Except where otherwise noted, the contents of Magic Squares, Spheres and Tori (https://carresmagiques.blogspot.com/), are licensed by the author William Walkington, under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence. For details of the licence and instructions how to attribute this work, please refer to the CC BY-NC-SA 4.0 link in the adjoining sidebar.
This licence lets others remix, tweak and build upon the author's work non-commercially, as long as they credit the author and license their new creations under the identical terms.
The author has uploaded some of the illustrations of this blog to public albums on Flickr.com . All of the images in this blog (whether they are on Flickr.com or not) are covered by the Creative Commons licence, unless stated otherwise in the exceptions listed below.
Excluded from the CC NY-NC-SA 4.0 licence, the copyright logo (alongside the header title of the blog) is part of a trademark application made at l'Institut National de la Propriété National (I.N.P.I.) of Paris on the 24th May 2005. This was given the national number 05 3 361 836 and was published and registered in the Bulletin Officiel de la Propriété Industrielle (B.0.P.I.) n° 05/27 Volume 1. However, the author may grant permission for the re-use of the logo.
The other exceptions to the Creative Commons Licence are detailed below:
The author has uploaded some of the illustrations of this blog to public albums on Flickr.com . All of the images in this blog (whether they are on Flickr.com or not) are covered by the Creative Commons licence, unless stated otherwise in the exceptions listed below.
Excluded from the CC NY-NC-SA 4.0 licence, the copyright logo (alongside the header title of the blog) is part of a trademark application made at l'Institut National de la Propriété National (I.N.P.I.) of Paris on the 24th May 2005. This was given the national number 05 3 361 836 and was published and registered in the Bulletin Officiel de la Propriété Industrielle (B.0.P.I.) n° 05/27 Volume 1. However, the author may grant permission for the re-use of the logo.
The other exceptions to the Creative Commons Licence are detailed below:
Post by post, the exceptions to the Creative Commons Licence are as follows:
Friday, 23 July 2010 (republished on Saturday, 4 April 2020)
Les sept carrés magiques d'Agrippa, ainsi que les trois autres images de carrés magiques qui sont dessinés par l'auteur à partir d'exemples connus, restent dans le Domaine Public.
L'image du tore sphérique, issue d'un travail en commun de Rémy C. et William W. est copyright tous droits réservés.
L'image du tore sphérique, issue d'un travail en commun de Rémy C. et William W. est copyright tous droits réservés.
Il n'y a pas d'autres exceptions à la licence CC BY-NC-SA 4.0.
Saturday, 24 July 2010 (republished on Sunday, 5 April 2020)
Il n'y a pas d'exceptions à la licence CC BY-NC-SA 4.0.
Tuesday, 10 August 2010 (republished on Sunday, 5 April 2020)
Le fichier image Dürer Melancholia [sic] I.jpg publié par Wikimedia Commons est dans le domaine public.
Le fichier image Durers [sic] Solid Square.PNG publié par Wikimedia Commons est dans le domain public.
Il n'y a pas d'autres exceptions à la licence CC BY-NC-SA 4.0.
Friday, 28 October 2011 (republished on Sunday, 5 April 2020)
Le fichier image Torus from rectangle.gif, par l'auteur Lucas Vieira, publié par Wikimedia Commons, est dans le domaine public.
Les tores partiellement pandiagonaux type T4.04 ont été identifiés par la programmation de Walter Trump. Les tores semi-magiques types T4.06, T4.07, T4.08, T4.09 et T4.10 ont été énumérés par la programmation de Walter Trump. Les images des tores semi-magiques (ou carrés semi-magiques) ont été dessinées par l'auteur utilisant les résultats de Walter Trump.
Walter Trump est d'accord avec les termes du licence CC BY-NC-SA 4.0.*
Walter Trump est d'accord avec les termes du licence CC BY-NC-SA 4.0.*
Il n'y a pas d'autres exceptions à la licence CC BY-NC-SA 4.0.
Friday, 11 November 2011 (republished on Sunday, 5 April 2020)
Le fichier image Inside-out torus (animated, small).gif, par l'auteur Surot, publié par Wikimedia Commons, est dans le domaine public.
Il n'y a pas d'autres exceptions à la licence CC BY-NC-SA 4.0.
