Editing Magic square (section)

===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}}