Magic squares of order 5
The magic sum of the order 5 squares is 65, and there are 12 equations to be satisfied (5 rows, 5 columns, 2 diagonals). One of these equations is redundant, so there are 14 free entries. If you set , e.g., the entries marked by "x", you can calculate the remaining entries explicitly.
Implementing such an algorithm, you can find out that the number of magic squares of order 5 is 2202441792.
Reordering Rows and Columns
When you know that the following square is magic,
Following these ideas, you can rearrange (by rotations, reflections and rows/colums interchanging) any magic square of order 5 in such a way that the smallest off-center diagonal element will be located in position (1,1) (8 possibilities), that a15 < a51 (by possible reflection, 2 possibilities) and that a22 < a44 (2 possibilities).
This reduces the effort to be taken by a factor 32. Additional computing time can be saved by taking into account that the square (26 - aij) is also magic. Be careful with double-counting when a33=13.
Not knowing the work of Richard Schroeppel (1973), I calculated the number of order 5 squares in the early 1990s on a MicroVax 3500 (see below, with 8MB of memory). My Fortran code took about 8 hours of CPU time.
Lessons learned for algorithm development
The above ideas to reduce the number of candidates have several similarities in Monte Carlo simulation for finance applications: From my point of view, interchanging rows and columns is related to antithetic variables in Monte Carlo simulation.
Exiting loops as soon as possible (magic squares) is somehow related to rejecting paths of Monte Carlo simulation in risk management, e.g., when you kno in advance that the path considered will not end in extreme percentiles.
Next week: The other way round: Monte Carlo simulation and magic squares.