• Transforms
• Associative
• Pandiagonal
• Reversible
• Most-perfect

AssocPan Notes

To ensure a unique result from each input square, it is necessary to choose an appropriate aspect of the square. AssocPan does this by rotating each input square if necessary so that the quarter with the smallest corner number, (of the 16 corner numbers of the quarters), is at top left.

If some such strategy is not used, more than one input square can transform to the same output square. For example:

• Transforming the 48 order 4 associative squares in Frénicle standard form, makes only 42 unique pandiagonal complete squares and 6 duplicates. Using any other same aspect of the associative squares results in only 38 to 42 unique pandiagonal complete squares.
  
  
• Similarly, using any same aspect of the 48 order 4 pandiagonal squares as input, results in only 34 to 38 unique associative squares.
  
  
• Transforming the 368,640 order 8 most-perfect squares in Frénicle form, makes only 297,472 unique associative squares. Using the squares rotated 90 degrees as input, makes only 260,368 unique associative squares.

ReversibleMost-perfect Notes

In their book:

MOST-PERFECT PANDIAGONAL MAGIC SQUARES
Their construction and enumeration

Ollerenshaw and Brée say that the given transformation of a reversible square is a unique most-perfect square. But, that requires regarding each of the 8 aspects as a different square. For example:

• Transforming the 16 order 4 reversible squares as shown on page 28 of the book makes only 14 unique most-perfect squares and 2 duplicates, (different aspects).

To get the 48 most-perfect squares it is necessary to transform all 8×48 = 384 aspects of the reversible squares and remove the duplicates.

ReversibleMost-perfect avoids this by choosing an aspect of each reversible square that makes a unique most-perfect square.