Area Magic Squares of Order-4

Further to the discoveries of the area magic squares of order-3, the first linear area magic square of order-4 was found by Walter Trump on the 14th January 2017.

On the 15th January 2017, Walter Trump's computer program was able to produce thousands of 4x4 linear area magic squares. Meanwhile, he was joined by fellow mathematician and programmer Hans-Bernhard Meyer, who on the 16th January 2017, found a different type of 4x4 linear area magic square (L-AMS). From then on Walter and Hans-Bernhard worked in collaboration, and they have kindly authorised me to illustrate samples of their findings below:

The linear area magic squares (L-AMS) of order-4 with vertical or horizontal lines have not only interesting geometric properties, but also surprising arithmetic features that are even more exciting. On the 8th February Hans-Bernhard Meyer informed me that the number and the structure of the fourth-order L-AMS with 3 horizontal or vertical lines had been totally clarified by parameterisations with 3 parameters, although this was not yet the case for L-AMS with only 2 horizontal or vertical lines. For full details of the mathematics behind these area magic squares please refer to the related links at the foot of this page.

On the 12th February 2017, in an e-mail correspondence, I asked Hans-Bernhard Meyer if it was possible to create a L-AMS of order-4 that had a perpendicular intersection of slanted lines. He responded on the 13th February 2017, sending the following L-AMS:

Congratulations to Hans-Bernhard for this interesting square!

Concerning the possible existence of L-AMS of order-4 without vertical or horizontal lines, on the 12th February 2017 Hans-Bernhard informed me that this question was still quite difficult to resolve.

Many other types of fourth-order AMS are also possible, and the following squares are area magic interpretations of some of the classical Frénicle squares of order-4:

Related links

Since the 3rd February 2017, full details of the findings of Walter Trump can be found in the chapter "Area Magic Squares" of his website: Notes on Magic Squares and Cubes.

Since the 25th January 2017, full details of the findings of Hans-Bernhard Meyer can be found in the article "Observations on 4x4 Area Magic Squares with Vertical Lines" of his website: Math'-pages.

Since the 25th January 2017, "Area Magic Squares of Order-6" relates the first findings of area magic squares of the sixth-order.

Since the 13th January 2017, "Area Magic Squares and Tori of Order-3" relates the first findings of area magic squares of the third-order.

In the N° 487 2018 May issue of "Pour La Science" (the French edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "Les Carrés Magiques d'Aires."

In the December 2018 issue of "Spektrum der Wissenschaft" (a Springer Nature journal, and the German edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "FLÄCHENMAGISCHE QUADRATE."

 

On the 21st May 2021, Yoshiaki Araki (面積魔方陣がテセレーションみたいな件 @alytile) tweeted several order-4 and order-3 solutions in a polyomino area magic square thread. These included (amongst others) an order-4 example constructed with 16 assemblies of 5 to 20 dominoes. For more information on polyomino area magic squares please check the links at the end of the article "Area Magic Squares of Order-3."

 

Since the 22nd June 2021, Inder Taneja has published a paper entitled "Creative Magic Squares: Area Representations" in which he explores polyomino area magic using perfect square magic sums.

 

A new post on "Polyomino Area Magic Tori" can be found in these pages since the 7th June 2022.

Read more ...

Complementary Number Patterns on Fourth-Order Magic Tori

The 880 fourth-order magic squares were first identified and listed by Bernard Frénicle de Bessy in his « Table générale des quarrez magiques de quatre, » which was published posthumously in 1693, in his book « Des quarrez magiques. » The census of these 880 4x4 squares is given in the appendix to the book « New Recreations With Magic Squares » by William H. Benson and Oswald Jacoby (Edition 1976), and also on the website of the late Harvey D. Heinz:  Frénicle n° 1 à 200, Frénicle n° 201 à 400, Frénicle n° 401 à 600, Frénicle n° 601 à 880.

Further to Bernard Frénicle de Bessy's initial findings, Henry Ernest Dudeney then classified the 880 fourth-order magic squares under 12 pattern types, in "The Queen" in 1910, (later republished in his book  "Amusements in Mathematics" in 1917). These patterns were determined through the observation of the relative positions of complementary pairs of numbers (which add up to N²+1 when N is the order of the square).

Although the different Dudeney pattern types can indeed be observed on fourth-order magic squares, they cannot be used for the classification of the fourth-order magic tori that display these squares. A Dudeney pattern type can be misleading because it depends on a bordered magic square viewpoint, whilst the magic torus that displays the magic square has a limitless surface...

In the study that follows the complementary number patterns are therefore tested, by comparing a developed torus surface and two traditional Frénicle magic square viewpoints for each of the 12 different types of fourth-order magic tori.

