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(English version / French version / German version)

1. "Site's news" map
2. "Multimagic squares" map
3. "Cubes and hypercubes" map
4. "To know more" map
5. "Contact" map
•  Downloadable files in red!

Site's news

• April 2006
• January 2006
• October 2005
• May 2005
• March 2005
• November 2004
• February 2004
• November 2003
• September 2003
• July 2003
• March 2003
• November 2002
• July 2002

Multimagic squares

• What's a multimagic square?

• Records table
• Bimagic square of order 8
• Solution of the first bimagic square, 8th-order, of Pfeffermann
• Trimagic square of order 128
• Tarry.zip 128th-order trimagic square of Gaston Tarry
• Trimagic square of order 64
• Cazalas.zip 64th-order trimagic square of General Cazalas
• Trimagic square of order 32
• Portrait of William Benson in 1966
• Biography of William Benson by the Carlisle Sentinel
• Portrait of William Benson in 1960
• Biography of William Benson by the president of the Dickinson College
• Portrait of William Benson in 1961
• Benson.zip Image of the 32nd-order trimagic square of William Benson
• BensonXLS.zip 32nd-order trimagic square of William Benson
• Other sample of a 32nd-order trimagic square
• Trimagic square of order 12
• Article published in the German newspaper Schwabacher Tagblatt
• Article published in the German newspaper Nürnberger Nachrichten
• English translation of the Nürnberger Nachrichten article
• Story of the smallest trimagic square, text written by Walter Trump
• Tetramagic square of order 512
• Tetra.zip 512th-order tetramagic square of Christian Boyer and André Viricel
• Eulogy to André Viricel, written (in French) by Yves Roussel
• Portrait of André Viricel in 1978
• Tetramagic square of order 256
• Tetra256.zip 256th-order tetramagic square of Christian Boyer
• Tetramagic square of order 243
• Tetra243.zip 243th-order tetramagic square of Pan Fengchu
• Pentamagic square of order 1024
• Penta.zip 1024th-order pentamagic square of Christian Boyer and André Viricel
• Pentamagic square of order 729
• Penta729.zip 729th-order pentamagic square of Li Wen
• Hexamagic square of order 4096
• Hexa4096.zip 4096th-order hexamagic square of Pan Fengchu
• Highly multimagic squares
• Wroblewski-V1.zip "Magic squares J(p)" paper by Jaroslaw Wroblewski
• Biography of Charles Devimeux (in French)
• Portrait of Charles Devimeux in 1990
• Portrait of Charles Devimeux young

• Magic squares of squares
• Beginning of the article "Some notes on the magic squares of squares problem" published in The Mathematical Intelligencer
• Supplement to the article
• Supplement in HTML format
• Supplement.pdf Supplement in PDF format
• Open problems from the article
• Lecture presenting the main points of the article
• Photo of Christian Boyer during the lecture
• LectureAmiens.pps English version in PowerPoint 2003 format
• ConferenceAmiens.pps Original French version in PowerPoint 2003 format
• List of figures of the article and its supplement
• References from the article
• Search.pdf A search for 3x3 magic squares having more than six square integers among their nine distinct integers, by Christian Boyer
• Bremner1.pdf On squares of squares, by Andrew Bremner, from Acta Arithmetica
• Bremner2.pdf On squares of squares II, by Andrew Bremner, from Acta Arithmetica
• Buell.pdf A search for a magic hourglass, by Duncan Buell
• Sallows.pdf The lost theorem, by Lee Sallows, from The Mathematical Intelligencer
• Thanks to
   
• First known 4x4 to 7x7 magic squares of squares
• 4x4 magic squares of squares
• 5x5 magic squares of squares
• 6x6 magic squares of squares
• 7x7 magic squares of squares

• Magic squares of cubes
• Open problems from the article
• Squares (CB10) (CB11) (CB12) from the article
• References from the article

• Smallest bimagic square
• 3rd or 4th-order bimagic square?
• Demonstration (in French) of the impossibility of 3rd and 4th-orders bimagic squares by Edouard Lucas
• Demonstration of the impossibility of non-normal 4th-order bimagic squares by Luke Pebody
• 5th-order bimagic square?
• 6th-order bimagic square?
• 7th-order bimagic square?
• nearlybi7.xls The 86 nearly bimagic squares of order 7 computed by Pan Fengchu
• D.N. Lehmer and the smallest bimagic square problem
   
• Smallest bimagic square using distinct integers
• 3rd or 4th-order bimagic square using distinc integers?
• 5th-order bimagic square using distinc integers?
• 6th-order bimagic square using distinc integers?
• First known 6th-order bimagic square, by Jaroslaw Wroblewski
• Photo of Jaroslaw Wroblewski in 2002
• WroblewskiBi6.xls The four 6th-order bimagic squares of Jaroslaw Wroblewski
• 7th-order bimagic square using distinc integers?

