Magic squares of squares
See also the 4x4 to 7x7 Magic squares of squares page
See also the Latest research on the 3x3 Magic squares of squares page
See also the Magic squares of cubes page

• Beginning of the article “Some notes on the magic squares of squares problem” published in , Springer, New-York
  
• Supplement to the article, and first 6x6 and 7x7 magic squares of squares constructed after the article and after the supplement
• Open problems from the article (your solutions or partial results are welcome, I offer a €100 prize + a bottle of champagne!)
• Lecture presenting the main points of the article (download it to see an easy-to-understand summary of the article, its supplement, and its open problems)
• List of figures of the article and its supplement
• References from the article, available from this site or from the Internet
• and Thanks to...

Beginning of the article
“Some notes on the magic squares of squares problem”
by Christian Boyer, and published in The Mathematical Intelligencer (Vol 27, N 2, Spring 2005, pages 52-64)

“Permettez-moi, Monsieur, que je vous parle encore d'un problème
qui me paraît fort curieux et digne de toute attention”
Leonhard Euler, 1770, sending his 4×4 magic square of squares to Joseph Lagrange

Can a 3x3 magic square be constructed with nine distinct square numbers? This short question asked by Martin LaBar^[38] in 1984 became famous when Martin Gardner republished it in 1996^{[25] [26]} and offered $100 to the first person to construct such a square. Two years later, Gardner wrote^[28]:

So far no one has come forward with a “square of squares” – but no one has proved its impossibility either. If it exists, its numbers would be huge, perhaps beyond the reach of today’s fastest computers.

Today, this problem is not yet solved. Several other articles in various magazines have been published^{[10] [11] [12] [27] [29] [30] [49] [51] [52]}. John P. Robertson^[51] showed that the problem is equivalent to other mathematical problems on arithmetic progressions, on Pythagorean right triangles, on congruent numbers and elliptic curves y² = x³ – n²x. Lee Sallows^[52] discussed the subject in The Mathematical Intelligencer presenting the nice (LS1) square, a near-solution with only one bad sum.

LS1. Three rows, three columns and one diagonal have the same magic sum S2=21609.
But unfortunately the other diagonal has a different magic sum S2=38307:

In the present article I add both old forgotten European works of XVIIIth/XIXth centuries that I am proud to revive (and to numerically complete for the first time^[9]) after years of oblivion, and very recent developments of the very last months on the problem - and more generally on multimagic squares, cubes and hypercubes. And I have highlighted 10 open subjects. An open invitation to number-lovers!

The magic square of squares problem is an important part of unsolved problem D15 of Richard K. Guy’s Unsolved Problems in Number Theory book^[30], third edition, 2004, summarizing the main published articles on this subject since 1984. I have organized my exposition around nine quotations from Guy’s text.

…For the continuation, read the complete article in The Mathematical Intelligencer
or read its summary in the presentation used for the lecture

Supplement to the article

A referenced^[9] (but non-published) supplement to the M.I. article is available from this site, including four new magic squares (CB15) through (CB18), a numerical analysis of Euler’s 4x4 and Lucas’s 3x3 squares of squares, and some results on the magic squares of prime squares problem. Two formats are available:

• See the supplement in HTML format
• Download the supplement in PDF format (31Kb, 4 pages)

Several 4x4 and 5x5 magic squares of squares are published in the M.I. article. The first 6x6 and 7x7 magic squares of squares were constructed unfortunately later, after the article and after the above supplement. They are available here:

• See the first 6x6 and 7x7 magic squares of squares

Open problems of the article

Each multimagic (bimagic, trimagic,…) square is a “magic square of squares” when its numbers are squared. But a “magic square of squares” is rarely a multimagic square because it is probably not magic when its numbers are not squared. This remark does not imply that magic squares of squares problems are easier than multimagic problems…

In the article, 10 open problems wait your answers, including my open problem 2: for its solution, I offer a €100 prize + a bottle of champagne!!!

• Read the 10 problems on the unresolved multimagic problems page.

If you arrive at solutions or partial results to any of these problems, send me a message. Your results will be added here.

Lecture presenting the main points of the article

If you want to see a summary of the article, this part is for you!

In a series organized by the University of Picardy, the ESIEE Amiens, the URISP-CNISF, and the ADCS in March-April 2005, several scientific lectures were presented in Amiens to mark the 100th anniversary of the death of Jules Verne.

