Polyomino Area Magic Tori

A magic torus can be found with any magic square, as has already been demonstrated in the article "From the Magic Square to the Magic Torus". In fact, there are n² essentially different semi-magic or magic squares, displayed by every magic torus of order-n. Particularly interesting to observe with pandiagonal (or panmagic) examples, a magic torus can easily be represented by repeating the number cells of one of its magic square viewpoints outside its limits. However, once we begin to look at area magic squares, it becomes much less evident to visualise and construct the corresponding area magic tori using repeatable area cells, especially when the latter have to be irregular quadrilaterals... The following illustration shows a sketch of an area magic torus of order-3 that I created back in January 2017. I call it a sketch because it may be necessary to use consecutive areas starting from 2 or from 3, should the construction of an area magic torus of order-3, using consecutive areas from 1 to 9, prove to be impossible. And while it can be seen that such a torus is theoretically constructible, many calculations will be necessary to ensure that the areas are accurate, and that the irregular quadrilateral cells can be assembled with precision:

At the time discouraged by the complications of such geometries, I decided to suspend the research of area magic tori. But since the invention of area magic squares, other authors have introduced some very interesting polyomino versions that open new perspectives: 

On the 20th May 2021, Morita Mizusumashi (盛田みずすまし @nosiika) tweeted a nice polyomino area magic square of order-3 constructed with 9 assemblies of 5 to 13 monominoes. 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. On the 24th May 2021, Yoshiaki Araki then tweeted an order-3 polyomino area magic square constructed using 9 assemblies of 1 to 9 same-shaped pentominoes! Edo Timmermans, the author of this beautiful square, had apparently been inspired by Yoshiaki Araki's previous posts! Since the 22nd June 2021, Inder Taneja has also published a paper entitled "Creative Magic Squares: Area Representations" in which he studies polyomino area magic using perfect square magic sums.

Intention and Definition

The intention of the present article is to explore the use of polyominoes for area magic torus construction, with the objective of facilitating the calculation and verification of the cell areas, while avoiding the geometric constraints of irregular quadrilateral assemblies. Here, it is useful to give a definition of a polyomino area magic torus:

1/ In the diagram of the torus, the entries of the cells of each column, row, and of at least two intersecting diagonals, will add up to the same magic sum. The intersecting magic diagonals can be offset or broken, as the area magic torus has a limitless surface, and can therefore display semi-magic square viewpoints.

2/ Each cell will have an area in proportion to its number. The different areas will be represented by tiling with same-shaped holeless polyominoes.

3/ The cells can be of any regular or irregular rectangular shape that results from their holeless tiling. 

4/ Depending on the order-n of the area magic torus, each cell will have continuous edge connections with contiguous cells (and these connections can be wrap-around, because the torus diagram represents a limitless curved surface).

5/ The vertex meeting points of four cells can only take place at four convex (i.e. 270° exterior  angled) vertices of each of the cells.

Polyomino Area Magic Tori (PAMT) of Order-3

Magic Torus index n° T3, of order-3. Magic sums = 15.
Please note that this is not a Polyomino Area Magic Torus,
but it is the Agrippa "Saturn" magic square, after a rotation of +90°, in Frénicle standard form.

Polyomino Area Magic Torus (PAMT) of order-3. Magic sums = 15. Tetrominoes.
Consecutively numbered areas 1 to 9, in an irregular rectangular shape of 180 units.

Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 18. Trominoes.
Consecutively numbered areas 2 to 10, in an irregular rectangular shape of 162 units.

Polyomino Area Magic Torus (PAMT) of order-3. Magic sums = 24. Trominoes.
Consecutively numbered areas 4 to 12, in an oblong 12 ⋅ 18 = 216 units.

Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 36. Trominoes Version 1.
Square 18 ⋅ 18 = 324 units.

Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 36. Trominoes Version 2.
Square 18 ⋅ 18 = 324 units.

Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 60. Pentominoes Version 1.
Square 30 ⋅ 30 = 900 units.

Polyomino Area Magic Torus (PAMT) of order-3. Magic sums = 24. Monominoes.
Consecutively numbered areas 4 to 12, in an irregular rectangular shape of 72 units.

Polyomino Area Magic Torus (PAMT) of order-3. Magic sums = 27. Monominoes Version 1.
Consecutively numbered areas 5 to 13, in a square 9 ⋅ 9 = 81 units.