• Smallest trimagic square
• 7th-order trimagic square? Or smaller order?
• 8th-order trimagic square?
• 9th-order trimagic square?
• 10th-order trimagic square?
• 11th-order trimagic square?
• TrumpTri.doc Demonstration of the non-existence of 11th-order trimagic squares by Walter Trump

• Smallest pentamagic square

• 9th-order bimagic squares
• Solution of the first 9th-order bimagic square of Pfeffermann

• 10th and 11th-order bimagic squares
• Story of the first 10th-order bimagic square, text written by Fredrik Jansson

• 12th to 16th-order bimagic and trimagic squares
• 12th-order bimagic and trimagic squares?
• 13th-order bimagic and trimagic squares?
• Gueron13.xls 13th-order bimagic square of Jacques Guéron, with construction method (in French)
• 14th-order bimagic and trimagic squares?
• 15th-order bimagic and trimagic squares?
• 16th-order bimagic and trimagic squares?
• 16th-order non-normal trimagic squares
• Collison.xls 16th-order non-normal trimagic square of David Collison
• Gueron.xls 16th-order non-normal trimagic square of Jacques Guéron
• MutianQinwu.xls 16th-order normal trimagic square of Chen Mutian and Chen Qinwu
• Tri16Story.doc Story of the 16th-order normal trimagic square of Chen Mutian and Chen Qinwu

• 17th to 64th-order bimagic and trimagic squares
• Jacques Guéron's bimagic squares: 17th and 19th-orders
• Gueron17.xls 17th-order bimagic square of Jacques Guéron
• Gueron19.xls 19th-order bimagic square of Jacques Guéron, with construction method (in French)
• Biography of Jacques Guéron (in French)
• Photo of Jacques Guéron in 2006
• Su Maoting's bimagic squares: 20th, 28th and 36th orders (and 24th-order)
• Maoting20_36.xls Bimagic squares of Su Maoting
• Chen Qinwu's bimagic square: 24th-order
• Qinwu24.xls 24th-order bimagic square of Chen Qinwu
• Li Wen's bimagic square: 35th-order
  Pan Fengchu's bimagic squares: 40th, 45th and 48th-orders (and 35th-order)
• LiWen35.xls 35th-order bimagic square of Li Wen
• Fengchu35_48.xls Bimagic squares of Pan Fengchu

• Pandiagonal bimagic squares
• Maoting18.xls 18th-order non-normal pandiagonal bimagic square of Su Maoting
• Maoting32.xls 32th-order normal pandiagonal bimagic square of Su Maoting

• Construction methods

• Multimagic formulas

• Multimagic series for squares
• Table
• Bima6.xls Bimagic series for squares of order 6
• Bima7.zip Bimagic series for squares of order 7
• Bima8.zip Bimagic series for squares of order 8
• Trima8.zip Trimagic series for squares of order 8
• Trima9.xls Trimagic series for squares of order 9
• Trima11.zip Trimagic series for squares of order 11
• Tetra12.xls Tetramagic series for squares of order 12
• Tetra13.xls Tetramagic series for squares of order 13
• Estimations, orders 13 to 20, by Walter Trump
• Proof on non-existence of tetramagic series of order 15, by Robert Gerbicz

• Unsolved multimagic problems
• Unsolved multiplicative magic squares and cubes problems
• Unsolved magic squares of squares problems

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• Sudokus and bimagic squares
• About Gaston Tarry

• Smallest possible pandiagonal Sudoku
• PanSudoku25x25.pdf A 25x25 example, the smallest possible pandiagonal Sudoku, letter published in Mathematics Today

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• The smallest possible multiplicative magic squares
• What are the smallest possible multiplicative magic squares?
• What is a multiplicative magic square?
• Smallest 3rd-order multiplicative magic squares
• List of smallest magic products of 3rd-order multiplicative magic squares
• Smallest 4th-order multiplicative magic squares
• List of smallest magic products of 4th-order multiplicative magic squares
• Smallest 5th-order multiplicative magic squares
• List of smallest magic products of 5th-order multiplicative magic squares

• Smallest multiplicative magic squares, 6th and 7th-order
• Smallest 6th-order multiplicative magic squares
• List of smallest magic products of 6th-order multiplicative magic squares
• Smallest 7th-order multiplicative magic squares
• List of smallest magic products of 7th-order multiplicative magic squares

• Additive-multiplicative magic squares, 8th and 9th-order
• Smallest multiplicative magic squares, 8th and 9th-order

• Smallest multiplicative magic squares, 10th-order and above
• MMS10-17.xls the 16 best known examples of multiplicative magic squares from order 10 to order 17

