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!

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Smart Filling

I wish to thank Francis Gaspalou for bringing to my attention a board game question n° J113 "Savant Remplissage (1er épisode)" that was asked on the site of Diophante:

In French, the question reads as follows:

"Prouver qu'il est possible de remplir les 81 cases d' un tableau 9 x 9 avec les entiers de 1 à 81 de sorte que les sommes des nombres contenus dans tous les carrés 3 x 3 sont identiques.

Pour les plus courageux: même question pour le remplissage des 256 cases d'.un tableau 16 x 16 avec les entiers de 1 à 256 et les sommes identiques des nombres contenus dans tous les carrés 4 x 4."

Translated into English, the question is as follows:

Show that it is possible to fill the 81 cells of a 9 x 9 grid with the whole numbers from 1 to 81 so that the sums of the numbers contained in each of the 3 x 3 squares are identical. For the more courageous: Show that it is possible to fill the 256 cells of a 16 x 16 grid with the whole numbers from 1 to 256 so that the sums of the numbers contained in each of the 4 x 4 squares are identical.

The Reply to the Question

Remembering earlier research carried out in 2016 when preparing "Magic Torus Coordinate and Vector Symmetries" I realised that the question could be answered using this modular coordinate equation:

The following pdf files (in English and French versions) show how the equation is used to produce perfect square order N pandiagonal tori that are always entirely covered by N² x [√N x √N] submagic squares:

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First-Order Panmagic Torus T1

As is clearly stated in the OEIS sequence A006052 "Number of magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections," the first and smallest magic square is of order-1.

The dotted red lines across this magic square represent its magic diagonals. For a basic magic square, each row, column, and main diagonal must sum to the magic constant. The magic constant of a magic square is equal to the division of the triangular number of its squared order by its order. The magic constant (mcN) of an Nth order magic square (or torus) can thus be calculated as follows:

mcN  =  (N²)² + N²  x  1    =    N(N² + 1)

                     2            N                2

For the first-order magic square (or magic torus) the magic constant should therefore be:

mc1   =  1(1² + 1)   =  1,  which is the case.

                    2

The blue border of the unique number cell illustrated above, is also the limit of the magic square itself. The video below shows how this blue border merges to form single latitude and longitude lines on the curved 2D surface of the first-order magic torus:

Gluing a Torus video by Geometric Animations - University of Hannover, hosted by YouTube

We can check some of the simple conditions that define a basic magic odd-order torus, previously deduced whilst observing the unique third-order T3 magic torus:

The number of N-order squares (both magic and semi-magic squares) that are displayed on each N-order magic torus = N². The first-order magic torus should therefore display 1² = 1 first-order square, which is the case.

Basic magic odd-N-order tori  display not only 1 basic N-order magic square, but also N²-1 semi-magic N-order squares. The first-order basic magic square should therefore display 1 basic order-1 magic square, and 1²-1 = 0 semi-magic order-1 squares, which is also the case.

What is more surprising, although quite logical once we consider the question, is that the order-1 magic torus even satisfies the basic conditions for pandiagonality!

If pandiagonal (magic along all of its diagonals), an N-order torus displays N² pandiagonal squares of order N. The first-order torus should therefore display 1² pandiagonal squares = 1 pandiagonal square, which once again is the case.

Taking into account the panmagic properties of this torus, the adjective "trivial," which is often used to describe the first-order square, now seems almost depreciatory, (notwithstanding the fact that for mathematical contexts, the dictionary definitions of trivial include: "simple, transparent, or immediately evident"). There is more to the order-1 torus than first meets the eye! Not only is it pandiagonal, but its number One also signifies mathematical creation and the very beginnings... The Pythagoreans referred to the number One as the "monad," which engendered the numbers, which engendered the point, which engendered all lines, etc. For Plotinus and other neoplatonists, the number One was the ultimate reality and the source of all existence. The first-order torus, together with the number One that it displays, both symbolise the "Big Bang."

I admit to having rather underestimated this first-order magic torus until Miguel Angel Amela kindly sent me a copy of one of his studies "Pandiagonal Latin Squares and Latin Schemes in the Torus Surface," on the 9th March 2016. By extrapolating his findings, I discovered for myself the proof of the pandiagonal characteristics of the first-order, and I wish to thank Miguel for this paper which has been an inspiration to me.

The pandiagonal torus T1 of order-1 comes first again in the new OEIS sequence A270876 "Number of magic tori of order n composed of the numbers from 1 to n^2."

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A Magic Square Tribute to Dürer, 500+ Years After Melencolia I

Albrecht Dürer's "Melencolia I" contains numerous mathematical and philosophical symbols amongst which we can see a magic square, a polyhedron and a sphere. These 3 symbols are partially interpreted in an earlier article written in French.

1514, the date of the engraving, appears in the lower line of the fourth-order magic square. 500 years later, is it possible to create a fourth-order magic torus tribute to Dürer?

In order to write 2014 we can subtract 1 from all the numbers (1 to 16), and thus obtain the zero that cannot be found in normal fourth-order magic squares. After the subtraction, we can use a flattened viewpoint of the basic magic torus T4.177 (group number T4.05.4.12) that displays not only the Frénicle squares n°99 and 618, but also 14 semi-magic squares. The numbers 0 to 15 are repeated beyond the edges of the square in order to represent the toroidal continuity. As the new figures contain the numbers from 0 to (N²-1) instead of 1 to N², their magic constant is 30 instead of 34.