Polyomino Area Magic Torus (PAMT) of order-3. Magic sums = 27. Monominoes Version 2.
Consecutively numbered areas 5 to 13, in a square 9 ⋅ 9 = 81 units.

Polyomino Area Magic Tori (PAMT) of Order-4

Magic Torus index n° T4.198, of order-4. Magic sums = 34.
Please note that this is not a Polyomino Area Magic Torus,
but it is a pandiagonal torus represented by a pandiagonal square that has Frénicle index n° 107.

The pandiagonal torus above displays 16 Frénicle indexed magic squares n° 107, 109, 171, 204, 292, 294, 355, 396, 469, 532, 560, 621, 691, 744, 788, and 839. It is entirely covered by 16 sub-magic 2x2 squares. The torus is self-complementary and has the magic torus complementary number pattern I. The even-odd number pattern is P4.1. This torus is extra-magic with 16 extra-magic nodal intersections of 4 magic lines. It displays pandiagonal Dudeney I Nasik magic squares. It is classified with a Magic Torus index n° T4.198, and is of Magic Torus type n° T4.01.2. Also, when compared with its two pandiagonal torus cousins of order-4, the unique Magic Torus T4.198 of the Multiplicative Magic Torus MMT4.01.1 is distinguished by its total self-complementarity.

Pandiagonal Polyomino Area Magic Torus (PAMT) of order-4. Magic sums = 34. Pentominoes.
Index PAMT4.198, Version 1, Viewpoint 1/16, displaying Frénicle magic square index n° 107.
Consecutively numbered areas 1 to 16, in an oblong 34 ⋅ 20 = 680 units.

Pandiagonal Polyomino Area Magic Torus (PAMT) of order-4. Magic sums = 50. Dominoes.
Version 1, Viewpoint 1/16.
Consecutively numbered areas 5 to 20, in a square 20 ⋅ 20 = 400 units.

Pandiagonal Polyomino Area Magic Torus (PAMT) of Order-4. Sums = 50. Dominoes.
Version 1, Viewpoint 16/16.
Consecutively numbered areas 5 to 20, in an irregular rectangular shape of 400 units.

Polyomino Area Magic Tori (PAMT) of Order-5

Pandiagonal Torus type n° T5.01.00X of order-5. Magic sums = 65.
Please note that this is not a Polyomino Area Magic Torus.

This pandiagonal torus of order-5 displays 25 pandiagonal magic squares. It is a direct descendant of the T3 magic torus of order-3, as demonstrated in page 49 of "Magic Torus Coordinate and Vector Symmetries" (MTCVS). In "Extra-Magic Tori and Knight Move Magic Diagonals" it is shown to be an Extra-Magic Pandiagonal Torus Type T5.01 with 6 Knight Move Magic Diagonals. Note that when centred on the number 13, the magic square viewpoint becomes associative. The torus is classed under type n° T5.01.00X (provisional number), and is one of 144 pandiagonal or panmagic tori type 1 of order-5 that display 3,600 pandiagonal or panmagic squares. On page 72 of "Multiplicative Magic Tori" it is present within the type MMT5.01.00x.

Pandiagonal Polyomino Area Magic Torus (PAMT) of order-5. Magic sums = 65. Hexominoes.
Index PAMT5.01.00X, Version 1, Viewpoint 1/25.
Consecutively numbered areas 1 to 25, in an oblong 65 ⋅ 30 = 1950 units.

Pandiagonal Polyomino Area Magic Torus (PAMT) of Order-5. Magic sums = 125. Monominoes.
Version 1, Viewpoint 1/25.
Consecutively numbered areas 13 to 37, in a square 25 ⋅ 25 = 625 units.

Pandiagonal Polyomino Area Magic Torus (PAMT) of Order-5. Magic sums = 125. Monominoes.
Version 2, Viewpoint 1/25.
Consecutively numbered areas 13 to 37, in a square 25 ⋅ 25 = 625 units.

Polyomino Area Magic Tori (PAMT) of Order-6

Partially Pandiagonal Torus type n° T6 of order-6. Magic sums = 111.
Please note that this is not a Polyomino Area Magic Torus.

Harry White has kindly authorised me to use this order-6 magic square viewpoint. With a supplementary broken magic diagonal (24, 19, 31, 3, 5, 29), this partially pandiagonal torus displays 4 partially pandiagonal squares and 32 semi-magic squares. In "Extra-Magic Tori and Knight Move Magic Diagonals" it is shown to be an Extra-Magic Partially Pandiagonal Torus of Order-6 with 6 Knight Move Magic Diagonals. This is one of 2627518340149999905600 magic and semi-magic tori of order-6 (total deduced from findings by Artem Ripatti - see OEIS A271104 "Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2").