Cubes and hypercubes

• Perfect magic cubes
• What is the smallest possible perfect cube?
• Image of the first perfect magic cube of order 5, of Walter Trump and Christian Boyer
• Perfect cubes of order 3 and 4 are impossible
• The four magic cubes of order 3
• Image of Fermat's cube of order 4
• Fermat's letter (in old French) to Mersenne including his cube of order 4
• Fermat.xls Cube of order 4, of Pierre de Fermat
• Demonstration that the center of a perfect magic cube of order 5 is 63, by Richard Schroeppel
• Biography of Richard Schroeppel
• Portrait of Richard Schroeppel in 2004
• perfect-5-cube.xls First perfect magic cube of order 5, of Walter Trump and Christian Boyer
• Articles published in the world after the discovery of the first perfect magic cube of order 5
• Image of the first perfect magic cube of order 5 published in Science & Vie
• Image of the first perfect magic cube of order 5 used on the cover of Mathematics for Elementary Teachers, 7th edition
• TST070104.pdf Federico Peiretti's article published in La Stampa
• LR373_CubesMagiques.pdf Christian Boyer's article published in La Recherche
• First perfect cube of order 6
• magic-order-6-cube.xls First perfect magic cube of order 6, of Walter Trump
• What is a "perfect" magic cube?
• What are the first published perfect cubes?
• First perfect cube of order 7
• Frost.xls Perfect cube of order 7 of Rev. Andrew H. Frost
• Biography of Rev. Andrew H. Frost
• Portrait of Rev. Andrew H. Frost in 1870
• Original text about the perfect cube of order 7 by Harry Langman
• Langman.xls Perfect cube of order 7 of Harry Langman
• First perfect cube of order 8
• Original text about the perfect cube of order 8 by Gustavus Frankenstein
• Frankenstein.xls Perfect cube of order 8 of Gustavus Frankenstein
• First perfect cube of order 9
• Cube_9-Planck.zip Perfect cube of order 9 of Charles Planck
• Biography of Charles Planck
• First perfect cube of order 10
• PerfectCube10.xls Perfect cube of order 10 of Li Wen
• First perfect cube of order 11
• First perfect cube of order 12
• Cube_12-BJ.zip Perfect cube of order 12 of William Benson

• Pandiagonal perfect magic cubes
• Biography (in French) of Gabriel Arnoux
• Other biography (in French) of Gabriel Arnoux, by Charles-Ange Laisant
• Cube_17-Arnoux.zip Pandiagonal perfect magic cube of order 17, of Gabriel Arnoux
• Biography of Frederick A. P. Barnard
• Other biography of Frederick A. P. Barnard, from the Proceedings of the AAAS
• Portraits of Frederick A.P. Barnard: 1, 2, 3, 4
• Cube_8-Barnard.zip Pandiagonal perfect magic cube of order 8, of Frederick A. P. Barnard
• Cube_11-Barnard-1.zip Pandiagonal perfect magic cube #1 of order 11, of Frederick A. P. Barnard
• Cube_11-Barnard-2.zip Pandiagonal perfect magic cube #2 of order 11, of Frederick A. P. Barnard

• Multimagic cubes
• bicube25.pdf Document by Holger Danielsson about the bimagic cube of order 25, of John-R. Hendricks
• Biography of John-R. Hendricks from Magic Square Lexicon: Illustrated
• Biography of John-R. Hendricks by the Journal of Recreational Mathematics
• BiCube16.xls Bimagic cube of order 16, of Christian Boyer
• BiCube25.zip Bimagic cube of order 25, of John-R. Hendricks
• BiCube27.zip Bimagic cube of order 27, of Christian Boyer
• BiCube32.zip Perfect bimagic cube of order 32, of Christian Boyer
• TriCube64.zip Trimagic cube of order 64, of Christian Boyer

• Smallest bimagic cube
• 3rd-order bimagic cube?
• 4th-order bimagic cube?
• 5th-order bimagic cube?
• 6th-order bimagic cube?
• Proof of the non-existence of 6th-order bimagic cube by Walter Trump
• 7th-order bimagic cube?
• 8th-order bimagic cube?

• Multimagic series for cubes
• Table
• Bima5C.xls Bimagic series for cubes of order 5
• Bima6C.zip Bimagic series for cubes of order 6
• Trima7C.xls Trimagic series for cubes of order 7
• Trima8C.zip Trimagic series for cubes of order 8

• Multimagic hypercubes
• BiHyper32d+.zip Bimagic hypercube (tesseract) of order 32, of Christian Boyer
• BiHyper32t+.zip Other bimagic hypercube (tesseract) of order 32, of Christian Boyer

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• Multiplicative magic cubes
• What are multiplicative magic cubes?
• Table
• MMC3-7.xls Best known multiplicative magic cubes, orders 3 to 7, of Christian Boyer
• MMC8-11.xls Best known multiplicative magic cubes, orders 8 to 11, of Christian Boyer
• 3rd-order multiplicative magic cubes
• 4th-order multiplicative magic cubes
• Image of 4th-order cube, P=6486480, Max nb=546
• Image of 4th-order cube, P=8648640, Max nb=416
• 5th-order multiplicative magic cubes
• Image of the best known 5th-order cube
• 6th-order multiplicative magic cubes
• 7th-order multiplicative magic cubes
• 8th to 11th-order multiplicative magic cubes

• Pandiagonal perfect multiplicative magic cubes
• What are pandiagonal perfect multiplicative magic cubes?
• What is the smallest possible magic product P?
• Image of the best known 8th-order pandiagonal perfect cube
• What is the smallest possible maximum number?

 To know more

• Site map
• Bibliography
• Links

Contact

• Christian Boyer
• Many thanks to Harvey Heinz for help with the English version
• Biography of Harvey Heinz from Magic Square Lexicon: Illustrated
• Many thanks to Holger Danielsson, Walter Trump and Peter Bartsch for their translation in German

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