Monday, 9 January 2012 (republished on Sunday, 5 April 2020)
The image file Torus from rectangle.gif, by the author Lucas Vieira, published by Wikimedia Commons, is in the public domain.
The partially pandiagonal tori type T4.04 have been identified thanks to Walter Trump's computing skills. The semi-magic tori types T4.06, T4.07, T4.08, T4.09 and T4.10 have been enumerated by Walter Trump's computing. The images of the semi-magic tori (or semi-magic squares) have been drawn by the author using Walter Trump's findings.
Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Friday, 9 March 2012 (republished on Sunday, 5 April 2020)
The image file Inside-out torus (animated, small).gif, by the author Surot, published by Wikimedia Commons, is in the public domain.
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 22 April 2012 (republished on Monday, 6 April 2020)
The torus type T4.04, (as illustrated by the author in the image of the partially pandiagonal torus with index n°T4.096 (type T4.04.1)), was identified thanks to Walter Trump's computing skills.
Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Friday, 28 September 2012 (republished on Monday, 6 April 2020)
Although the text and illustrations are by the author, the findings come from Walter Trump's computing.
Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Friday, 5 October 2012 (republished on Monday, 6 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
Monday, 15 October 2012 (republished on Monday, 6 April 2020)
Although the text and illustrations are by the author, the findings come from Walter Trump's computing.
Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 17 October 2012 (republished on Monday, 6 April 2020)
Although the text and illustrations are by the author, the findings come from Walter Trump's computing.
Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 12 December 2012
The interlocked torus image is part of a trademark application for "Magic Torus" made at l'Institut National de la Propriété National (I.N.P.I.) of Paris on the 10th August 2012. This was given the national number 12 3 940 264 and was published and registered in the Bulletin Officiel de la Propriété Industrielle (B.0.P.I.) n° 12/36 Volume 1 on the 7th September 2012. This image is consequently excluded from the CC BY-NC-SA 4.0 licence.
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 13 January 2013 (republished on Monday, 6 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
Saturday, 23 March 2013 (republished on Monday, 6 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
Monday, 25 March 2013
The fourth-order partially pandiagonal tori type T4.04 were first identified thanks to Walter Trump's computing.
Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Tuesday, 26 March 2013
There are no exceptions to the licence CC BY-NC-SA 4.0.
Thursday, 28 March 2013
There are no exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 17 April 2013
The images drawn by the author that show examples of semi-magic tori with sub-magic 3x3 squares and sub-magic 2x2 diamonds are illustrations of Dwane Campbell's findings.
Dwane Campbell agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Tuesday, 26 December 2013 (republished on Monday, 6 April 2020)
The Cuboctahedron image file Kuboctaëder.png, by the author Svdmolen, published by Wikimedia Commons, is in the public domain.
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Tuesday, 18 March 2014
There are no exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 19 March 2014
There are no exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 26 March 2014 (republished on Monday, 6 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 7 December 2014
The magic square close-up detail of Albrecht Dürer's "first state" Melencolia I, cropped from the image hosted by Google's Cultural Institute and by the National Gallery of Victoria, Melbourne, is in the public domain: See the download conditions of the National Gallery of Victoria, Melbourne.
The revised version of Albrecht Dürer's Melencolia I magic square, illustrated by the image file Albrecht Dürer - Melencolia I (detail).jpg, and published by Wikipedia Commons, is in the public domain.
The commemorative magic square with 20 14 written in the centre cells of its bottom row, is the author's illustration of a square created by Miguel Angel Amela. The corresponding water retention diagram of the same square is Craig Knecht's creation.
The magic square that celebrates the 544th anniversary of Albrecht Dürer is Miguel Angel Amela's creation.
The two semi-pandiagonal squares that are tributes to Dûrer, 500 years after Melencolia I, are created by Lorenzo D. Sisican Jr.
Miguel Angel Amela, Craig Knecht, and Lorenzo D. Sisican Jr. agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 27 March 2016 (republished on Monday, 6 April 2020)
The Gluing a Torus video, by Geometric Animations of the University of Hannover, and hosted by YouTube, is subject to the Terms of Service (ToS) YouTube API.