Partially Pandiagonal Polyomino Area Magic Torus (PAMT) of order-6. Magic sums = 111.
Heptominoes, Version 1, Viewpoint 1/36.
Consecutively numbered areas 1 to 36, in an oblong 111 ⋅ 42 = 4662 units.

Partially Pandiagonal Polyomino Area Magic Torus (PAMT) of order-6. Sums = 147.
Dominoes, Version 1, Viewpoint 1/36.
Consecutively numbered areas 7 to 42, in a square 42 ⋅ 42 = 1764 units.

Observations

As they are the first of their kind, these Polyomino Area Magic Tori (PAMT) can most likely be improved: The examples illustrated above are all constructed with their cells aligned horizontally or vertically; and though it is convenient to do so, because it allows their representation as oblongs or squares, this method of constructing PAMT is not obligatory. Representations of PAMT that have irregular rectangular contours may well give better results, with less-elongated cells and simpler cell connections. 

While the use of polyominoes has the immense advantage of allowing the construction of area magic tori with easily quantifiable units, it also introduces the constraint of the tiling of the cells. It has been seen in the examples above that the PAMT can be represented as oblongs or as squares, while other irregular rectangular solutions also exist. A normal magic square of order-3 displays the numbers 1 to 9 and has a total of 45, which is not a perfect square. As the smallest addition to each of the nine numbers 1 to 9, in order to reach a perfect square total is four (45 + 9 ⋅ 4 = 81), this implies that when searching for a square PAMT with consecutive areas of 1 to 9, in theory the smallest polyominoes for this purpose will be pentominoes.

But to date, in the various shaped examples of PAMT shown above, the smallest cell area used to represent the area 1 is a tetromino, as this gives sufficient flexibility for the connections of a nine-cell PAMT of order-3 with consecutive areas of 1 to 9. Edo Timmermans has already constructed a Polyomino Area Magic Square of order-3 using pentominoes for the consecutive areas of 1 to 9, but it seems that such polyominoes cannot be used for the construction of a same-sized and shaped PAMT of order-3. Straight polyominoes are always used in the examples given above, as these facilitate long connections, but other polyomino shapes will in some cases be possible.

We should keep in mind that the PAMT are theoretical, in that, per se, they cannot tile a torus: As a consequence of Carl Friedrich Gauss's "Theorema Egregium", and because the Gaussian curvature of the torus is not always zero, there is no local isometry between the torus and a flat surface: We can't flatten a torus without distortion, which therefore makes a perfect map of that torus impossible. Although we can create conformal maps that preserve angles, these do not necessarily preserve lengths, and are not ideal for our purpose. And while two topological spheres are conformally equivalent, different topologies of tori can make these conformally distinct and lead to further mapping complications. For those wishing to know more, the paper by Professor John M. Sullivan, entitled "Conformal Tiling on a Torus", makes excellent reading.

Notwithstanding their theoreticality, the PAMT nevertheless offer an interesting field of research that transcends the complications of tiling doubly-curved torus surfaces, while suggesting interesting patterns for planar tiling: For those who are not convinced by 9-colour tiling, 2-colour pandiagonal tiling can also be a good choice for geeky living spaces:

Tiling with irregular rectangular shaped PAMT of order-3. Monominoes. S=24.

Tiling with irregular rectangular shaped PAMT of order-3. Tetrominoes. S=15.

Tiling with irregular rectangular shaped PAMT of order-3. Trominoes. S=18.

Tiling with oblong PAMT of order-3. Trominoes. S=24.

Tiling with oblong pandiagonal PAMT of order-4. Pentominoes. S=34.

Tiling with oblong pandiagonal PAMT of order-5. Hexominoes. S=65.

Tiling with square pandiagonal PAMT of order-5. Monominoes. S=125.

Tiling with oblong partially pandiagonal PAMT of order-6. Heptominoes. S=111.

Tiling with square partially pandiagonal PAMT of order-6. Dominoes. S=147.

There are still plenty of other interesting PAMT that remain to be found, and I hope you will authorise me to publish or relay your future discoveries and suggestions!

Read more ...

