Editing Magic square (section)

==Variations of the magic square==

===Extra constraints===
[[File:Magic prime square 12.svg|thumb|Magic square of primes]]
Certain extra restrictions can be imposed on magic squares.

If raising each number to the ''n''th power yields another magic square, the result is a bimagic (n&nbsp;=&nbsp;2), a trimagic (n&nbsp;=&nbsp;3), or, in general, a [[multimagic square]].

A magic square in which the number of letters in the name of each number in the square generates another magic square is called an [[alphamagic square]].

There are magic squares consisting [[prime magic square|entirely of primes]]. [[Rudolf Ondrejka]] (1928–2001) discovered the following 3×3 magic square of [[prime number|primes]], in this case nine [[Chen prime]]s:

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:7em;height:7em;table-layout:fixed;"
|-
| 17 || 89 || 71
|-
| 113 || 59 || 5
|-
| 47 || 29 || 101
|}

The [[Green–Tao theorem]] implies that there are arbitrarily large magic squares consisting of primes.

The following "reversible magic square" has a magic constant of 264 both upside down and right way up:<ref>Karl Fulves, [http://www.markfarrar.co.uk/othmsq01.htm#reversible Self-working Number Magic (Dover Magic Books)]</ref>
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
| 96 || 11 || 89 || 68
|-
| 88 || 69 || 91 || 16
|-
| 61 || 86 || 18 || 99
|-
| 19 || 98 || 66 || 81
|}

[[File:Ramanujan_magic_square_construction.svg|thumb|Construction of Ramanujan's magic square from a [[Latin square]] with distinct diagonals and day (D), month (M), century (C) and year (Y) values, and Ramanujan's birthday example]]
When the extra constraint is to display some date, especially a birth date, then such magic squares are called birthday magic square. An early instance of such birthday magic square was created by [[Srinivasa Ramanujan]]. He created a 4×4 square in which he entered his date of birth in D&ndash;M&ndash;C-Y format in the top row and the magic happened with additions and subtractions of numbers in squares. Not only do the rows, columns, and diagonals add up to the same number, but the four corners, the four middle squares (17, 9, 24, 89), the first and last rows two middle numbers (12, 18, 86, 23), and the first and last columns two middle numbers (88, 10, 25, 16) all add up to the sum of 139.

===Multiplicative magic squares===
Instead of ''adding'' the numbers in each row, column and diagonal, one can apply some other operation. For example, a multiplicative magic square has a constant ''product'' of numbers. A multiplicative magic square can be derived from an additive magic square by raising 2 (or any other integer) to the power of each element, because the [[logarithm]] of the product of 2 numbers is the sum of logarithm of each. Alternatively, if any 3 numbers in a line are 2<sup>''a''</sup>, 2<sup>''b''</sup> and 2<sup>''c''</sup>, their product is 2<sup>''a''+''b''+''c''</sup>, which is constant if ''a''+''b''+''c'' is constant, as they would be if ''a'', ''b'' and ''c'' were taken from ordinary (additive) magic square.<ref>{{citation
 | last = Stifel | first = Michael | author-link = Michael Stifel
 | language=la
 | pages = 29–30
 | title = Arithmetica integra
 | url = https://books.google.com/books?id=fndPsRv08R0C
 | year = 1544}}.</ref> For example, the original Lo-Shu magic square becomes:

{| class="wikitable" style="margin:0.5em auto;text-align:center;width:9em;height:9em;table-layout:fixed;"
|-
|+ ''M'' = 32768
|-
| 16 || 512 || 4
|-
| 8 || 32 || 128
|-
| 256 || 2 || 64
|}

Other examples of multiplicative magic squares include:

{{col-begin|width=auto;margin:0.5em auto}}
{{col-break|valign=bottom}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
|-
|+ ''M'' = 216
|-
| 2 || 9 || 12
|-
| 36 || 6 || 1
|-
| 3 || 4 || 18
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
|+ ''M'' = 6720
|-
| 1 || 6 || 20 || 56
|-
| 40 || 28 || 2 || 3
|-
| 14 || 5 || 24 || 4
|-
| 12 || 8 || 7 || 10
|}
{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:18em;height:12em;table-layout:fixed;"
|-
|+ ''M'' = 6,227,020,800
|-
|27 ||50 ||66 ||84 ||13 ||2 ||32
|-
|24 ||52 ||3 ||40 ||54 ||70 ||11
|-
|56 ||9 ||20 ||44 ||36 ||65 ||6
|-
|55 ||72 ||91 ||1 ||16 ||36 ||30
|-
|4 ||24 ||45 ||60 ||77 ||12 ||26
|-
|10 ||22 ||48 ||39 ||5 ||48 ||63
|-
|78 ||7 ||8 ||18 ||40 ||33 ||60
|}
{{col-end}}

