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Abstract

A Pantactic Square is a square colored in two or more colors such that all the possible 2x2 subsquares (blocks) you can extract, form all the possible combinations.

In this article, we consider the 5x5 Pantactic Squares colored in just two colors (black and white) such that the 16 possible 2x2 subsquares (blocks) you can extract, form all the possible combinations.

This web page summarizes my discoveries (re-discoveries, really) in the area of Pantactic Squares.

Like the experts quoted here, I found 800 Pantactic Squares, which form 50 classes of 16 squares each.  The 16 squares in each class are formed by the 8 rotations and reflections (all of which make unique squares, since no Pantactic Square is symmetric about any axis, and all rotations can be described as an even number of reflections about one or more axes) and one color inversion.

I added my own original work, which is a classification scheme of the 50 Pantactic Squares, which allows any Square to be quickly classified into one of eleven "categories" based on the relative positions of the two "solid blocks" (2x2 subsquares that are all black or all white).

It is interesting that the relative positions of the two solid blocks can be (and are, in my categories) described in a way that makes the description invariant under any of the rotations, reflections, and inversions that distinguish members of a class.  This fact makes my classification scheme very useful, in that it allows one to quickly identify the class from among the 50 to which any particular Pantactic Square belongs.

I also discuss the torroidal and cylindrical symmetries, and I cite a number of articles, including one that appeared in the Dutch publication, CFF: Pantactic Patterns and Puzzles, by Jacques Haubrich.

Pantactic Squares

A Pantactic Square is a 5x5 square colored in just two colors (e.g. black and white) such that the 16 possible 2x2 subsquares (blocks) you can extract, form all the possible combinations.

In 1970, C. J. Bouwkamp, P. Janssen and A. Koene used a computer to generate all pantactic patterns [Bouwkamp, C. J., P. Janssen, A. Koene, "Note on Pantactic Squares", Math. Gazette, 54 (1970), pp. 348-351].  They found 800 different patterns.  These can be arranged into 50 classes of 16 each.  Patterns in the same class can be obtained from each other by rotation, reflection and/or color inversion.  (To see the 16 rotations, reflections, and color inversions of a particular Square, click here.)

I did a similar "brute force" search for pantactic squares by first searching all 1024 combinations of the first two rows of the square to find those "good combinations" that contain four different blocks.  Then, for each "good combination" I checked the 32 possible third rows of the pantactic square to find the "good combinations" of three rows that contain eight different blocks.  Continuing this way, I added a row at a time, and maintained a list of "good combinations".  When I finished, I had found all 800 Pantactic Squares.

No Pantactic Square is symmetric about a horizontal, vertical, or diagonal axis.  To see this for the horizontal and vertical axes, consider the location of the black block.  It has to be on one side of the axis or the other.  If the black block has a mirror image on the other side of the axis, then there would be two black blocks.  To see this for a diagonal axis, consider that there are eight different blocks that are symmetric about a particular diagonal axis.  All eight of these blocks would have to be on the diagonal of the Pantactic Square, but only four blocks can be on the diagonal.  For my examination of Pantactic Squares, I used three reflections -- one about a vertical axis, one about a horizontal axis, and one about a diagonal axis that extends from the top-left corner to the bottom-right corner.  By combining these three reflections, all eight different reflections and rotations are possible.  For example, rotating 90ยบ clockwise is the same as flipping about the diagonal and then reflecting about the vertical axis.

To begin organizing the 800 Pantactic Squares, I started by pairing the squares that differ only in that one is a color inversion of the other.  Each Pantactic Square A has an inverse A' in which every white square of A is black in A', and vice-versa.  So I selected 400 representatives of each pair such that the center square is black.

Each of the 400 representative Pantactic Squares with a black center square has a mirror image about the vertical axis, and only one of each pair has the black block on the left side, so I'll keep these 200 Pantactic Squares as representatives of their class.