Complementary number patterns of each fourth-order magic torus type

General information on the magic torus index and type numbers

In addition to the Frénicle index numbers and the Dudeney types that have already been mentioned above, the present study also uses the index and type numbers of the fourth-order magic tori. For readers who may not be acquainted with the subject, essential information can be found in the following pages:

In a previous article "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus," the fourth-order magic tori have been classed in 12 types, and each magic torus has received a specific type number.

In a previous article "Table of Fourth-Order Magic Tori," so as to facilitate the identification of a magic torus beginning with any magic square, a standard torus form has also been defined, and each of the 255 fourth-order magic tori has received a specific index number.

Complementary number patterns of the pandiagonal tori type T4.01

The 3 pandiagonal tori type T4.01 are here represented by the pandiagonal torus with index n° T4.01.1 (torus type n° T4.01.1). This pandiagonal torus displays 16 pandiagonal squares with Frénicle index n°s 102, 104, 174, 201, 279, 281, 365, 393, 473, 530, 565, 623, 690, 748, 785 and 828. All of these squares, (including the Frénicle index n°s 102 and 174 illustrated here), show complementary number patterns that are Dudeney type I.

Complementary number patterns of the semi-pandiagonal tori type T4.02.1

The 24 semi-pandiagonal tori type T4.02.1 are here represented by the semi-pandiagonal torus with index n° T4.115 (torus type n° T4.02.1.09). This semi-pandiagonal torus displays 8 semi-pandiagonal squares with Frénicle index n°s 48, 192, 255, 400, 570, 734, 763 and 824. Half of these squares, (including the Frénicle index n° 192 illustrated here), show complementary number patterns that are Dudeney type IV. The other half, (including the Frénicle index n° 48 illustrated here), show complementary number patterns that are Dudeney type VI.

Complementary number patterns of the semi-pandiagonal tori type T4.02.2

The 12 semi-pandiagonal tori type T4.02.2 are here represented by the semi-pandiagonal torus with index n° T4.059 (torus type n° T4.02.2.01). This semi-pandiagonal torus displays 8 semi-pandiagonal squares with Frénicle index n°s 21, 176, 213, 361, 445, 591, 808 and 860. Half of these squares, (including the Frénicle index n° 21 illustrated here), show complementary number patterns that are Dudeney type II. The other half, (including the Frénicle index n° 176 illustrated here), show complementary number patterns that are Dudeney type III.

Complementary number patterns of the semi-pandiagonal tori type T4.02.3

The 12 semi-pandiagonal tori type T4.02.3 are here represented by the semi-pandiagonal torus with index n° T4.085 (torus type n° T4.02.3.01). This semi-pandiagonal torus displays 8 semi-pandiagonal squares with Frénicle index n°s 32, 173, 228, 362, 425, 577, 798 and 853. All of these squares, (including the Frénicle index n°s 173 and 228 illustrated here), show complementary number patterns that are Dudeney type V.

Complementary number patterns of the partially pandiagonal tori type T4.03.1

The 6 partially pandiagonal tori type T4.03.1 are here represented by the partially pandiagonal torus with index n° T4.108 (torus type n° T4.03.1.1). This partially pandiagonal torus displays 4 partially pandiagonal squares with Frénicle index n°s 46, 50, 337 and 545. All of these squares, (including the Frénicle index n°s 46 and 545 illustrated here), show complementary number patterns that are Dudeney type VI.

As already mentioned in "A New Census of Fourth-Order Magic Squares," Dudeney overlooked the partially pandiagonal characteristics, and mistakenly classified these squares as "simple."

Complementary number patterns of the partially pandiagonal tori type T4.03.2

The 12 partially pandiagonal tori type T4.03.2 are here represented by the partially pandiagonal torus with index n° T4.195 (torus type n° T4.03.2.06). This partially pandiagonal torus displays 4 partially pandiagonal squares with Frénicle index n°s 208, 525, 526 and 693. Half of these squares, (including the Frénicle index n° 693 illustrated here), show complementary number patterns that are Dudeney type VII. The other half, (including the Frénicle index n° 208 illustrated here), show complementary number patterns that are Dudeney type IX.

As already mentioned in "A New Census of Fourth-Order Magic Squares," Dudeney overlooked the partially pandiagonal characteristics, and mistakenly classified these squares as "simple."

Complementary number patterns of the partially pandiagonal tori type T4.03.3

The 2 partially pandiagonal tori type T4.03.3 are here represented by the partially pandiagonal torus with index n° T4.217 (torus type n° T4.03.3.1). This partially pandiagonal torus displays 4 partially pandiagonal squares with Frénicle index n°s 181, 374, 484 and 643. All of these squares, (including the Frénicle index n°s 484 and 643 illustrated here), show complementary number patterns that are Dudeney type XI.

