Editing Magic square (section)

===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
|}