===Multiplicative magic squares of complex numbers===
Still using [[Ali Skalli]]'s non iterative method, it is possible to produce an infinity of multiplicative magic squares of [[complex numbers]]<ref>"[http://sites.google.com/site/aliskalligvaen/home-page/-multiplicative-of-complex-numbers-8x8 8x8 multiplicative magic square of complex numbers]" Ali Skalli's magic squares and magic cubes</ref> belonging to <math>\mathbb C</math> set. On the example below, the real and imaginary parts are integer numbers, but they can also belong to the entire set of real numbers <math>\mathbb R</math>.
The product is: '''−352,507,340,640 − 400,599,719,520 ''i'''''.

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:left;"
|+ Skalli multiplicative 7×7 of [[complex numbers]]
|-
| style="text-align:right;border-right:none;padding-right:0;width:5ex;"|       21 ||style="border-left:none;padding-left:0;width:6ex;"|      +14''i''
| style="text-align:right;border-right:none;padding-right:0;width:5ex;"|&minus;70 ||style="border-left:none;padding-left:0;width:6ex;"|      +30''i''
| style="text-align:right;border-right:none;padding-right:0;width:5ex;"|&minus;93 ||style="border-left:none;padding-left:0;width:6ex;"|&minus;9''i''
| style="text-align:right;border-right:none;padding-right:0;width:5ex;"|&minus;105||style="border-left:none;padding-left:0;width:6ex;"|&minus;217''i''
| style="text-align:right;border-right:none;padding-right:0;width:5ex;"|       16 ||style="border-left:none;padding-left:0;width:6ex;"|      +50''i''
| style="text-align:right;border-right:none;padding-right:0;width:5ex;"|       4  ||style="border-left:none;padding-left:0;width:6ex;"|&minus;14''i''
| style="text-align:right;border-right:none;padding-right:0;width:5ex;"|       14 ||style="border-left:none;padding-left:0;width:6ex;"|&minus;8''i''
|-
| style="text-align:right;border-right:none;padding-right:0;"|       63 ||style="border-left:none;padding-left:0;"|&minus;35''i''
| style="text-align:right;border-right:none;padding-right:0;"|       28 ||style="border-left:none;padding-left:0;"|      +114''i''
| style="text-align:right;border-right:none;padding-right:0;"|          ||style="border-left:none;padding-left:0;"|&minus;14''i''
| style="text-align:right;border-right:none;padding-right:0;"|       2  ||style="border-left:none;padding-left:0;"|      +6''i''
| style="text-align:right;border-right:none;padding-right:0;"|       3  ||style="border-left:none;padding-left:0;"|&minus;11''i''
| style="text-align:right;border-right:none;padding-right:0;"|       211||style="border-left:none;padding-left:0;"|      +357''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;123||style="border-left:none;padding-left:0;"|&minus;87''i''
|-
| style="text-align:right;border-right:none;padding-right:0;"|       31 ||style="border-left:none;padding-left:0;"|&minus;15''i''
| style="text-align:right;border-right:none;padding-right:0;"|       13 ||style="border-left:none;padding-left:0;"|&minus;13''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;103||style="border-left:none;padding-left:0;"|      +69''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;261||style="border-left:none;padding-left:0;"|&minus;213''i''
| style="text-align:right;border-right:none;padding-right:0;"|       49 ||style="border-left:none;padding-left:0;"|&minus;49''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;46 ||style="border-left:none;padding-left:0;"|      +2''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;6  ||style="border-left:none;padding-left:0;"|      +2''i''
|-
| style="text-align:right;border-right:none;padding-right:0;"|       102||style="border-left:none;padding-left:0;"|&minus;84''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;28 ||style="border-left:none;padding-left:0;"|&minus;14''i''
| style="text-align:right;border-right:none;padding-right:0;"|       43 ||style="border-left:none;padding-left:0;"|      +247''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;10 ||style="border-left:none;padding-left:0;"|&minus;2''i''
| style="text-align:right;border-right:none;padding-right:0;"|       5  ||style="border-left:none;padding-left:0;"|      +9''i''
| style="text-align:right;border-right:none;padding-right:0;"|       31 ||style="border-left:none;padding-left:0;"|&minus;27''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;77 ||style="border-left:none;padding-left:0;"|      +91''i''
|-
| style="text-align:right;border-right:none;padding-right:0;"|&minus;22 ||style="border-left:none;padding-left:0;"|&minus;6''i''
| style="text-align:right;border-right:none;padding-right:0;"|       7  ||style="border-left:none;padding-left:0;"|      +7''i''
| style="text-align:right;border-right:none;padding-right:0;"|       8  ||style="border-left:none;padding-left:0;"|      +14''i''
| style="text-align:right;border-right:none;padding-right:0;"|       50 ||style="border-left:none;padding-left:0;"|      +20''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;525||style="border-left:none;padding-left:0;"|&minus;492''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;28 ||style="border-left:none;padding-left:0;"|&minus;42''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;73 ||style="border-left:none;padding-left:0;"|      +17''i''
|-
| style="text-align:right;border-right:none;padding-right:0;"|       54 ||style="border-left:none;padding-left:0;"|      +68''i''
| style="text-align:right;border-right:none;padding-right:0;"|       138||style="border-left:none;padding-left:0;"|&minus;165''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;56 ||style="border-left:none;padding-left:0;"|&minus;98''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;63 ||style="border-left:none;padding-left:0;"|      +35''i''
| style="text-align:right;border-right:none;padding-right:0;"|       4  ||style="border-left:none;padding-left:0;"|&minus;8''i''
| style="text-align:right;border-right:none;padding-right:0;"|       2  ||style="border-left:none;padding-left:0;"|&minus;4''i''
| style="text-align:right;border-right:none;padding-right:0;"|       70 ||style="border-left:none;padding-left:0;"|&minus;53''i''
|-
| style="text-align:right;border-right:none;padding-right:0;"|       24 ||style="border-left:none;padding-left:0;"|      +22''i''
| style="text-align:right;border-right:none;padding-right:0;"|&minus;46 ||style="border-left:none;padding-left:0;"|&minus;16''i''
| style="text-align:right;border-right:none;padding-right:0;"|       6  ||style="border-left:none;padding-left:0;"|&minus;4''i''
| style="text-align:right;border-right:none;padding-right:0;"|       17 ||style="border-left:none;padding-left:0;"|      +20''i''
| style="text-align:right;border-right:none;padding-right:0;"|       110||style="border-left:none;padding-left:0;"|      +160''i''
| style="text-align:right;border-right:none;padding-right:0;"|       84 ||style="border-left:none;padding-left:0;"|&minus;189''i''
| style="text-align:right;border-right:none;padding-right:0;"|       42 ||style="border-left:none;padding-left:0;"|&minus;14''i''
|}