Even and Odd Number Patterns on Magic Tori of Orders 3 and 4

Ancient references to the pattern of even and odd numbers of the 3 × 3 magic square appear in the "I Ching" or "Yih King" (Book of Changes). In the introduction to the Chou edition, the scroll of the river Loh or "Loh-Shu", which is represented by the magic square of order-3, is written with black dots (the yin symbol and emblem of earth) for the even numbers, and white dots (the yang symbol and emblem of heaven) for the odd numbers. The "Loh-Shu" is incorporated in the writings of Ts'ai Yuän-Ting who lived from 1135 to 1198:

The Scroll of Loh, according to Ts'ai Yüang-Ting

A normal magic torus (or magic square) of order-n uses consecutive numbers from 1 to n² and consequently, for the order-3, the magic constant (MC) = 15. The Loh-Shu Magic Torus of Order-3 is presented below, together with its corresponding even and odd number pattern P3:

The "Loh-Shu" Magic Torus of Order-3 and its Even and Odd Number (x mod 2) Pattern P3

The magic torus T3 of order-3 is shown here as seen from the Scroll of Loh magic square viewpoint.

The even and odd number pattern P3 has reflection symmetry (with 6 lines of symmetry); rotational symmetry (of order 4); and point symmetry (over number 5, between numbers 3 and 7, between numbers 1 and 9, and between numbers 4, 6, 2, and 8).

The Even and Odd Number Patterns of the Magic Tori of Order-4

Only consecutively numbered magic tori (with numbers from 1 to n²) are studied here, and therefore, for order-4, the magic constant MC = 34.

The "Table of Fourth-Order Magic Tori" lists and describes each magic torus of order-4 using index numbers, while "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus" gives a group categorisation of the magic tori of order-4 using type numbers. Both of these classifications give the details of the Frénicle indexed magic squares that are displayed by each torus.

In the study that follows it can be seen that there are 4 essentially different even and odd number patterns on the magic tori of order-4:

The Order-4 Even and Odd Number Pattern P4.1
(represented by the pandiagonal torus T4.198)

The Pandiagonal Torus T4.198 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.1

The pandiagonal torus of order-4 (with index n° T4.198 and type n° T4.01.2) is shown here as seen from the Frénicle n° 107 magic square viewpoint.

The even and odd number pattern P4.1 has reflection symmetry (with 6 lines of symmetry); rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The pattern P4.1 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.1 concerns 79 out of the 255 magic tori, and 404 out of the 880 magic squares of order-4. These include the 3 pandiagonal tori that display 48 pandiagonal squares; 30 semi-pandiagonal tori that display 240 semi-pandiagonal squares; 12 partially pandiagonal tori that display 48 partially pandiagonal squares; and also 34 basic magic tori that display 68 basic magic squares.

The Order-4 Even and Odd Number Pattern P4.2
(represented by the semi-pandiagonal torus T4.059)

The Semi-Pandiagonal Torus T4.059 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.2

The semi-pandiagonal torus of order-4 (with index n° T4.059 and type n° T4.02.2.01) is shown here as seen from the Frénicle n° 21 magic square viewpoint.

The even and odd number pattern P4.2 has reflection symmetry (with 8 lines of symmetry); rotational symmetry (of order 4); and multiple point symmetry when reading as a torus. The pattern P4.2 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.2 concerns 72 out of the 255 magic tori, and 256 out of the 880 magic squares of order-4. These include 18 semi-pandiagonal tori that display 144 semi-pandiagonal squares; 6 partially pandiagonal tori that display 16 partially pandiagonal squares; and also 48 basic magic tori that display 96 basic magic squares.

The Order-4 Even and Odd Number Pattern P4.3
(represented by the partially pandiagonal torus T4.186)

The Partially Pandiagonal Torus T4.186 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.3

The partially pandiagonal torus (with index n° T4.186 and type n° T4.03.1.3) is shown here as seen from the Frénicle n° 100 magic square viewpoint.

The even and odd number pattern P4.3 has reflection symmetry (with 8 lines of symmetry); rotational symmetry (of order 4); and multiple point symmetry when reading as a torus. The pattern P4.3 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.3 concerns 16 out of the 255 magic tori, and 44 out of the 880 magic squares of order-4. These include 6 partially pandiagonal tori that display 24 partially pandiagonal squares; and also 10 basic magic tori that display 20 basic magic squares.

