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”
by Christian
Boyer, and published in The
Mathematical Intelligencer (Vol 27, N 2, Spring 2005, pages 5264)
“Permettezmoi,
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 y^{2} = x^{3} – n^{2}x. Lee Sallows^{[52]} discussed the subject in The Mathematical Intelligencer presenting the nice (LS1) square, a nearsolution with only one bad sum.
127² 
46² 
58² 
2² 
113² 
94² 
74² 
82² 
97² 
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 numberlovers!
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
A referenced^{[9]} (but nonpublished) 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:
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:
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…
Order 
Magic squares of squares 
Bimagic squares 
Normal bimagic squares 
3x3 
Who? Or proof of its impossibility? 
Impossible. E. Lucas (1891) 

4x4 
L. Euler (1770) 
Impossible. 
Impossible. 
5x5 
C. Boyer (2004) 
Who? Or proof of its impossibility? 
Impossible. 
6x6 
C. Boyer (2005) 
J. Wroblewski (2006) 

7x7 
C. Boyer (2005) 
L. Morgenstern (2006) 

8x8 
G. Pfeffermann
(8x8 in 1890, 9x9 in 1891). 
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!!!
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 URISPCNISF, and the ADCS in MarchApril 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”:
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:
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
Part 1
1² 
68² 
44² 
76² 
16² 
23² 
28² 
41² 
64² 
Part 2
373² 
289² 
565² 
360721 
425² 
23² 
205² 
527² 
222121 
Part 4
Part 5
68² 
29² 
41² 
37² 
17² 
31² 
79² 
32² 
59² 
28² 
23² 
61² 
11² 
77² 
8² 
49² 
48² 
23² 
6² 
19² 
21² 
26² 
33² 
32² 
1² 
36² 
13² 
42² 
22² 
27² 
44² 
9² 
(2k + 42)² 
(4k + 11)² 
(8k  18)² 
(k + 16)² 
(k  24)² 
(8k + 2)² 
(4k + 21)² 
(2k  38)² 
(4k  11)² 
(2k  42)² 
(k  16)² 
(8k + 18)² 
(8k  2)² 
(k + 24)² 
(2k + 38)² 
(4k  21)² 
1² 
2² 
31² 
3² 
20² 
22² 
16² 
13² 
5² 
21² 
11² 
23² 
10² 
24² 
7² 
12² 
15² 
9² 
27² 
14² 
25² 
19² 
8² 
6² 
17² 
Bimagic squares
Part 7
Part 8
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, 98102
Download the PDF file (168Kb)
(see also www.multimagie.com/English/Tetrapenta.htm)
[5] Christian Boyer, Les
cubes magiques, Pour La Science (the
French edition of Scientific American),
N°311 September 2003, 9095
(see also www.multimagie.com/English/Cube.htm#PLS)
[6] Christian Boyer, Le plus
petit cube magique parfait, La Recherche,
N°373 March 2004, 4850
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)
289297
Download the PDF file (99Kb)
(see also slide 7 of the lecture)
[11] Andrew Bremner, On squares of squares II, Acta Arithmetica, 99(2001)
289308
Download the PDF file (172Kb)
(see also slides 1112 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, RoseHulman Institute
of Technology Undergraduate Math Journal, 4(2003), N.1www.rosehulman.edu/mathjournal/v4n1.php
(now https://scholar.rosehulman.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, 5154
Download
the PDF file (76Kb)
(see also slides 67 of the lecture)
[54] Richard Schroeppel, The center cell of a magic 5^{3} 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, GauthierVillars,
Paris, 2(1894), 186194
(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
Christian Boyer. 
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