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 Skip Navigation LinksMath Help > Counting > Pantactic Squares > Pantactic Squares - Members of a class

Pantactic Squares -- Members of a Class

There are 50 classes of Pantactic Squares, and within each class there are 16 different forms of the Square that differ from one another by rotations, reflections and color inversions.

I have taken one of these 50 classes -- the one I designate Class 9a -- and applied all 16 combinations of these transformations to show you the 16 squares that belong to this one class.

To create the 16 members of Class 9a, I either applied or didn't apply the following four actions, in this order, to the original square:

1000 - Inversion -- changing every black square to white, and vice-versa
0100 - Left-right reflection
0010 - Top-bottom reflection
0001 - Reflection about the diagonal axis that goes from upper-left to lower-right

There are 16 ways to apply or not apply each of these transformations individually.  I numbered the 16 ways by adding up the code numbers listed for each of the transformations.  You will see this sum, printed as a caption under each of the Squares, below.

The one labeled "0000" is the original square, the representative of Class 9a.  The one labeled 0110, for example, is the one that has had the 0100 (left-right reflection) and 0010 (top-bottom reflection) done to it, in that order.

The 16 Pantactic Squares in Class 9a


9a 0000

9a 0001

9a 0010

9a 0011

9a 0100

9a 0101

9a 0110

9a 0111

9a 1000

9a 1001

9a 1010

9a 1011

9a 1100

9a 1101

9a 1110

9a 1111

Every rotation and reflection of the 9a class representative is included here, even though no rotation was explictly performed.  For example, to perform a 90ยบ counterclockwise rotation, all you need to do is a left-right reflection followed by a diagonal reflection (9a 0101, above).  You will see that 9a 0011 and 9a 0110 are other rotations of the original Square.

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The webmaster and author of this Math Help site is Graeme McRae.