{{Short description|Unsolved problem in mathematics}}
{{unsolved|field=mathematics|explanation=Is it possible to construct a three-by-three magic square from nine distinct integer squares?}} The '''magic square of squares''' is an unsolved problem in mathematics which asks whether it is possible to construct a three-by-three magic square, the elements of which are all square numbers. The problem was first posed anonymously by Martin LaBar in 1984, before being included in Richard Guy's ''Unsolved problems in number theory (2nd edition)'' in 1994.<ref name = Robertson>{{Cite journal |last=Robertson |first=John P. |date=1996-10-01 |title=Magic Squares of Squares |journal=Mathematics Magazine |volume=69 |issue=4 |pages=289–293 |doi=10.1080/0025570X.1996.11996457 |issn=0025-570X}}</ref>
The problem has been a popular choice for recreational mathematicians following two articles Martin Gardner published in ''Quantum Magazine'' on the problem, offering a prize of {{currency|100|US}} in 1996.<ref>{{cite journal |last1=Gardner |first1=Martin |title=The magic of 3x3 |journal=Quantum |date=January 1996 |volume=6 |issue=3 |pages=24–26 |url=https://static.nsta.org/pdfs/QuantumV6N3.pdf |access-date=6 January 2024 |issn=1048-8820}}</ref><ref>{{cite journal |last1=Gardner |first1=Martin |title=The latest magic |journal=Quantum |date=March 1996 |volume=6 |issue=4 |page=60 |url=https://static.nsta.org/pdfs/QuantumV6N4.pdf |access-date=6 January 2024 |issn=1048-8820}}</ref> Other prizes have subsequently been offered for the first solution.<ref>{{cite magazine|author = <!-- not stated -->|date = October 2014|title = Can You Solve a Puzzle Unsolved Since 1996?|url = https://www.scientificamerican.com/article/can-you-solve-a-puzzle-unsolved-since-1996/|magazine = Scientific American}}</ref>
==Background== {{main|Magic square}} [[Image:Magicsquareexample.svg|thumb|The smallest (and unique up to rotation and reflection) non-trivial case of a magic square, order 3]] A magic square is a square array of integer numbers in which each row, column, and diagonal sums to the same number.<ref>{{cite book |first = Steven |last = Schwartzman |url = {{google books |id=iuoZSkSOBQsC |page=130 |plainurl=yes}} |title = The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English |page = 130 |publisher = MAA |date = 1994 }}</ref> The order of the square refers to the number of integers along each side.<ref>[https://mathworld.wolfram.com/MagicSquare.html Wolfram MathWorld: Magic Square] Weisstein, Eric W.</ref> A '''trivial magic square''' is a magic square which has at least one repeated element, and a '''semimagic square''' is a magic square in which the rows and columns, but not both diagonals, sum to the same number.
==Problem== The problem asks whether it is possible to construct a third-order magic square such that every element is itself a square number.<ref>{{cite journal |last1=LaBar |first1=Martin |title=Problems |journal=College Mathematics Journal |date=January 1984 |volume=15 |pages=68–74 |doi=10.1080/00494925.1984.11972754 |doi-broken-date=12 July 2025 |url=http://www.tandfonline.com/doi/abs/10.1080/00494925.1984.11972754 |access-date=6 June 2025}}</ref> A square which solves the problem would thus be of the form {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;" |- | <math>x_1^2</math> || <math>x_2^2</math> || <math>x_3^2</math> |- | <math>x_4^2</math> || <math>x_5^2</math> || <math>x_6^2</math> |- | <math>x_7^2</math> || <math>x_8^2</math> || <math>x_9^2</math> |} and satisfy the following equations<ref name = Alvarado>{{cite AV media|last = Várilly-Alvarado|first=Anthony|title = Magic Squares of Squares (are PROBABLY impossible) - Numberphile|url = https://www.youtube.com/watch?v=Kdsj84UdeYg|via=YouTube|collaboration=Numberphile}}</ref>
<math>\begin{align} x^2_1+x^2_2+x^2_3&=x^2_4+x^2_5+x^2_6\\ &=x^2_7+x^2_8+x^2_9\\ &=x^2_1+x^2_5+x^2_9\\ &=x^2_3+x^2_5+x^2_7\\ &=x^2_1+x^2_4+x^2_7\\ &=x^2_2+x^2_5+x^2_8\\ &=x^2_3+x^2_6+x^2_9 \end{align}</math>