Similarly, each of the 200 Pantactic Squares with a black center square and the black block on the left side has a mirror image about the horizontal axis, and only one of each pair has the black block in the upper left quadrant, so I'll keep these 100 Pantactic Squares as representatives of their class.

Last, I flipped each of the 100 Pantactic Squares with a black center square and the black block in the upper left quadrant about the diagonal that goes from upper left to lower right.  This reflection keeps the center square black, and it keeps the black block in the upper left quadrant.  I kept as a representative of the class the reflection in which the white block is closest to the upper right corner.  (For the four pairs in which the diagonal reflection doesn't change the position of the white block, I just picked randomly.)

Now I have 50 Pantactic Squares, each with a black center square, a black block in the upper left quadrant, and a white block on or above the upper-left/lower-right diagonal.  Next, I will want to organize the 50 Pantactic Squares to see if there are any more groupings or patterns that will help me further understand them.  There are no more reflections or rotations I can use, though.  So I chose to invent categories based on the relative positions of the solid blocks.  I chose this method because there are always two solid blocks, and their positions relative to each other and relative to the edges and corners of the Pantactic Square are unchanged by any inversion, reflection, or rotation.

Here are the 50 class representatives, organized into categories by their characteristics.

Category 1: The two solid blocks abut each other and an edge of the Square, with "stripe" blocks in that row:


Class 1a

Class 1b

Class 1c

Class 1d

Class 1e

Class 1f

Category 2: The two solid blocks abut in an edge row, along with an L-block in the same row.


Class 2a

Class 2b

Class 2c

Class 2d

Category 3: The two solid blocks abut in an interior row, with both "stripe" blocks in that row.


Class 3a

Class 3b

Class 3c

Class 3d

Class 3e

Class 3f

Class 3g

Category 4: The two solid blocks abut in an interior row, with an L-block in the same row.


Class 4a

Class 4b

Class 4c

Class 4d

Category 5: The two solid blocks share a short edge, and both solid blocks are at the edge of the Square.


Class 5a

Class 5b

Class 5c

Class 5d

Class 5e

Category 6: The two solid blocks share a short edge, one is in the interior of the Square, and the other is on an edge of the Square.


Class 6a

Class 6b

Class 6c

Class 6d

Class 6e

Class 6f

Category 7: The two solid blocks share a short edge, one is in the corner of the Square.


Class 7a

Class 7b

Class 7c

Class 7d

Class 7e

Category 8: One solid block is in the corner of the Square, and the other is on a far edge of the Square.


Class 8a

Class 8b

Class 8c

Class 8d

Category 9: Both solid blocks are on opposite edges of the Square.


Class 9a

Category 10: The solid blocks are touching at one corner, with one in a corner of the Square.


Class 10a

Class 10b

Class 10c

Class 10d

Category 11: The solid blocks are touching at one corner, both on adjacent edges of the Square.


Class 11a

Class 11b

Class 11c

Class 11d

"Cylindrical" Symmetry

If the leftmost and rightmost columns are the same (or if the top and bottom rows are the same) then the Pantactic Square has what I call "Cylindrical" Symmetry.  That is, the square can be painted on a cylinder, with the identical columns overlapping.  Then there are four different ways to peel a Pantactic Square off the cylinder, corresponding to four rotations of the cylinder.

For example, consider the Pantactic Square that I've labeled "8a":

   Class 8a


8a,
rotation 0

8a,
rotation 1

8a,
rotation 2

8a,
rotation 3

In each case I have "rotated" (as if on a cylinder) the four distinct columns one position to the left in each successive image.  The resulting four Pantactic Squares are not the representatives of their class (as I've defined them), however, because the center square isn't black, and the black block isn't in the upper-left quadrant.  So by inverting, reflecting, and rotating "8a rotation 1" we see that it is in class 7c.  (Perhaps I should digress for a moment here, and point out how my classification scheme helps find the representative Pantactic Squares of each class.  Looking at 8a, rotation 1, above, you can see the solid blocks share a short edge, and one of them is in the corner, which puts it in one of the "category 7" classes.)  Similarly, 8a, rotation 2, is in class 6d, and 8a, rotation 3, is in class 5b.