The Order-4 Even and Odd Number Pattern P4.4
(represented by the basic magic torus T4.062)

The Basic Magic Torus T4.062 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.4

The basic magic torus (with index n° T4.062 and type n° T4.05.1.12) is shown here as seen from the Frénicle n° 37 magic square viewpoint.

The even and odd number pattern P4.4 has reflection symmetry (with 8 lines of symmetry); rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The pattern P4.4 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.4 concerns 88 out of the 255 magic tori, and 176 out of the 880 magic squares of order-4. All of these 88 magic tori and the displayed 176 magic squares are basic magic.

Observations on the Even and Odd Number Patterns of the Magic Tori and Magic Squares of Order-4

The findings of the even and odd number patterns of the magic tori of order-4 are recapitulated and analysed in comparative tables, (together with the Dudeney complementary number patterns and the Magic Torus self-complementary and paired complementary number patterns), in the file below:

The Even and Odd Number Patterns of the Semi-Magic Tori of Order-4

Only consecutively numbered semi-magic tori (with numbers from 1 to n²) are studied here, and consequently, for the order-4, the magic constant MC = 34.

In "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus" the 4,038 semi-magic tori of order-4 have been sorted, and type numbers have been attributed taking into account the arrangements of their magic diagonals (if any). 

Each of the four even and odd number patterns already found above on the magic tori of order-4, also occur on semi-magic tori of order-4. For example, from left to right below, we find a semi-magic torus type T4.08.0W with pattern P4.1; a semi-magic torus type T4.08.0X with pattern P4.2; a semi-magic torus type T4.07.0Y with pattern P4.3; and a semi-magic torus type T4.09.00Z with pattern P4.4:

Semi-Magic Tori of Order-4 with Even and Odd Number (x mod 2) Patterns P4.1, P4.2, P4.3 and P4.4

The even and odd number patterns P4.1 to P4.4 are not the only ones that can be found, but the patterns which are specific to the semi-magic tori of order-4 have neither been fully investigated nor precisely counted, and the examples that follow only represent some of the different varieties.

The Order-4 Even and Odd Number Pattern P4.5
(represented by the semi-magic torus type T4.06.0A)

The Semi-Magic Torus type T4.06.0A of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.5

This semi-magic torus type n° T4.06.0A was found by Walter Trump, and has been previously illustrated in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The even and odd number pattern P4.5 has diagonal reflection symmetry (with 4 lines of symmetry); and negative (evens to odds and vice versa) rotational symmetry (of order 2). The blocks of even and odd numbers are translations of each other.

The Order-4 Even and Odd Number Pattern P4.6
(represented by the semi-magic torus type T4.07.0A)

The Semi-Magic Torus type T4.07.0A of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.6

This semi-magic torus type n° T4.07.0A was found by Walter Trump, and has been previously illustrated in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The even and odd number pattern P4.6 has reflection symmetry (with 4 lines of symmetry); rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The blocks of even and odd numbers are translations of each other.

The Order-4 Even and Odd Number Pattern P4.7
(represented by the semi-magic torus type T4.09.00M)

The Semi-Magic Torus type T4.09.00M of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.7

This semi-magic torus type n° T4.09.00M was found by Walter Trump, and has been previously illustrated in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The even and odd number pattern P4.7 has both positive and negative reflection symmetry; negative rotational symmetry (of order 2); and multiple negative point symmetry when reading as a torus. The blocks of even and odd numbers are translations of each other.

The Order-4 Even and Odd Number Pattern P4.8
(represented by the semi-magic torus type T4.10.000X)

The Semi-Magic Torus type T4.10.000X of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.8

3,726 semi-magic tori of the type 10 have been found by Walter Trump, and his findings are detailed in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The T4.10.000X's even and odd number pattern P4.8 has negative reflection symmetry (2 lines of symmetry); negative rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The blocks of even and odd numbers are translations of each other.

General Observations

The "Table of Fourth-Order Magic Tori" has been revised to include the even and odd number pattern references for each of the magic tori of order-4 (listed together with the Frénicle indexed magic squares that each magic torus displays).

Other interesting websites that examine, but do not enumerate, the different even and odd number patterns on magic squares of order-4, include "4x4 Magic Squares" by Dan Rhett and "Pattern in Magic Squares" by Vipul Chaskar.

A brief look at higher-orders suggests that the order-5 has many irregular patterns; the order-6 has many patterns with negative reflection symmetry (evens to odds and vice versa); the order-7 has many patterns with rotational symmetry; and the order-8 has patterns very like those of the order-4 examined above. All of these orders merit an in-depth analysis of their even and odd number patterns.