==Current research== It has been shown that the problem is equivalent to several other problems.<ref name = Robertson/> # Do there exist three arithmetic progressions such that each has three terms, each has the same difference between terms as the other two, the terms are all perfect squares, and the middle terms of the three arithmetic progressions themselves form an arithmetic progression? # Do there exist three rational right triangles with the same area, such that the squares of the hypotenuses are in arithmetic progression? # Does there exist an elliptic curve, <math>y^2=x^3-n^2x</math>, where <math>n</math> is a congruent number, with three rational points on the curve, <math>(x_1,y_1)</math>, <math>(x_2,y_2)</math>, <math>(x_3,y_3)</math>, such that each point is "double" another rational point on the curve ("double" in the sense of the group structure for points on an elliptic curve), and <math>x_1</math>, <math>x_2</math> and <math>x_3</math> are in arithmetic progression?
=== Properties of a magic square of squares === Let <math>S </math> denotes the sum of each row, column or main diagonals of a ''primitive'' (all cells are pairwise coprime) third-order magic square of squares. If such a magic square of squares exists, it must satisfy the following properties.
* All cells must be odd.<ref name=":0">{{Cite journal |last=Rabern |first=Landon |date=2017-01-15 |title=Properties of Magic Squares of Squares |url=https://scholar.rose-hulman.edu/rhumj/vol4/iss1/3 |journal=Rose-Hulman Undergraduate Mathematics Journal |volume=4 |issue=1}}</ref> * All cells must be of the form <math>24k+1</math>.<ref name=":1">{{cite web |last=Zimmermann |first=Paul |date=2015 |title=Magic Squares of Squares |url=https://members.loria.fr/PZimmermann/papers/squares.pdf |access-date=April 30, 2026 |publisher=Inria / LORIA |format=PDF}}</ref> * No cell can have a prime divisor <math>p</math> of the form <math>p = 8k+3</math>.<ref name=":0" /> * <math>S </math> must be of the form <math>S = 3a^2</math> where <math>a^2</math>is the middle cell.<ref>{{Cite book |last=Gardner |first=Martin |url=https://doi.org/10.5948/upo9780883859476 |title=Riddles of the Sphinx |date=1987 |publisher=American Mathematical Society |isbn=978-0-88385-632-1 |series=Anneli Lax New Mathematical Library |location=Providence, Rhode Island |pages=136-137}}</ref> * <math>S </math> must be of the form <math>S = 72 k+3</math>.<ref name=":1" /> * All prime divisors <math>p</math> of the middle cell must be of the form <math>p = 4k+1</math>.<ref name=":0" /> * A prime <math>p</math> of the form <math>p = 8k+5</math> cannot divide a non-corner edge cell.<ref name=":0" /> * If a prime <math>p</math> of the form <math>p = 8k+3</math> divides a non-center cell then <math>p</math> must also divide the center cell and the other cell in that line.<ref name=":0" /> * If a prime <math>p</math> of the form <math>p = 4k+3</math> divides a corner cell then it must also divide the two non-corner edge cells that are not adjacent to that corner.<ref name=":0" /> * If a prime <math>p</math> of the form <math>p = 8k+5</math> divides a corner cell then it must also divide the center cell and the opposite corner cell.<ref>{{cite web |last=Morgenstern |first=Maurice |date=2007 |title=Properties of a 3x3 Magic Square of Squares |url=http://www.multimagie.com/MorgensternMssProperties.pdf |access-date=April 30, 2026 |website=Multimagie.com |publisher=Christian Boyer |format=PDF}}</ref>
Brute force searches for solutions have been unsuccessful, and suggest that if a solution exists, it would consist of numbers greater than at least <math>10^{14}</math>.<ref name="latest">{{cite web |last1=Boyer |first1=Christian |title=Latest research on the "3x3 magic square of squares" problem |url=http://www.multimagie.com/English/SquaresOfSquaresSearch.htm |website=Multimagie.com |access-date=19 June 2025}}</ref>
Rice University professor of mathematics Anthony Várilly-Alvarado has expressed his doubt as to the existence of the magic square of squares.<ref name= Alvarado/>
==Notable attempts== There have been a number of attempts to construct a magic square of squares by recreational mathematicians.