As already mentioned in "A New Census of Fourth-Order Magic Squares," Dudeney overlooked the partially pandiagonal characteristics, and mistakenly classified these squares as "simple."

Complementary number patterns of the partially pandiagonal tori type T4.04

The 4 partially pandiagonal tori type T4.04 are here represented by the partially pandiagonal torus with index n° T4.096 (torus type n° T4.04.1). This partially pandiagonal torus displays 2 partially pandiagonal squares with Frénicle index n°s 40 and 522. The square with Frénicle index n° 40 (illustrated here) shows a complementary number pattern that is Dudeney type VIII, whilst the other square with Frénicle index n° 522 (illustrated here) shows a complementary number pattern that is Dudeney type X.

As already mentioned in "A New Census of Fourth-Order Magic Squares," Dudeney overlooked the partially pandiagonal characteristics, and mistakenly classified these squares as "simple."

Complementary number patterns of the basic magic tori type T4.05.1

The 92 basic magic tori type T4.05.1 are here represented by the basic magic torus with index n° T4.002 (torus type n° T4.05.1.01). This basic magic torus displays 2 basic magic squares with Frénicle index n°s 1 and 458. Both of these squares, illustrated here, show complementary number patterns that are Dudeney type VI.

Complementary number patterns of the basic magic tori type T4.05.2

The 32 basic magic tori type T4.05.2 are here represented by the basic magic torus with index n° T4.049 (torus type n° T4.05.2.01). This basic magic torus displays 2 basic magic squares with Frénicle index n°s 23 and 767. The square with Frénicle index n° 767 (illustrated here) shows a complementary number pattern that is Dudeney type VII, whilst the square with Frénicle index n° 23 (illustrated here) shows a complementary number pattern that is Dudeney type IX.

Complementary number patterns of the basic magic tori type T4.05.3

The 4 basic magic tori type T4.05.3 are here represented by the basic magic torus with index n° T4.005 (torus type n° T4.05.3.1). This basic magic torus displays 2 basic magic squares with Frénicle index n°s 3 and 613. Both of these squares, illustrated here, show complementary number patterns that are Dudeney type XII.

Complementary number patterns of the basic magic tori type T4.05.4

The 52 basic magic tori type T4.05.4 are here represented by the basic magic torus with index n° T4.019 (torus type n° T4.05.4.1). This basic magic torus displays 2 basic magic squares with Frénicle index n°s 8 and 343. The square with Frénicle index n° 8 (illustrated here) shows a complementary number pattern that is Dudeney type VIII, whilst the square with Frénicle index n° 343 (illustrated here) shows a complementary number pattern that is Dudeney type X.

Conclusions

We can see that the Dudeney pattern types I, V, XI, and XII are not only valid for magic squares, but also for the magic tori that display these squares. On the other hand, the Dudeney pattern types II and III are shown to be partial viewpoints of a single pattern on the fourth-order magic tori. Again, the Dudeney pattern types IV and VI, the Dudeney pattern types VII and IX, and also the Dudeney pattern types VIII and X, are shown to be partial viewpoints of single patterns on the fourth-order magic tori.

We also notice that some of the detected complementary number patterns have contrasting hemitorus arrangements.

It is necessary to adapt the Dudeney pattern types, and add new symbols that not only reveal the real nature of the complementary number relationships, but also facilitate their comprehension. With this in mind, the 8 figures that follow have been selected to illustrate the essentially different complementary number pattern types that occur on fourth-order magic tori:

To be read with a previous table that compared the 12 Dudeney pattern types with the 12 Magic torus types, published in "A New Census of Fourth-Order Magic Squares," the new table that follows, recapitulates the latest findings, and shows the repartition of the 8 complementary number patterns on the magic squares of the fourth-order magic tori:

Further Developments

New articles entitled "Self-Complementary and Paired Complementary Magic Tori of Order-4" and "Multiplicative Magic Tori" now extend the above findings.

Read more ...

Fourth-Order Magic Torus Chart

The fourth-order magic tori have already been identified and classified by type in a previous article "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus." The fourth-order magic tori have also been indexed in numerical order using normalised squares for convenient reference in the "Table of Fourth-Order Magic Tori."

The latest study below shows how the 255 4th-order magic tori are linked, and how they interrelate. The Fourth-Order Magic Torus Chart proposes the division of the 255 magic tori into 25 groups, and indicates the position of each magic torus in the system. All of Bernard Frénicle de Bessy's 880 4x4 magic squares, and all of Henry Dudeney's 12 square types are accounted for. A sample page of the Fourth-Order Magic Torus Chart is shown here:

New developments!