In conclusion, it is interesting to note that the Arab mathematicians of the tenth century constructed odd-order magic squares with striking even and odd number patterns (as for example the magic squares in figures b44 and b49 on pages 243 and 248 of Magic Squares in the Tenth Century: Two Arabic treatises by Anṭākī and Būzjānī, translated by Jacques Sesiano). And in the same context, Paul Michelet has recently brought to my attention a magnificent 15 x 15 magic square by Ali b. Ahmad al-Anṭākī (d. 987). Al-Anṭākī's bordered magic square is solid evidence of the Middle Eastern mathematicians' mastery of even and odd number patterns!

Read more ...

Area Magic Squares and Tori of Order-3

In December 2016 I made a sketch design of an area magic square to illustrate a seasonal greetings card for 2017. Using the traditional order-3 magic square, I attempted to use continuous straight lines for the area cell borders. I quickly realised that such an assembly was impossible to achieve, but that with approximate areas I could nevertheless make a pleasing graphic pattern:

Sending this image with my best wishes to a circle of magic square enthusiasts on the 30th December 2016, I added the following postscript: "The areas are approximate, and I don't know if it is possible to obtain the correct areas with 2 vertically slanted straight lines through the square. Perhaps someone will be able to work this out in 2017?"

Lee Sallows was the first to respond on the 4th and 5th January 2017, proposing an approach based on his previous work on geomagic squares. Developing the ideas that can be found in the figure 4.1 on page 5 of his book "Geometric Magic Squares: A Challenging New Twist Using Colored Shapes Instead of Numbers," his area square is illustrated below:

Area square by Lee Sallows

Lee Sallows points out that, when comparing the initial square of his book with this new puzzle square, "in many (but not all) cases adjacent piece outlines have been made to complement each other. For every gain in area at one position there is an identical loss of area at another. In this way the areas of all 9 pieces remain as they were, so that the square remains a magical dissection." Thank you for your work Lee! I particularly like your puzzle solution.

Meanwhile, I was wondering if a solution could be found for a third-order area magic torus, that is to say, one where the area cell intersections would meet correctly when wrapped. With this purpose in mind I set the following new rules for myself:

1/ Each cell will have an area that corresponds with its number from 1 to 9. The total areas will therefore be 45.

2/ The connections between the cells will remain unchanged, but as these cells will be of different sizes, the resulting latitudes, longitudes and diagonals will not necessarily be straight lines. The initially square cells will become regular or irregular quadrilaterals, (excluding complex quadrilaterals).

3/ The distances measured orthogonally between opposite sides of the resulting magic or semi-magic square viewpoints will always be √45, (the circumferences of the magic torus).

This third rule took into account the flattened square that we are accustomed to looking at, but it also implied that the torus would degenerate into a sphere... With these new constraints, on the 6th January 2017 I was able to come up with the area magic torus illustrated below:

These extended best wishes, expressed through the complete set of third-order magic and semi-magic squares, include messages that reflect the multiple nationalities of the members of the magic square circle. The magic square viewpoint of the torus is placed at the top left. The patterns still need adjusting in order to achieve the correct areas, but as there is more flexibility, I am fairly confident that at least one accurate solution can be found.

On the 6th January 2017 Walter Trump also sent us a new area magic design that used 4 continuous straight dissection lines. Although this schema could not be adapted to the number sequence 1 to 9, it was area magic:

Area Magic Schema by Walter Trump

Inder Jeet Taneja then suggested that the problem with the sequence 1 to 9 was that the total areas did not add up to a perfect square area. He proposed that instead of using the numbers 1 to 9, perhaps we should try using the numbers 5 + 6 + ... + 13 = 81 = 9² ?

On the same day, following Inder's suggestion, Walter Trump amazed us all with this first third-order linear area magic square! :

Bravo Walter for your achievement following Inder's suggestion! I hardly dared to believe that such a simple area pattern could produce a magic square!

Having other commitments to honour, Walter Trump then requested that somebody else searched for solutions using lower sequences. No other volunteers came forward, so I decided to look for myself, and came up with this second linear area magic square of the third-order using the sequence 3 to 11:

As I am not a programmer, I used Autocad to construct this linear area magic square manually. The areas of this provisional square are therefore only accurate up to two decimal places, before computer verification and area optimisation.

