# ORDER-16 PAN-MAGIC SQUARES

There are features that are pan within pan-magic square. They are patterns whose members add to a constant regardless of the position of the pattern within the pan-magic square, including positions that require wraparound. Among those features are the rows, columns, and diagonals of the square. These three are the minimum requirement for a pan-magic square. The complete and the {2compact2} features are the most commonly mentioned additional pan features. Dame Kathleen Ollerenshaw called pan-magic squares that also have both the complete and {2compact2} features most-perfect.

Within the order-16 pan-magic squares it is also possible to have the corners of all 3x3, 5x5 and/or the 9x9 squares equal S/2, but only two of these in any given square. {Zigzag2} and {zigzag3} lines can also be pan within the magic square. Zigzag lines go back and forth between two or three rows or columns in the magic square. A 2 indicates that the zigzag is between two adjacent lines. A {zigzag3} line involves three adjacent lines. Zigzags will have an abababab pattern for the two lines, a and b, whereas Franklin W diagonals will have an abbaabba pattern.

Other features can be expressed within pan-magic squares that are not pan within the square. These features are only expressed in the magic square that is shown or in a limited number of other transformed squares. The associated property is the most commonly cited example of this type. Inlaid squares are also not pan within the larger pan-magic square but they may themselves be pan-magic and they may have different features than the larger square. Franklin bent diagonals are pan in one direction only. They are pan in the direction the V or W points but not laterally. Both V,W, and WW shaped Franklin bent diagonals are possible in an order-16 square.

Other patterns, both pan and non-pan within the magic square, are present in some order-16 pan-magic squares. Some of these may be accidentally expressed in squares that are generated below but they are not directly supported.

## ORDER-16 PAN-MAGIC SQUARE GENERATOR

 monagonals associated diagonals Franklin V half rows Franklin W complete FranklinWW {2compact2} 16 order-4 {2compact3} 4 inlaid 4 A {2compact5} 4 inlaid 4 B {2compact9} 16 order-8 {zigzag2} 4 inlaid 8's {zigzag3} 4 center 8's