===Sallows' Square=== Following Gardner's prize offer for anyone who could find a magic square of squares in 1996, Lee Sallows published his attempt in ''The Mathematical Intelligencer''. His attempt is unique in that all of the rows and columns, and one of the diagonals, all sum to the same square number.<ref name = Sallows>{{cite journal|last=Sallows|first=Lee|journal=The Mathematical Intelligencer|title=The Lost Theorem|year=1997|url=http://www.multimagie.com/Sallows.pdf}}</ref><ref name = Alvarado/> {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;table-layout:fixed;width:19em;height:8em;border-style:none;" |+Sallows' Square<ref name = Sallows/> |- | style="background-color: white; border-style: none;" | || 127{{sup|2}}|| 46{{sup|2}}|| 58{{sup|2}} || '''147{{sup|2}}''' |- | style="background-color: white; border-style: none;" | || 2{{sup|2}} || 113{{sup|2}}|| 94{{sup|2}} || '''147{{sup|2}}''' |- | style="background-color: white; border-style: none;" | || 74{{sup|2}} || 82{{sup|2}}|| 97{{sup|2}} || '''147{{sup|2}}''' |- | '''147{{sup|2}}''' || '''147{{sup|2}}''' || '''147{{sup|2}}''' || '''147{{sup|2}}''' || '''38307''' |}
===Bremner Square=== In 1999, Andrew Bremner published his attempt at the problem, and further research surrounding magic squares of squares.<ref name = Bremner>{{cite journal|last=Bremner|first=Andrew|journal=Acta Arithmetica|title=On squares of squares|year=1999|url=https://pdfs.semanticscholar.org/1b0d/bd7d764ece84e9b22b462927979c6e635676.pdf}}</ref> Bremner's attempt differs from others in that not all elements of the square are square numbers, while all the rows, columns and diagonals sum to the same number.<ref name=Alvarado/> {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;table-layout:fixed;width:20em;height:8em;border-style:none;" |+Bremner Square<ref name = Bremner/> |- | style="background-color: white; border-style: none;" | || 373{{sup|2}}|| 289{{sup|2}}|| 565{{sup|2}} || '''541875''' |- | style="background-color: white; border-style: none;" | || 360721 || 425{{sup|2}}|| 23{{sup|2}} || '''541875''' |- | style="background-color: white; border-style: none;" | || 205{{sup|2}} || 527{{sup|2}}|| 222121 || '''541875''' |- | '''541875''' || '''541875'''|| '''541875'''|| '''541875'''|| '''541875''' |}
===Parker square=== The '''Parker square'''<ref>{{cite arXiv|eprint=1908.03236|quote=Some ’near misses’ have been found such as the Parker Square [2] |last1=Cain |first1=Onno |title=Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares |date=2019 |class=math.RA }}</ref> is an attempt by Matt Parker to solve the problem. His solution is a trivial, semimagic square of squares, as <math>41^2</math>, <math>29^2</math> and <math>1^2</math> all appear twice, and the diagonal <math>23^2+37^2+47^2</math> sums to 4107 instead of 3051.<ref>{{Cite AV media |url=https://www.youtube.com/watch?v=aOT_bG-vWyg|title=The Parker Square - Numberphile |date=April 18, 2016|author=Matt Parker|collaboration=Numberphile|access-date=June 6, 2025|via=YouTube}}</ref><ref name = latest/>