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Monday, 29 August 2016 (republished on Monday, 6 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
The seven traditional Agrippa magic squares (figures 8, 10, 12, 13, 24, 25, and 28) are in the public domain.
There are no other exceptions to the licence CC BY-NC-SA 4.0.
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 30 November 2016 (republished on Tuesday, 7 April 2020)
The order-9 pandiagonal and associative magic square discussed in the paper is Miguel Angel Amela's creation.
Miguel Angel Amela agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Friday, 13 January 2017 (republished on Tuesday, 7 April 2020)
The puzzle area square illustrated by WilliamWalk4.png was created by Lee Sallows.
Lee Sallows agrees to the terms of the CC BY-NC-SA 4.0 licence.*
Except for the area magic schema by Walter Trump, all of the other jpeg files were created by the author. As noted in the text, several of these illustrate squares that were found by Walter Trump. The algorithm and integer area square pdf files were created by Walter Trump.
Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
Since the 12th June 2017, the first linear area magic square has been released under a CC BY-SA 4.0 licence in Wikimedia Commons.
The third order area magic squares equations pdf file was created by Francis Gaspalou.
Francis Gaspalou agrees to the terms of the CC BY-NC-SA 4.0 licence.*
The palprime and prime area magic squares dated 3rd and 4th February 2017 were both found and illustrated by Jan van Delden.
Jan van Delden agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 25 January 2017 (republished on Tuesday, 7 April 2020)
The parameterisation of the 6x6 square was created by Hans-Bernhard Meyer after the square had been constructed by the author of this blog.
The 6x6 L-AMS dated 26th and 29th January 2017 were both found by Hans-Bernhard Meyer.
Hans-Bernhard Meyer agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 8 February 2017 (republished on Tuesday, 7 April 2020)
Although all of the graphics are by the author of this blog, several of the illustrated area magic squares were found by Walter Trump and Hans-Bernhard Meyer (as indicated in the blue texts below each figure).
Both Hans-Bernhard Meyer and Walter Trump agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Thursday, 9 March 2017 (republished on Tuesday, 7 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
Monday, 11 September 2017 (republished on Tuesday, 7 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 21 January 2018 (republished on Tuesday, 7 April 2020)
The MMT6 with 8 magic diagonals, and the Pandiagonal Bimagic MMT8 are from Francis Gaspalou's collection. The Partially Pandiagonal Bimagic MMT8 is from Walter Trump's collection.
Both Francis Gaspalou and Walter Trump agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 8 August 2018 (republished on Tuesday, 7 April 2020)
As stated, the article refers to and reproduces the contents of a paper that was initially published by Miguel Angel Amela on the 8th April 2018. Calculated by Miguel Angel Amela, the illustrations of the Magic Tetragonal Octahedra and the table of polar sums have only been redrawn for an improved resolution on the web. Miguel Angel Amela agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 26 August 2018 (republished on Tuesday, 7 April 2020)
Alan Grogono has contributed an improved version of the proof that a tetrahedron is isosceles when all of its faces have equal perimeters. Alan Grogono agrees to the terms of the CC BY-NC-SA 4.0 licence.*
The Integer Heronian Magic Triangular Pyramid was jointly discovered by Walter Trump and William Walkington.
Walter Trump has additionally contributed the list of all Heronian non isosceles triangles, together with a simplification of the proof that the totals of the edge lengths of each of the 3 base vertices of MTP are always equal, and the proof of another interesting property of MTP (links in the acknowledgements). Walter Trump has also largely participated in the definition of the schema for the edge length notation definitions that is detailed in the observations. Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 6 February 2019 (republished on Tuesday, 7 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
Thursday, 25 April 2019 (republished on Tuesday, 7 April 2020)
The T6 8md(a) with 8 magic diagonals, and the Pandiagonal Bimagic T8 2bd(a) are from Francis Gaspalou's collection. The Partially Pandiagonal Bimagic T8 4bd(a) is from Walter Trump's collection.
Both Francis Gaspalou and Walter Trump agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Thursday, 20 June 2019 (republished on Tuesday, 7 April 2020, and latest development published on Monday, 3rd July 2023)
The "Dürer" magic square is in the public domain.
