Pandiagonal Magic Squares of Order Five

permute 5 × (0 1 2 3 4)

1

× 0

permute 1 × (1 2 3 4 5)

1

× 0

subtract 13

rotate left/right

-2

-1

0

1

2

rotate up/down

-2

-1

0

1

2

1	7	13	19	25
14	20	21	2	8
22	3	9	15	16
10	11	17	23	4
18	24	5	6	12

A magic square has the same sums for the numbers in the rows, columns, and main diagonals. In a pandiagonal magic square, the square can be rotated as if the edges were wrapped around (like a rubber square sheet can be made into a torus), and the main diagonals will still add up to the same magic sum (65 here). All of the 5×5 pandiagonal magic squares can be generated by adding together two sets of permutations.

Details

By one count, there are 3600 order-5 pandiagonal magic squares[1]. If you would like to generate all 275,305,224 order-5 magic squares, try[2].

References

[1] H. Heinz. "Pandiagonal 5×5." (Feb 17, 2016) recmath.org/Magic%20 Squares/pandiag5.htm.

[2] Netstaff Co., Inc. "Magic Squares." (Feb 17, 2016) www.netstaff.co.jp/msq/msqe.htm.

External Links

Fermat's Magic Cube

The Lucas Magic Square

Magic Cube of Order 5

Magic Square (Wolfram MathWorld)

Magic Squares and Designs

Torus (Wolfram MathWorld)

Permanent Citation

Ed Pegg Jr

"Pandiagonal Magic Squares of Order Five"
http://demonstrations.wolfram.com/PandiagonalMagicSquaresOfOrderFive/
Wolfram Demonstrations Project
Published: February 18, 2016