===Additive-multiplicative magic and semimagic squares===

Additive-multiplicative magic squares and semimagic squares satisfy properties of both ordinary and multiplicative magic squares and semimagic squares, respectively.<ref>{{cite web|title=Multimagie.com – Additive-Multiplicative magic squares, 8th and 9th-order|url=http://multimagie.com/English/Multiplicative8_9.htm|access-date=26 August 2015}}</ref>
{| width="100%"
| valign="top"|
{| class="wikitable" style="text-align:center;"
|+
First known<br/>additive-multiplicative magic square<br />
{{nobold|1=
8×8 found by W. W. Horner in 1955<br />
Sum = 840<br />
Product = {{val|2058068231856000}}}}
|-
| 162 || 207 || 51 || 26 || 133 || 120 || 116 || 25
|-
| 105 || 152 || 100 || 29 || 138 || 243 || 39 || 34
|-
| 92 || 27 || 91 || 136 || 45 || 38 || 150 || 261
|-
| 57 || 30 || 174 || 225 || 108 || 23 || 119 || 104
|-
| 58 || 75 || 171 || 90 || 17 || 52 || 216 || 161
|-
| 13 || 68 || 184 || 189 || 50 || 87 || 135 || 114
|-
| 200 || 203 || 15 || 76 || 117 || 102 || 46 || 81
|-
| 153 || 78 || 54 || 69 || 232 || 175 || 19 || 60
|}
| valign="top"|
{| class="wikitable" style="text-align:center;"
|+
Smallest known additive-multiplicative semimagic square<br />
{{nobold|1=4×4 found by L. Morgenstern in 2007<br />
Sum = 247<br />
Product = {{val|3369600}}}}
|-
| 156 || 18 || 48 || 25
|-
| 30 || 144 || 60 || 13
|-
| 16 || 20 || 130 || 81
|-
| 45 || 65 || 9 || 128
|}
|}
It is unknown if any additive-multiplicative magic squares smaller than 7×7 exist, but it has been proven that no 3×3 or 4×4 additive-multiplicative magic squares and no 3×3 additive-multiplicative semimagic squares exist.<ref>{{cite web|title=Multimagie.com – Smallest additive-multiplicative magic square|url=http://multimagie.com/English/SmallestAddMult.htm|access-date=16 January 2024}}</ref>