In the meantime Walter Trump continued writing a computer program to handle the equations and search iteratively for solutions. This was able to give orthogonal coordinates having at least 14 decimals after the commas. At first unable to solve the non-linear equations explicitly, Walter progressively improved his approach, and on the 8th January 2017 he sent us a message stating that for some sequences there were two solutions, and that the lowest number sequence ranged from 2 to 10!

On the 10th January 2017, Walter Trump created an algorithm for "Third-Order Linear Area Magic," which he has kindly authorised me to publish below. This explains the method of construction and gives the reason why two solutions exist for most, but not all of the sequences studied:

Using the coordinates generated by Walter Trump's computer program, both solutions of the 5 lowest sequences of the third-order linear area magic squares are illustrated in square form below. It will be seen that the 1 to 9 sequence fails, and the linear area magic squares are only possible from the sequence 2 to 10 upwards (when the continuous straight dissection lines produce no more than 4 intersections inside the squares):

During this time my first question concerning the sequence 1 to 9 remained unanswered. So, on the 9th January 2017, I created the first area magic square that used the numbers of the classical magic square of order-3, as illustrated below:

More recreational than mathematical, this "Mont Saint Michel" area magic square demonstrates that it is possible to use 2 continuous straight dissection lines - thus resolving the initial 2017 greetings card challenge. The design is an area magic square, because all of the cells are quadrilaterals (even if near triangular for the area 1), and all of their connections are preserved.

Inder Taneja has since followed up on his initial suggestions for perfect square sequences. On the 11th February 2017 he published some very interesting results in his new paper "Magic Squares with Perfect Square Number Sums."

This is an ongoing story, and there will probably be new developments in the near future. Apart from Inder Taneja, Lee Sallows and Walter Trump who I have already mentioned, I also wish to thank the other participants: These include Craig Knecht for his kind encouragements, Miguel Angel Amela for his encouragements and suggestions concerning accuracy, Dwane Campbell for his encouragements and suggestions, and last but not least, Francis Gaspalou for his encouragements and interventions: For the area magic squares of the third-order, Francis Gaspalou wrote the equations of the 4 straight lines using 4 parameters which were the 4 slopes of those lines. He found the exact conditions which had to be fulfilled by these 4 parameters in order to obtain a solution. Unfortunately it was not possible to solve the system and write explicit formulae for the solutions. Francis was nevertheless able to check that the two solutions per sequence found by Walter Trump's algorithm fulfilled the conditions. Thus by using this different method, Francis was able to confirm the validity of Walter's computer results. Francis Gaspalou has kindly authorised me to publish his equations here:



On the 13th January 2017 Bob Ziff intervened, stating that if we agree that the system of equations is soluble, then it is soluble to any number of digits, and proved his point by calculating the slopes with 1000 digits, using the professional software Mathematica and the four equations of Francis Gaspalou.

Also, on the 13th January 2017, Walter Trump created a third-order linear area nearly-magic square with integer coordinates. This square is semi-magic with a magic constant of 940800! Walter points out that when you divide all the areas by 140, then the coordinates are no longer integers, but irrational numbers that can be exactly described using fractions and square roots. Walter has kindly authorised me to publish his findings below:

Francis Gaspalou has suggested that the next step might be to find a fourth-order area magic square with 6 continuous straight dissection lines, or even a concentric fifth-order area magic square. Why not imagine a contest, open to all, so as to find higher-order solutions of an equal mathematical importance to that of the third-order linear area magic squares?

If anyone wishes to contribute linear or other strict geometrical constructions of higher-order area magic squares, then please send me the x and y coordinates of the cell intersections that define the areas correctly up to two or more decimals after the commas. No prizes can be given, but the authors of pertinent solutions will, if they wish, have their solutions published here, and thus be able to bask in the reflected glory...

Latest developments

On the 19th January 2017 Greg Ross published an article on the subject of Area Magic Squares in his Futility Closet - An Idler's Miscellany of Compendious Amusements. This article has since been relayed by Reddit Mathpics, Simplementenumeros, Prime Puzzles, and by Thermally Stressed Dairy Cows, amongst others.