Jules Verne (Nantes 1828 - Amiens 1905), Edouard Lucas (Amiens 1842 - Paris 1891)

During this event, I gave a lecture titled “Resort to computing on a problem of Euler and of Lucas”:

• Slides 1-3: Jules Verne and Edouard Lucas were contemporaries, and both men lived for many years in Amiens
• Slides 4-13: summary of previously published articles on the “magic squares of squares” problem
• Slides 14-27: summary of my own studies on the problem (M.I. article, search^[8], supplement^[9], Euler’s and Lucas’s works, ...)
• Slides 28-30: open problems and conclusion

C. Boyer during the lecture, slide 8.
Photo by Marc Lecoester, President of the URISP. Click on the image to enlarge it (JPG file, 1.2Mb).

You can download the presentation used for this lecture:

• Download the PowerPoint 2003 file in English version (30 slides, 2.3Mb)
• Download the PowerPoint 2003 file in original French version (30 slides, 2.3Mb) used during this lecture

If you do not have the PowerPoint 2003 product, there is a free viewer downloadable from the Microsoft site which will allow you to see the presentation. Go to Google or Yahoo!, and type “PowerPoint Viewer 2003”: you will immediately get the Microsoft link to download this tool. Do not use PowerPoint Viewer 97, the sequences will be incorrectly displayed.

Congratulations and thanks to Yves Roussel for organizing the series of lectures commemorating the Jules Verne centenary.

List of figures of the article and its supplement

Introduction

LS1 (and lecture slides 6 and 21). Example of a 3×3 square with seven magic sums by Lee Sallows

Part 1

EL1 (and lecture slide 20, and supplement). Edouard Lucas’s 3×3 semi-magic squares of squares family
EL2 (and lecture slide 20). The example of a 3×3 semi-magic square of squares by Edouard Lucas in 1876

LE1 (and lecture slide 25). The smallest 3×3 example published by Leonhard Euler in 1770
LE1cb (and lecture slide 21). The smallest semi-magic square of squares from LE1

Part 2

AB1 (and lecture slide 7). 3×3 magic square with seven square entries by Andrew Bremner

Part 4

MS1, MS2, MS3. Examples of 3×3 squares with seven magic sums by Michael Schweitzer
AB2. 3×3 semi-magic square of squares by Andrew Bremner
MS4. 3×3 example with a non-square magic sum by Michael Schweitzer

Part 5

AB3 (and lecture slide 12). 4×4 magic square of squares by Andrew Bremner
LE2 (and lecture slides 22 and 23). The first known magic square of squares, sent in 1770 by L. Euler to J. Lagrange:

LE3 (and lecture slides 24 and 26, and supplement). Euler’s 4×4 magic squares of squares family
CB1 (and lecture slides 16 and 27). The smallest magic square of squares of Euler’s 4×4 family, not found by Euler

 CB2 (and lecture slide 16, and supplement). Sub-family of 4×4 magic squares of squares. S2 = 85(k² + 29):

CB3. A small game : will you quickly locate the error in this member of Euler's family?
LE3cb. Transition from Euler’s 4×4 to Lucas’s 3×3 magic square of squares
CB4 (and lecture slide 17). The smallest 5×5 magic square of squares. S2 = 1375:

CB5. The second smallest 5×5 magic square of squares

Bimagic squares

CB6. A putative 3×3 semi-bimagic square
CB7. The smallest 4×4 semi-bimagic square without magic diagonal
CB8. The smallest 4×4 semi-bimagic square with one magic diagonal
CB9. The smallest 5×5 semi-bimagic square with two magic diagonals
GP1. A 6×6 semi-bimagic square with two magic diagonals, by G. Pfeffermann in 1894
RVI 1. A 6×6 semi-bimagic square family with two magic diagonals, by R. Venkatachalam Iyer in 1961

Part 6

CB10 (and lecture slide 19). A 4×4 magic square of cubes. S3 = 0
CB11 (and lecture slide 19). A 5×5 magic square of cubes. S3 = 0
CB12. The smallest 5×5 semi-magic square of cubes using positive integers

Part 7

WT1. The first 12×12 trimagic square, by Walter Trump in 2002

Part 8

PF1. The 4×4×4 nearly perfect magic cube, sent in 1640 by Pierre de Fermat to Mersenne
WT2CB13. The first 5×5×5 perfect magic cube, by Walter Trump and Christian Boyer in 2003
CB14. The 8192×8192×8192 perfect tetramagic cube compared to the cathedral Notre-Dame de Paris