Posted on the 24th April 2024, a new article entitled "Plus or Minus Groups of Magic Tori of Order 4" demonstrates that the 255 magic tori of order 4 come from 137 ± Groups!

Although "Plus or Minus Groups of Magic Tori of Order 4" proposes an alternative way of looking at things, the "Fourth-Order Magic Torus Chart" is still a valid basic solution, and therefore remains available below.

Please note that when you click on the button that appears at the top right hand side of the pdf viewer, a new window will open and full size pages will then be displayed, with options for zooming.

Read more ...

Sub-Magic 2x2 Squares on Fourth-Order Magic Tori

My recent studies have shown that 2x2 sub-magic squares are fundamental in the way the fourth-order magic tori work.

It has long been known that fourth-order magic squares have 2x2 sub-magic squares at their centres and that the 4 corner numbers of the squares also add up to 34. However, once considered as magic tori, the 4 corner numbers of each 4th-order magic square are in fact another sub-magic 2x2 square, and more subsquares are to be found overlapping the former boundary of the initial square.

I have therefore tried to determine the number of magic sub-magic squares that are displayed on each type of the fourth-order magic tori, and my first results are illustrated below. Using simple observation and manual input I have not verified all of the 880 Frénicle fourth-order squares and may have missed something. Your comments and corrections are welcome.

Surprisingly the fourth-order type 4 partially panmagic torus displays less sub-magic 2x2 squares than the basic fourth-order type 5 magic torus.

Hypothesis

Each number of a fourth-order magic torus (or magic square) comes from at least one 2x2 sub-magic square. Or, expressed otherwise, the surface of every fourth-order magic torus is entirely covered by 2x2 sub-magic squares.

Conclusion

The above remained to be verified by a computer programme until the day after this article was published when Harry White very kindly checked and validated the hypothesis: His results confirm that every 4th-order magic torus (or magic square) is entirely covered by 2x2 sub-magic squares. For all the 880 Frénicle squares he has found 7,712 sub-squares and has confirmed that all the numbers of the different magic squares come from at least one 2x2 sub-magic square.

Harry White's results show that within a same type of tori the patterns and numbers of sub-squares can vary. For example, some of the type 5 tori have a reduced cover of 4 sub-magic squares instead of the 8 subsquares shown above (for example the Frénicle square index 277). This will need to be investigated further.

Please note that I have just published (the 28th March 2013) a new article that takes into account the different arrangements of the sub-magic 2x2 squares, and which attempts to complete the picture.

As for 4th-order semi-magic squares I have started to look into the question and have discovered some examples which are totally covered by 4 2x2 sub-magic squares whilst other examples have no subsquares at all: The phenomenon is not just limited to the magic tori.

Read more ...

Fourth-Order Panmagic Torus T4.194

The illustration shows the interchangeability of horizontal or vertical representations of a same panmagic torus. By twisting one side of the interlocked torus the other side turns, and vice versa. A simultaneous twisting and turning movement is also possible. The contact between the two sides of the interlocked torus takes place along perpendicularly connected circular tangents.

Interlocked Pandiagonal Torus - (excluded from CC licence)

 What happens at the intersection of these tangents? Here are two ways of seeing things:

The interlocked magic torus could symbolise the interaction between opposites such as behind and in front, outside and inside, etc. Alternatively, if one side of the torus represents the past and the other the future, the present could take place along the two perpendicular circular tangents.

Leaving philosophical considerations, and returning to mathematics, the above illustration portrays a pandiagonal or panmagic torus classified by type n°T4.01.1 - see the previous article "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus" The torus displays 16 fourth-order pandiagonal squares (Frénicle index numbers 102, 104, 174, 201, 279, 281, 365, 393, 473, 530, 565, 623, 690, 748, 785, and 828). Since the publication of a new "Table of Fourth-Order Magic Tori" this torus is now also indexed and listed as the n° T4.194 in normalised square form:

Numbered following the Bernard Frénicle de Bessy index, the pandiagonal squares that are displayed on this pandiagonal torus are as follows:  

The illustration below shows a way to spot the similar number sequences in the different Frénicle squares showing that they come from the same magic torus:

In order to better visualise how the T4.194 torus works I have produced the diagrams below. The magic torus is a symmetrically stable number system. When the torus is twisted through 360° changing number couples produce continuously balanced tensions. I have indicated some of the mathematical properties but you will notice others when you contemplate this beautiful counting machine. Further below, the results of the study are illustrated by patterns on the panmagic square.

Read more ...