Order 5 Magic Squares


Program Order5 makes the 275,305,224 order 5 magic squares.


Here are the first and last 5x5 magic squares, (in Frénicle standard form sorted in ascending order):

Type Summary

This is the type summary from program GetType:

Note: Pandiagonal 1-way does not include pandiagonal.


There are 101,774,553 complement pair pattern groups. The squares of 4,845,739 groups have center 13. See all groups, center 13 groups, center not 13 groups.

Counting the Squares

Order5count counts the squares in 4 minutes on a personal computer at 3.0GHz, (see counting detail):


  Manufacturer:       Dell
  Model:              Alienware Area-51 R2
  Processor:          Intel® Core™i7-5960X CPU @ 3.00GHz 3.00GHz
  Memory(RAM):        32.0 GB
  System type:        64-bit Operating System

  Windows edition:    Windows 10 Home
  Environment:        Microsoft® Visual C++® 2010 Express Edition

Cell Number Position Counts

The counts of squares for each cell number by position are: .txt

For example, there are 35,472,326 squares with 1 in a corner cell, another 35,472,326 squares with 1 in a cell one in from a corner along a main diagonal, 4,365,792 squares with 1 in the center cell, and so on.

Since each number from 1 to 25 is somewhere in every square, each row in the table totals 275,305,224. The column totals, (including row 1 to 12 numbers twice), are 275,305,224 multiplied by the number of green cells at the top.