouping the numbers into runs that follow the same move. Use Next and Back to move between stages. Nothing plays automatically and nothing animates. The caption under each grid explains the moves, and the new cells are bold so you do not have to rely on colour.

Check the finished square

After stage 8 the grid is full. Add every line before trusting it: the table shows the five row sums on the right, the five column sums underneath and the two diagonal sums in the corners.

Why there is no list of 5x5 squares

The 4x4 guide can show every square because there are only 880 of them (Wikipedia). For order 5 the count is 275,305,224 different squares, not counting rotations and reflections (Wikipedia), and MathWorld credits R. Schroeppel with computing it in 1973.

Each of those squares also has rotated and mirrored copies. Counting those separately multiplies the total by 8, which comes to 2,202,441,792 grids. Nobody can browse or print a list that size, and the Siamese square above is one point in it.

The magic square solver covers 3x3 and 4x4 only. For a partly filled 5x5, find a row, column or diagonal with one blank and subtract the other four numbers in that line from 65.

Other sizes

The same rules build any odd-order square. The how to make a magic square guide covers other sizes, and the magic square generator builds squares and checks their lines.

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