Order 5 Magic Square Count

Counting the Squares

Distinct Squares

Consider the numbers A, B, C, D at the corners of the center 3x3 square:

   o o o o o
   o A o B o
   o o o o o
   o C o D o
   o o o o o

The 24 permutations of these numbers can be arranged in 3 groups:

	        AB CA DC BD CD BA DB AC
	        CD DB BA AC AB DC CA BD

	        AB DA CD BC DC BA CB AD
	        DC CB BA AD AB CD DA BC

	        AC DA BD CB DB CA BC AD
	        DB BC CA AD AC BD DA CB

It is sufficient to count squares that have 1 permutation from each of these groups; the others are rotations and reflections.

Suppose that D > C > B > A. Then one strategy for counting only the distinct squares is to count only those with the first configuration in each group. That is, for the main diagonals, the second row, and the second column in the 5x5 square, count only permutations in which the fourth number is greater than the second.

Square Transformations

The number of squares to count can be reduced to 1/4 of the distinct squares by not counting the squares for these three transformations:

Swap rows 1 and 5; swap columns 1 and 5.
Swap rows 1 and 2, rows 4 and 5; swap columns 1 and 2, columns 4 and 5.
Transformation 2 of transformation 1.

Consider a square X with these diagonals:

   E o o o F
   o A o B o
   o o o o o
   o C o D o
   G o o o H

Transformation 1 will change X to a square with these diagonals:

   H o o o G
   o A o B o
   o o o o o
   o C o D o
   F o o o E

Transformation 2 will change X to a square with these diagonals:

   A o o o B
   o E o F o
   o o o o o
   o G o H o
   C o o o D

One way to avoid counting these is:

transformation 1: count only squares having the bottom left corner number greater than the top right corner number
transformations 2 and 3: disable the diagonal pair (AEoHD, BFoGC).

Complement Pairs

Consider the number K in the center cell. It is sufficient to count squares for which K is one number of each complement pair, plus squares for which K is 13. For example, count squares with center 1 .. 12 and 13. The count of squares for center 25 is the same as the count for center 1, ..., the count for center 14 is the same as the count for center 12.

Count squares for center K from 1 .. 13. Add the count for each of centers 1 .. 12 twice in the total.

Counting - Preparation

Note: Permutations are of combinations of 5 numbers chosen from 1 .. 25 that total to the magic constant.

Make arrays:

CenterKPermus: permutations that have K as the center number, (with K omitted)
CenterKUPermus: pointers to only CenterKPermus in which the third number is greater than the second
CenterKPermus14: pointers to CenterKPermus, indexed by the first and fourth numbers
CenterKPermus23: .. indexed by the second and third numbers
OKPermus: permutations that do not contain K
OKPermus125: pointers to OKPermus, indexed by the first, second, and fifth numbers
OKUPermus24: pointers to only OKPermus in which the fourth number is greater than the second, indexed by the second and fourth numbers
OKUPermus234: .. indexed by the second, third, and fourth numbers
OKPermus1235: the fourth number in OKPermus, indexed by the first, second, third, and fifth numbers
OKPermus1245: the third number in OKPermus, indexed by the first, second, fourth, and fifth numbers

Counting - Procedure

   x
     x
       x      Get a diagonal, (CenterKUPermus loop).    
         x  
           x


   o       x
     o   x  
       o      Get the other diagonal, (CenterKUPermus loop).    
     x   o
   x       o

Check for cell[2,4] greater than cell[2,2].
Check if disagonals disabled.
Flag diagonals to avoid for above transformations 2 and 3.
Check cell[5,1] greater than cell[1,5] to disable above transformation 1.

   o       o
   x o x o x  Get row 2, (OKUPermus24 loop).
       o    
     o   o  
   o       o


   o       o
   o o o o o
   x   o    
   x o   o    Get column 1, (OKPermus125 loop).
   o       o

   o       o
   o o o o o
   o   o   x  Get column 5, (OKPermus125 loop).  
   o o   o x
   o       o

   o       o
   o o o o o
   o   o   o
   o o x o o  Get the third number of row 4, (OKPermus1245).
   o       o


   o       o
   o o o o o
   o x o x o  Get row 3, (CenterKPermus14).
   o o o o o
   o       o


   o   x   o
   o o o o o
   o o o o o  Get column 3, (CenterKPermus23).
   o o o o o
   o   x   o


   o x o   o
   o o o o o
   o o o o o  Get column 2, (OKUPermus234).
   o o o o o
   o x o   o


   o o o x o  Get the fourth number of row 1, (OKPermus1235).
   o o o o o
   o o o o o
   o o o o o
   o o o x o  Check the fourth number of row 5, (OKPermus1235).