8a R0 = 8a

8a R1 = 7c

8a R2 = 6d

8a R3 = 5b

"Toroidal" Symmetry

Two of the fifty classes of Pantactic Squares have both left-right and top-bottom cylindrical symmetry.  I call this Toroidal Symmetry, because it is possible to paint the surface of a torus, overlapping both the left-right edges and the top-bottom edges.  Then there are 16 different rotations of the torus that would allow one to peel a Pantactic Square off the torus.

Jacques Haubrich writes in his article, "Pantactic Patterns and Puzzles", in CFF 34, pp.19-21.

Among these 50 pantactic patterns, only one has the property that not only the left side is equal to the right side, but at the same time the top side is also equal to the bottom side.  [Graeme's note: actually, two of the 50 pantactic patterns have this property.]  Hence, this particular pattern can be reproduced on a torus, producing a 4x4 (!) pantactic pattern that produces all 2x2 basic squares.


10d

11d

Here are the 16 rotations of 10d.  (If you check, you'll see that you get the same 16 patterns -- or inversions/reflections of those patterns -- if you toroidally rotate 11d.  Compare 11d to 10d R13 to see what I mean.)


10d R00

10d R01

10d R02

10d R03

10d R10

10d R11

10d R12

10d R13

10d R20

10d R21

10d R22

10d R23

10d R30

10d R31

10d R32

10d R33

Remarkably, every one of these 16 rotations belongs to class 10d or class 11d -- that is, any one of these rotations can be transformed into the representative of class 10d or the representative of class 11d by a sequence of the three ordinary reflection and/or the one inversion.

My classification scheme helps you pick out which class each of these toroidal rotations belongs to: if a solid block is in a corner, then it belongs to class 10d; if not, then it belongs to class 11d.

Internet references

http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/pantactic-squares/ 

The Internet reference, above, contains scanned images of Pantactic Patterns and Puzzles, by Jacques Haubrich.  This article appeared in CFF 34, pp.19-21.  CFF is Cubism For Fun, a publication of the Dutch Cubist Club (Nederlandse Kubus Club).  The journal appears three times per year.  The first 10 issues were in Dutch.  Numbers 11 on are in English.  The CCF website has an index of all issues.  The journal is available by subscription, but is not available online.

Summary of contents of CFF 34 -- www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/Dik_T._Winter__CFF_34,_summary_of_contents.html

Jacques Haubrich: Pantactic Patterns and Puzzles.
This discusses an extension of the memory wheel. On the wheel the digits 0 and 1 are written such that when you look at 3 consecutive digits, all 8 different configurations can be created.  This can be generalized to n consecutive digits.  It is well known (since N.G. de Bruijn) that 2n digits are needed.  A 2-dimensional extension was made by Brian Astle who had a 5x5 square with a black-white pattern such that when you look at the 16 different 2x2 subsquares you will find all 16 different configurations.  C.J. Bouwkamp made this into a puzzle (in the early 70's) as follows:  You have 16 2x2 squares with all possible patterns.  The puzzle is to put them together in a larger square such that the borders match.  Rotation is not permitted.

http://www.wpr3.co.uk/gazette/1960-69.html 

Brian Astle wrote an article called "Pantactic Squares" on p.144 of Mathematical Gazette, May 1965 issue.   Alas, this reference tells us nothing of the article.  If anyone has access to this issue, please email me the contents!

http://www.york.cuny.edu/~malk/biblio/geo-fabrics-biblio.html 

Has the B. Astle reference, above, plus the following:

Bouwkamp, C., and P. Janssen, A. Koene, Note on pantactic squares, Math. Gazette 54 (1970) 348-351.

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