Paul Michelet devised the method of using 2 x 2 squares to create his eighth-order pan-zigzag magic square. In the Latest Development section, Peter Loly calculated the percentage of compression factors C. Both Paul Michelet and Peter Loly agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Tuesday, 13 August 2019 (republished on Tuesday, 7 April 2020)
The two appended pdf files of the papers on magic and semi-magic squares with "rectangular pandiagonality" are by Miguel Angel Amela.
Miguel Angel Amela agrees to the terms of the CC BY-NC-SA 4.0 licence.*
Although all of the magic line graphs that are illustrated are original, many are based on magic squares or magic tori constructed by others. Please refer to the Acknowledgements section of this post to find the names of the authors of the magic squares that are used for the magic lines graphs. Some of the magic squares are in the public domain, and others are by authors of today who agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Monday, 2 September 2019 (republished on Tuesday, 7 April 2020)
The Scroll of Loh diagram drawn by Ts'ai Yüang-Ting is in the public domain.
Although all of the even and odd number patterns that are illustrated are original, some are based on semi-magic squares that were found by Walter Trump (as indicated in the text). Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Saturday, 14 September 2019 (republished on Tuesday, 7 April 2020)
Although the 880 Frénicle Magic Squares and the 12 Dudeney pattern types of order-4 are in the public domain, all of the other findings are original. The partially pandiagonal tori type T4.04 were identified thanks to Walter Trump's computing skills. Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 1 March 2020 (republished on Tuesday, 7 April 2020)
The illustrations of this post are transformations of an 18-sided polygonal tetrad with bilateral symmetry that was found independently by Robert Ammann, Greg Frederickson and Jean L. Loyer in 1977.
All content is covered by the CC BY-NC-SA 4.0 licence, except for the simple tetrad jigsaw puzzle piece with 3 tabs and 3 blanks (Tetrad Jigsaw Puzzle n°1) which is released under a Creative Commons Attribution-ShareAlike 3.0 Licence (CC BY-SA 3.0).
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Thursday, 30 April 2020
The design of the "in memoriam" plaque is an adaptation of an idea proposed by Miguel Angel Amela. Miguel Angel Amela agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 2 August 2020
The initial idea of a bimagic queen's tour came from Joachim Brügge. Walter Trump has given the answer to the "open question" formulated in my conclusion notes and has submitted the corresponding PDF file. Walter Trump has also found another 105 bimagic queen's tours which are illustrated in our co-authored paper “106 Bimagic Queen’s Tours on an 8x8 Board.” Both Joachim Brügge and Walter Trump agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Monday, 20 September 2021
Saturday, 10 September 2016 (republished on Tuesday, 7 April 2020)
Concerning the file MTCVS 160910.pdf, the fourth-order partially pandiagonal tori type T4.04 (represented by the torus T4.098 in section 4.4.1) and the semi-magic tori type T4.06 (represented by the torus T4.06.0A in section 4.6.1) were first identified thanks to Walter Trump's computing skills. The semi-magic torus T4.06.0A (figure 434) has been drawn by the author using Walter Trump's findings. Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.The seven traditional Agrippa magic squares (figures 8, 10, 12, 13, 24, 25, and 28) are in the public domain.
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Tuesday, 25 October 2016 (republished on Tuesday, 7 April 2020)
After consulting Peter D. Loly, there are no restrictions on the reuse of the dko9a magic square, as long as the original authors and publications are correctly cited.There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 30 November 2016 (republished on Tuesday, 7 April 2020)
The order-9 pandiagonal and associative magic square discussed in the paper is Miguel Angel Amela's creation.
Miguel Angel Amela agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Friday, 13 January 2017 (republished on Tuesday, 7 April 2020)
The puzzle area square illustrated by WilliamWalk4.png was created by Lee Sallows.
Lee Sallows agrees to the terms of the CC BY-NC-SA 4.0 licence.*
Except for the area magic schema by Walter Trump, all of the other jpeg files were created by the author. As noted in the text, several of these illustrate squares that were found by Walter Trump. The algorithm and integer area square pdf files were created by Walter Trump.
Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
Since the 12th June 2017, the first linear area magic square has been released under a CC BY-SA 4.0 licence in Wikimedia Commons.