On the 20th January 2017 I was aware that Walter Trump, (now joined by Hans-Bernhard Meyer), had already found that there are no cases of order-4 area magic squares with sequences of consecutive numbers. So I therefore began searching for non-consecutive number sequences that might also yield solutions in order-3, and produced the draft linear area magic square illustrated below:

I sent this square to Walter Trump and the other members of the magic square circle asking if it could be verified using a computer program. Walter was quick to react, sending me precise coordinates for two solutions. Drawn-up using his coordinates, the two versions of this square are illustrated below:

On the 3rd February 2017, responding to a Prime Puzzle challenge, Jan van Delden contributed the following palprime linear area magic square solution:

The palindromic primes used here are symmetrical numbers that remain the same when their digits are reversed. The magic sum of these 11-digit palindromic primes is 377,024,295,63. This area magic square is the conversion of a square that was originally sent by Carlos Rivera and Jaime Ayala to Harvey Heinz on the 22nd May 1999. Jan van Delden mentions that the direction coefficients of the lines are indicated in white, and that the quadrilaterals are numbered in the style of Francis Gaspalou. Full details of Jan van Delden's approach to the equations can be found in Prime Puzzles.

On the 4th February 2017 Jan van Delden contributed this second prime linear area magic square to Prime Puzzles:

Jan states that this prime area magic square has a minimal magic sum of S=213. Congratulations Jan, for these prime achievements!

Please note, that since the 5th March 2017, Jan has published a new paper entitled "Area Magic Squares of Order 3," in which he extends his findings.

Seemingly, the number of order-3 linear area squares is infinite!

Related links

A new post on Area Magic Squares of Order-6 can be found in these pages since the 25th January 2017.

On the same day Hans-Bernhard Meyer published an article entitled Observations on 4x4 Area Magic Squares with vertical lines in his website: Math'-pages.

Since the 3rd February 2017, Walter Trump has published a chapter entitled Area Magic Squares in his website: Notes on Magic Squares. This chapter includes many analyses and examples of area magic squares of the third and fourth-orders.

Since the 8th February 2017, "Area Magic Squares of Order-4" relates the first findings of area magic squares of the fourth-order.

Since the 11th February 2017, Inder Jeet Taneja, inspired by the research in area magic squares, has published a new paper entitled "Magic Squares with Perfect Square Number Sums."

Since the 5th March 2017, Jan van Delden has published a paper entitled "Area Magic Squares of Order 3" in which he presents an improved algorithm. His work also includes new findings in area semi-magic squares of order-3, and a shoelace formula to measure the deviation in area.

Since the 8th March 2017, following a tweet by Simon Gregg, Microsiervos published an article entitled "Cuadrados de áreas mágicas." This article has been relayed by Inoreader amongst others.

Since the 22nd March 2017, writing for EL PAÍS, Miguel Ángel Morales has included the subject of area magic squares in an article entitled "No solo de números consecutivos vive el cuadrado mágico."

Since the 19th April 2017, writing for "Geogebra," Georg Wengler has included an article entitled "Magisches Flächen-Quadrat."

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."

Since the 25th March 2019, Akehiko Takahashi, the webmaster of "Math Dojo," includes Area Magic Squares in his article entitled "Magic Squares and Beyond."

On the 17th July 2019, a discussion in Japanese, about Area Magic Squares, began on the forum of "Japanese Traditional Mathematical Calculator."

 

Circa 2019, on the French recreational mathematics site "Diophante," Michel Lafond published the first linear area magic square in a problem n° B138, entitled "Carré magique géométrique," and asked mathematicians to prove its existence. Several solutions are proposed.

On the 1st January 2020, (exactly three years after the original area magic square greetings card!), an article about area magic patchwork entitled "Flächenmagische Quadrate - aus Stoff," was published by the German blogger "Siebensachen-zum-Selbermachen."

On the 12th January 2020, the recreational mathematician Ed Pegg Jr., published a "Magic Square with areas" on his site "Mathpuzzle." 

 

On the 20th May 2021, Morita Mizusumashi (盛田みずすまし @nosiika) tweeted a nice polyomino area magic square of order-3 constructed with 9 assemblies of 5 to 13 monominoes. 

 

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.

 

On the 24th May 2021, Yoshiaki Araki then tweeted an order-3 polyomino area magic square constructed using assemblies of 1 to 9 same-shaped pentominoes! Edo Timmermans, the author of this beautiful square, had apparently been inspired by Yoshiaki Araki's previous posts!

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.

 

On the 3rd October 2024, Gianni A. Sarcone published an article entitled "Geometric Magic Square," presenting the area magic square concept in the pages of the Archimedes Lab Project.

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