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;table-layout:fixed;width:19em;height:8em;border-style:none;" |+The Parker Square, with sums shown in bold. |- | style="background-color: white; border-style: none;" | || 29{{sup|2}}|| 1{{sup|2}}|| 47{{sup|2}} || '''3051''' |- | style="background-color: white; border-style: none;" | || 41{{sup|2}} || 37{{sup|2}}|| 1{{sup|2}} || '''3051''' |- | style="background-color: white; border-style: none;" | || 23{{sup|2}} || 41{{sup|2}}|| 29{{sup|2}} || '''3051''' |- | '''4107''' || '''3051'''|| '''3051'''|| '''3051'''|| '''3051''' |} ==Non third-order magic squares of squares== Magic squares of squares of orders greater than 3 have been known since as early as 1770, when Leonhard Euler sent a letter to Joseph-Louis Lagrange detailing a fourth-order magic square.<ref name = Boyer1>{{cite journal |last1=Boyer |first1=Christian |title=Some Notes on the Magic Squares of Squares Problem |journal=The Mathematical Intelligencer |date=12 November 2008 |volume=27 |issue=2 |pages=52–64 |doi=10.1007/BF02985794}}</ref> {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;table-layout:fixed;width:16em;height:8em;border-style:none;" |+ Euler's magic square of squares |- | 68{{sup|2}}|| 29{{sup|2}}|| 41{{sup|2}}|| 37{{sup|2}} |- | 17{{sup|2}}|| 31{{sup|2}}|| 79{{sup|2}}|| 32{{sup|2}} |- | 59{{sup|2}}|| 28{{sup|2}}|| 23{{sup|2}}|| 61{{sup|2}} |- | 11{{sup|2}}|| 77{{sup|2}}|| 8{{sup|2}}|| 49{{sup|2}} |}
'''Multimagic squares''' are magic squares which remain magic after raising every element to some power. In 1890, Georges Pfeffermann published a solution to a problem he posed involving the construction of an eighth-order 2-multimagic square.<ref>{{cite web |last1=Boyer |first1=Christian |title=Bimagic squares |url=http://www.multimagie.com/English/Bimagic.htm |website=Multimagie.com |access-date=6 June 2025}}</ref>
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;table-layout:fixed;width:24em;height:4em;border-style:none;" |+Pfeffermann's eighth order 2-multimagic square<ref>{{cite web |last1=Boyer |first1=Christian |title=Solution of the first bimagic square, 8th-order, of Pfeffermann |url=http://www.multimagie.com/English/Pfeffermann8.htm |website=Multimagie.com |access-date=6 June 2025}}</ref> |- | style="background-color: white; border-style: none;" | ||56 ||34 ||8 ||57 ||18 ||47 ||9 ||31 ||'''260''' |- | style="background-color: white; border-style: none;" | ||33 ||20 ||54 ||48 ||7 ||29 ||59 ||10 ||'''260''' |- | style="background-color: white; border-style: none;" | ||26 ||43 ||13 ||23 ||64 ||38 ||4 ||49 ||'''260''' |- | style="background-color: white; border-style: none;" | ||19 ||5 ||35 ||30 ||53 ||12 ||46 ||60 ||'''260''' |- | style="background-color: white; border-style: none;" | ||15 ||25 ||63 ||2 ||41 ||24 ||50 ||40 ||'''260''' |- | style="background-color: white; border-style: none;" | ||6 ||55 ||17 ||11 ||36 ||58 ||32 ||45 ||'''260''' |- | style="background-color: white; border-style: none;" | ||61 ||16 ||42 ||52 ||27 ||1 ||39 ||22 ||'''260''' |- | style="background-color: white; border-style: none;" | ||44 ||62 ||28 ||37 ||14 ||51 ||21 ||3 ||'''260''' |- ||'''260''' ||'''260''' ||'''260''' ||'''260''' ||'''260''' ||'''260''' ||'''260''' ||'''260''' ||'''260''' ||'''260''' |}
==References== {{reflist}}
Category:Recreational mathematics Category:Unsolved problems in mathematics Category:Discrete mathematics Category:Number theory Category:Unsolved problems in number theory