The third order area magic squares equations pdf file was created by Francis Gaspalou.
Francis Gaspalou agrees to the terms of the CC BY-NC-SA 4.0 licence.*
The palprime and prime area magic squares dated 3rd and 4th February 2017 were both found and illustrated by Jan van Delden.
Jan van Delden agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 25 January 2017 (republished on Tuesday, 7 April 2020)
The parameterisation of the 6x6 square was created by Hans-Bernhard Meyer after the square had been constructed by the author of this blog.
The 6x6 L-AMS dated 26th and 29th January 2017 were both found by Hans-Bernhard Meyer.
Hans-Bernhard Meyer agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 8 February 2017 (republished on Tuesday, 7 April 2020)
Although all of the graphics are by the author of this blog, several of the illustrated area magic squares were found by Walter Trump and Hans-Bernhard Meyer (as indicated in the blue texts below each figure).
Both Hans-Bernhard Meyer and Walter Trump agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Thursday, 9 March 2017 (republished on Tuesday, 7 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
Monday, 11 September 2017 (republished on Tuesday, 7 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 21 January 2018 (republished on Tuesday, 7 April 2020)
The MMT6 with 8 magic diagonals, and the Pandiagonal Bimagic MMT8 are from Francis Gaspalou's collection. The Partially Pandiagonal Bimagic MMT8 is from Walter Trump's collection.
Both Francis Gaspalou and Walter Trump agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 8 August 2018 (republished on Tuesday, 7 April 2020)
As stated, the article refers to and reproduces the contents of a paper that was initially published by Miguel Angel Amela on the 8th April 2018. Calculated by Miguel Angel Amela, the illustrations of the Magic Tetragonal Octahedra and the table of polar sums have only been redrawn for an improved resolution on the web. Miguel Angel Amela agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 26 August 2018 (republished on Tuesday, 7 April 2020)
Alan Grogono has contributed an improved version of the proof that a tetrahedron is isosceles when all of its faces have equal perimeters. Alan Grogono agrees to the terms of the CC BY-NC-SA 4.0 licence.*
The Integer Heronian Magic Triangular Pyramid was jointly discovered by Walter Trump and William Walkington.
Walter Trump has additionally contributed the list of all Heronian non isosceles triangles, together with a simplification of the proof that the totals of the edge lengths of each of the 3 base vertices of MTP are always equal, and the proof of another interesting property of MTP (links in the acknowledgements). Walter Trump has also largely participated in the definition of the schema for the edge length notation definitions that is detailed in the observations. Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Wednesday, 6 February 2019 (republished on Tuesday, 7 April 2020)
There are no exceptions to the licence CC BY-NC-SA 4.0.
Thursday, 25 April 2019 (republished on Tuesday, 7 April 2020)
The T6 8md(a) with 8 magic diagonals, and the Pandiagonal Bimagic T8 2bd(a) are from Francis Gaspalou's collection. The Partially Pandiagonal Bimagic T8 4bd(a) is from Walter Trump's collection.
Both Francis Gaspalou and Walter Trump agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Thursday, 20 June 2019 (republished on Tuesday, 7 April 2020, and latest development published on Monday, 3rd July 2023)
The "Dürer" magic square is in the public domain.
Paul Michelet devised the method of using 2 x 2 squares to create his eighth-order pan-zigzag magic square. In the Latest Development section, Peter Loly calculated the percentage of compression factors C. Both Paul Michelet and Peter Loly agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Tuesday, 13 August 2019 (republished on Tuesday, 7 April 2020)
The two appended pdf files of the papers on magic and semi-magic squares with "rectangular pandiagonality" are by Miguel Angel Amela.
Miguel Angel Amela agrees to the terms of the CC BY-NC-SA 4.0 licence.*
Although all of the magic line graphs that are illustrated are original, many are based on magic squares or magic tori constructed by others. Please refer to the Acknowledgements section of this post to find the names of the authors of the magic squares that are used for the magic lines graphs. Some of the magic squares are in the public domain, and others are by authors of today who agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Monday, 2 September 2019 (republished on Tuesday, 7 April 2020)
The Scroll of Loh diagram drawn by Ts'ai Yüang-Ting is in the public domain.