Supplement

CB15. Another sub-family of 4×4 magic squares of squares
CB16. The smallest 4×4 semi-bimagic square of prime numbers
CB17 (and lecture slide 18). The smallest 4×4 magic square of squares of prime numbers
CB18 (and lecture slide 18). The smallest 5×5 magic square of squares of prime numbers

References from the article
available from this site or from the Internet

[4] Christian Boyer, Les premiers carrés tétra et pentamagiques, Pour La Science (the French edition of Scientific American), N°286 August 2001, 98-102
Download the PDF file (168Kb)
(see also www.multimagie.com/English/Tetra-penta.htm)

[5] Christian Boyer, Les cubes magiques, Pour La Science (the French edition of Scientific American), N°311 September 2003, 90-95
(see also www.multimagie.com/English/Cube.htm#PLS)

[6] Christian Boyer, Le plus petit cube magique parfait, La Recherche, N°373 March 2004, 48-50
Download the PDF file (955Kb)
(see also www.multimagie.com/English/Perfectcubes.htm)

[7] Christian Boyer, Multimagic squares, cubes and hypercubes web site,
www.multimagie.com/indexengl.htm

[8] Christian Boyer, A search for 3x3 magic squares having more than six square integers among their nine distinct integers, preprint, September 2004
Download the PDF file (60Kb)
(see also slide 15 of the lecture)

[9] Christian Boyer, Supplement to the “Some notes on the magic squares of squares problem” article, 2005
Download the PDF file (31Kb)
or www.multimagie.com/English/Supplement.htm
(see also slide 18 of the lecture)

[10] Andrew Bremner, On squares of squares, Acta Arithmetica, 88(1999) 289-297
Download the PDF file (99Kb)
(see also slide 7 of the lecture)

[11] Andrew Bremner, On squares of squares II, Acta Arithmetica, 99(2001) 289-308
Download the PDF file (172Kb)
(see also slides 11-12 of the lecture)

[12] Duncan A. Buell, A search for a magic hourglass, preprint, 1999
Download the PDF file (101Kb)
(see also slide 10 of the lecture)

[42] Edouard Lucas, Sur le carré de 3 et sur les carrés à deux degrés, Les Tablettes du Chercheur, March 1st 1891, p.7 (reprint in [44] and in
www.multimagie.com/Francais/Lucas.htm)

[49] Landon W. Rabern, Properties of magic squares of squares, Rose-Hulman Institute of Technology Undergraduate Math Journal, 4(2003), N.1
www.rose-hulman.edu/mathjournal/v4n1.php (now https://scholar.rose-hulman.edu/rhumj/vol4/iss1/3/)

[50] Carlos Rivera, www.primepuzzles.net
www.primepuzzles.net/puzzles/puzz_079.htm (Puzzle 79 « The Chebrakov’s Challenge »)
www.primepuzzles.net/puzzles/puzz_287.htm (Puzzle 287 « Multimagic prime squares »)
www.primepuzzles.net/puzzles/puzz_288.htm (Puzzle 288 « Magic square of (prime) squares »)
(on Puzzle 288, see also slide 17 of the lecture, and part 3 of the supplement)

[52] Lee Sallows, The lost theorem, The Mathematical Intelligencer, 19(1997), n°4, 51-54
Download the PDF file (76Kb)
(see also slides 6-7 of the lecture)

[54] Richard Schroeppel, The center cell of a magic 5³ is 63, (1976),
www.multimagie.com/English/Schroeppel63.htm

[57] Neil Sloane, Multimagic sequences A052457, A052458, A090037, A090653, A092312, ATT Research’s Online Encyclopaedia of Integer Sequences,
www.research.att.com/~njas/sequences/ (now http://oeis.org)
(see also www.multimagie.com/English/Series.htm)

[58] Paul Tannery and Charles Henry, Lettre XXXVIIIb bis, Fermat à Mersenne, Toulouse, 1 avril 1640, Œuvres de Fermat, Gauthier-Villars, Paris, 2(1894), 186-194
(partial reprint of the letter at www.multimagie.com/Francais/Fermat.htm)

[60] Walter Trump, Story of the smallest trimagic square, January 2003,
www.multimagie.com/English/Tri12Story.htm
(see also www.multimagie.com/English/Trimagic12.htm
and www.multimagie.com/English/Smallesttri.htm)

[62] Eric Weisstein, Magic figures, MathWorld,
http://mathworld.wolfram.com/topics/MagicFigures.html

[63] Eric Weisstein, Perfect magic cube, MathWorld,
http://mathworld.wolfram.com/PerfectMagicCube.html

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