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 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]:

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.

127²

46²

58²

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 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 Guys 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 Eulers 4x4 and Lucass 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:


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

First inventors of magic squares of squares and bimagic squares,
and remaining open problems

Order

Magic squares of squares

Bimagic squares
using distinct integers

Normal bimagic squares
(using consecutive integers)

3x3

Who? Or proof of its impossibility?
Open problem 1, or open sub-problem 2.
See the current status of the research here.

Impossible. E. Lucas (1891)

4x4

L. Euler (1770)
See LE2  in the M.I. article.

Impossible.
L. Pebody (2004) / J.-C. Rosa (2004)

Impossible.
E. Lucas (1891)

5x5

C. Boyer (2004)
See CB4 in the M.I. article.

Who? Or proof of its impossibility?
Open problem 3.
See the current status of the research here.

Impossible.
C. Boyer - W. Trump (2002)

6x6

C. Boyer (2005)
See here, constructed after the article.

J. Wroblewski (2006)
See here, constructed after the article.

7x7

C. Boyer (2005)
See here, constructed after the article.

L. Morgenstern (2006)
See here, constructed after the article.

8x8
and +

G. Pfeffermann (8x8 in 1890, 9x9 in 1891).
First bimagic squares, using consecutive integers.
Various other orders are known (10x10, 11x11, 12x12, 13x13,...)

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 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”:

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

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

49²

48²

23²

19²

21²

26²

33²

32²

36²

13²

42²

22²

27²

44²

(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

31²

20²

22²

16²

13²

21²

11²

23²

10²

24²

12²

15²

27²

14²

25²

19²

17²

Bimagic squares

Part 6

Part 7

Part 8

Supplement


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

[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

Many thanks to :

  • Andrew Bremner, Arizona State University, Duncan Buell, University of South Carolina, and Lee Sallows, University of Nijmegen, for their positive feedback and remarks based on their reading of early drafts of the article. Their excellent previous papers on the subject are downloadable from multimagie.com: [10], [11], [12], [52].
  • Chandler Davis, University of Toronto, and Editor-in-Chief of The Mathematical Intelligencer, for his very attentive reading and for his approval to publish the article in his magazine.
  • Susan Hannan, USA, for her kind and very efficient checking of my English!

Christian Boyer.


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