Although all of the even and odd number patterns that are illustrated are original, some are based on semi-magic squares that were found by Walter Trump (as indicated in the text). Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Saturday, 14 September 2019 (republished on Tuesday, 7 April 2020)
Although the 880 Frénicle Magic Squares and the 12 Dudeney pattern types of order-4 are in the public domain, all of the other findings are original. The partially pandiagonal tori type T4.04 were identified thanks to Walter Trump's computing skills. Walter Trump agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 1 March 2020 (republished on Tuesday, 7 April 2020)
The illustrations of this post are transformations of an 18-sided polygonal tetrad with bilateral symmetry that was found independently by Robert Ammann, Greg Frederickson and Jean L. Loyer in 1977.
All content is covered by the CC BY-NC-SA 4.0 licence, except for the simple tetrad jigsaw puzzle piece with 3 tabs and 3 blanks (Tetrad Jigsaw Puzzle n°1) which is released under a Creative Commons Attribution-ShareAlike 3.0 Licence (CC BY-SA 3.0).
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Thursday, 30 April 2020
The design of the "in memoriam" plaque is an adaptation of an idea proposed by Miguel Angel Amela. Miguel Angel Amela agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Sunday, 2 August 2020
The initial idea of a bimagic queen's tour came from Joachim Brügge. Walter Trump has given the answer to the "open question" formulated in my conclusion notes and has submitted the corresponding PDF file. Walter Trump has also found another 105 bimagic queen's tours which are illustrated in our co-authored paper “106 Bimagic Queen’s Tours on an 8x8 Board.” Both Joachim Brügge and Walter Trump agree to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Monday, 20 September 2021
The main illustration, (including the cropped bat-like creature close-up detail of Albrecht Dürer's "first state" Melencolia I), are from the image hosted by Google's Cultural Institute and by the National Gallery of Victoria, Melbourne, in the public domain: See the download conditions of the National Gallery of Victoria, Melbourne.
The writings of Bonnie James (see Acknowledgement) have inspired "The Historical Context" section, but her text has been rewritten. The pdf file of the paper "A Hidden Love Story" is by Miguel Angel Amela (see Latest Development). Miguel Angel Amela agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.The writings of Bonnie James (see Acknowledgement) have inspired "The Historical Context" section, but her text has been rewritten. The pdf file of the paper "A Hidden Love Story" is by Miguel Angel Amela (see Latest Development). Miguel Angel Amela agrees to the terms of the CC BY-NC-SA 4.0 licence.*
Tuesday, 07 June 2022
Polyomino Area Magic Tori
In the public domain we find the Lo Shu magic square of order-3 by Anonymus, and a pandiagonal magic squares of order-4, once again by Anonymus (but first listed by Bernard Frénicle de Bessy).
In the public domain we find the Lo Shu magic square of order-3 by Anonymus, and a pandiagonal magic squares of order-4, once again by Anonymus (but first listed by Bernard Frénicle de Bessy).
The order-6 magic square is by Harry White. Harry White agrees to the terms of the CC BY-NC-SA 4.0 licence.*
There are no other exceptions to the licence CC BY-NC-SA 4.0.
Thursday, 15 June 2023
440 Torus-Opposite Pairs of the 880 Frénicle Magic Squares of Order-4
The image "Tesseract Torus" by Tilman Piesk is issued under a creative commons CC-BY-4.0 licence. For the details, please refer to https://commons.wikimedia.org/w/index.php?curid=101975795.
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Wednesday, 24 April 2024
Plus or Minus Groups of Magic Tori of Order 4
All of the content (including both of the enclosed pdf papers) is original, and covered by the CC BY-NC-SA 4.0 licence. The second paper (Examples of Partial Groups of Magic Tori of Orders N > 4) contains magic squares by others - as indicated in the acknowledgements: The authors Francis Gaspalou, Walter Trump and Miguel Angel Amela have agreed to the terms of the CC BY-NC-SA 4.0 licence.*
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Saturday, 11 May 2024
137 Plus-or-Minus Groups of Magic Tori of Order 4, Linked by Plus-and-Minus Operations
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Wednesday, 25 September 2024
Magic Box Cubes, Rubik's Cubes and Twisty Puzzles
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