{| class="wikitable" style="text-align:center;"
|+
Smallest known additive-multiplicative magic square<br />
{{nobold|1=7×7 found by Sébastien Miquel(Sébastien Miquel<!--Q103036398-->) in August 2016<br />
Sum = 465<br />
Product = {{val|150885504000}}}}
|-
|126
|66
|50
|90
|48
|1
|84
|-
|20
|70
|16
|54
|189
|110
|6
|-
|100
|2
|22
|98
|36
|72
|135
|-
|96
|60
|81
|4
|10
|49
|165
|-
|3
|63
|30
|176
|120
|45
|28
|-
|99
|180
|14
|25
|7
|108
|32
|-
|21
|24
|252
|18
|55
|80
|15
|}

===Geometric magic squares===
[[File:Geomagic square - Diamonds.jpg|thumb|right| text-bottom |160px|A geometric magic square.]]
Magic squares may be constructed which contain geometric shapes instead of numbers. Such squares, known as [[geometric magic square]]s, were invented and named by [[Lee Sallows]] in 2001.<ref>[https://www.theguardian.com/science/2011/apr/03/magic-squares-geomagic-lee-sallows Magic squares are given a whole new dimension], [[The Observer]], April 3, 2011</ref>

In the example shown the shapes appearing are two dimensional. It was Sallows' discovery that ''all'' magic squares are geometric, the numbers that appear in numerical magic squares can be interpreted as a shorthand notation which indicates the lengths of straight line segments that are the geometric 'shapes' occurring in the square. That is, numerical magic squares are that special case of a geometric magic square using one dimensional shapes.<ref name=PourlaScience>[https://www.pourlascience.fr/sd/mathematiques/les-carres-magiques-geometriques-7372.php Les carrés magiques géométriques] by Jean-Paul Delahaye, ''Pour La Science'' No. 428, June 2013</ref>

=== Area magic squares ===
[[File:Trump Walkington Taneja first linear area magic square 170106.jpg|thumb|168x168px|The first linear area magic square
]]
In 2017, following initial ideas of [http://oeis.org/search?q=william%20walkington&sort=created William Walkington] and [https://www.researchgate.net/profile/Inder_Taneja Inder Taneja], the first linear area magic square (L-AMS) was constructed by [[Walter Trump]].<ref>{{Cite web|url=https://www.futilitycloset.com/2017/01/19/area-magic-squares/|title=Area Magic Squares|date=2017-01-19|website=[[Futility Closet]]|access-date=2017-06-12}}</ref>

===Other magic shapes===
Other two dimensional shapes than squares can be considered. The general case is to consider a design with ''N'' parts to be magic if the ''N'' parts are labeled with the numbers 1 through ''N'' and a number of identical sub-designs give the same sum. Examples include [[Magic circle (mathematics)|magic circle]]s, magic rectangles, [[Magic triangle (mathematics)|magic triangle]]s<ref name=Ely>Magic Designs, Robert B. Ely III, Journal of Recreational Mathematics volume 1 number 1, January 1968</ref> [[magic star]]s, [[magic hexagon]]s, magic diamonds. Going up in dimension results in magic spheres, magic cylinders, [[magic cube]]s, magic parallelepiped, magic solids, and other [[magic hypercube]]s.

Possible magic shapes are constrained by the number of equal-sized, equal-sum subsets of the chosen set of labels. For example, if one proposes to form a magic shape labeling the parts with {1, 2, 3, 4}, the sub-designs will have to be labeled with {1,4} and {2,3}.<ref name=Ely />