A magic torus tribute to Dürer

Detail of the "first state" version of the magic square that appears on Dürer's "Melencolia I"
(National Gallery of Victoria)

Perhaps Dürer hesitated when engraving the left hand side of his square: The number 5 appears to replace a former number 6. It should also be noted that the print detailed in the first version above is, according to The National Gallery of Victoria is "the very rare first state." The reversed number 9 was later corrected in the revised version below. On page 12 of the 1961 Aukland City Art Gallery catalogue "Albrecht DÜRER forty engravings and woodcuts," it is written that "There are two states - the first where the nine on the numerical table is reversed, the second where the nine is corrected." The contemplation of the two versions of Dürer's magic square may well remind the reader of the duality of the “De Umbrarum Regis Novum Portis” engravings in Roman Polanski's film "The Ninth Gate." David Fritz Finkelstein has written that "Dürer's engraving MELENCOLIA I was circulated in two versions not previously distinguished. Besides their conspicuous early Renaissance scientific instruments and tools, they contain numerous apparently unreported concealments whose detection reveals heresies expressed in the work." Frickelstein believes that the second version of the engraving is the one in which the number 9 is reversed, and suggests that the numerals 9, 6, and 0 are curled serpents. To find out more please consult Finkelstein's "Melencolia I.1*"

Detail of the revised version of the magic square that appears on "Melencolia I"
Albrecht Dürer [Public domain], via Wikimedia Commons

Dürer's square, (Frénicle n° 175), is one of 8 semi-pandiagonal squares displayed by the semi-pandiagonal torus T4.077 (torus type n° T4.02.2.03). The seven other semi-pandiagonal squares that are displayed by the same "Dürer torus" are the Frénicle index n° 27, 233, 360, 421, 583, 803, and 850. The semi-pandiagonal "Dürer torus" T4.077 also displays 8 semi-magic squares, and is completely covered by 8 sub-magic squares. More details of the fourth-order magic tori can be found in earlier articles such as:

"255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus," "Table of Fourth-Order Magic Tori," and "Fourth-Order Magic Torus Chart." Please note that since the 21st January 2018 "Fourth-Order Magic Torus Chart" has now been superseded by "Multiplicative Magic Tori."

A magic torus tribute to Dürer

Miguel Angel Amela has very kindly authorised me to publish an extraordinary pandiagonal square he created for this anniversary. Again, this is not a normal 4x4 square, as the magic constant of all the columns, rows, and diagonals is not 34, but 1514 - the year that Dürer engraved Melencolia I! Miguel has also managed to write 2014 at the centre of the bottom row, in just the same cells that Dürer used to date his square.

Miguel Amela's tribute to Dürer : 2014 - 1514 = 500 years.

The following figure shows a water retention diagram of Miguel Amela's square that was created by Craig Knecht.

A water retention diagram of Miguel Angel Amela's tribute to Dürer - diagram by Craig Knecht

Craig Knecht had already devised a 14 x 14 magic square in 2013 to honour the 500th anniversary of Durer's famous "Melencolia I" magic square. The animation of this water retention square can be found at the following link.

Further to the first edition of the present article, on the 21st May 2015, Miguel Angel Amela sent me this very original magic square that he created to celebrate the 544^th birthday of Albrecht Dürer (1471-1528):

Miguel Angel Amela's 544th birthday tribute to Albrecht Dürer, created on 21st May 2015

This is not a normal 4 x 4 magic square as the magic constant of all the columns, rows, and magic diagonals is not 34, but 544. Miguel Angel Amela's creation (using the reversed nine version of Dürer's writing style) is the product of the Dürer square integers and 16 (which is the number of integers). Miguel Angel Amela points out that the divisors of 544 are: 1, 2, 4, 8, 16, 17, 32, 34, 68, 136, 272, and 544:

1, 2, 4, 8, and 16 all figure on Dürer's fourth-order magic square.

17 is the total of the complementary numbers 1 and 16, 2 and 15, 3 and 14, etc. on Dürer's square.

32 is either the product of 4 and 2 times 4, or it is twice the number of the integers on Dürer's square.

34 is the magic constant of Dürer's square.

68 is twice the magic constant (2 x 34) of Dürer's square.

136 is 4 times the magic constant (4 x 34), and therefore the total of all the numbers on Dürer's square.

272 is 8 times the magic constant (8 x 34) of Dürer's square.

544 is 16 times the magic constant (16 x 34) of Dürer's square.

The square includes the numbers 16 (4 to the power of 2), 64 (4 to the power of 3) and 256 (4 to the power of 4) in its corners, and 4x4 magic square enthusiasts will appreciate this subtle wink! Thank you Miguel for allowing me to publish your very original creation!

Lorenzo D. Sisican, Jr.has kindly authorised me to publish two non-normal, but semi-pandiagonal squares that he has created: The first of these squares has a magic constant of 500 in its rows, columns and alternate diagonals (500 years after Melencolia I).  

Lorenzo D. Sisican Jr.'s first tribute to Dürer, 500 years after Melencolia I

The date of the creation of Melencolia I, the year 1514, is obtained by the numbers 15 & 14. The anniversary year 2014 is obtained by the numbers 20 & 14. Dürer’s date of birth, the 21^st May 1471 (21/5/1471), is obtained by the following numbers: (20 + 28 + 167) &14 & 71 = 215 14 71. Dürer died on April 6^th, 1528 (4/6/1528), and his date of death is obtained by the following numbers: (161 + 245) & 15 & 28 = 4 06 1528.

The second of Lorenzo D. Sisican Jr.'s squares is also semi-pandiagonal, and has a magic constant of 500 in its rows, columns and alternate diagonals (500 years after Melencolia I). 

Lorenzo D. Sisican Jr.'s second tribute to Dürer, 500 years after Melencolia I

The date of the creation of Melencolia I, the year 1514,  is obtained by the numbers 151 & 4. The anniversary year 2014 is obtained by the numbers 201 & 4. Melencolia I was created 14 years before Dürer’s death, and the number 14 is also prominent on this square.

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