Editing Magic square (section)

==Enumeration of magic squares==
{{unsolved|mathematics|How many magic tori and  magic squares of order {{mvar|n}} are there for <math>n > 5</math> and  <math>n > 6</math>, respectively? }}

;Low-order squares
There is only one (trivial) magic square of order 1 and no magic square of order 2. As mentioned above, the set of normal squares of order three constitutes a single [[equivalence class]]-all equivalent to the Lo Shu square. Thus there is basically just one normal magic square of order 3.

The number of different ''n'' × ''n'' magic squares for ''n'' from 1 to 6, not counting rotations and reflections is:
: 1, 0, 1, 880, 275305224, 17753889197660635632. {{OEIS|id=A006052}}
The number for ''n'' = 6 had previously been estimated to be {{nowrap|1=(1.7745 ± 0.0016) × 10<sup>19</sup>.}}<ref>{{cite journal | last1 = Pinn | first1 = K. | last2 = Wieczerkowski | first2 = C. | year = 1998 | title = Number of Magic Squares From Parallel Tempering Monte Carlo | journal = Int. J. Mod. Phys. C | volume = 9 | issue = 4| page = 541 | doi=10.1142/s0129183198000443| arxiv = cond-mat/9804109 | bibcode = 1998IJMPC...9..541P | s2cid = 14548422 }}</ref><ref name=Arxiv>[https://arxiv.org/abs/cond-mat/9804109 "Number of Magic Squares From Parallel Tempering Monte Carlo], arxiv.org, April 9, 1998. Retrieved November 2, 2013.</ref><ref name=Loly>{{cite journal | last = Loly | first = Peter | title = The invariance of the moment of inertia of magic squares | journal = [[Mathematical Gazette]] | volume = 88 | issue = 511 | date = March 2004 | pages = 151–153 | doi = 10.1017/S002555720017456X | orig-year = 1 August 2016 <!-- date published online --> | url = http://home.cc.umanitoba.ca/~loly/MathGaz.pdf | citeseerx = 10.1.1.552.7296 | s2cid = 125989925 | access-date = 5 June 2017 | archive-date = 14 November 2017 | archive-url = https://web.archive.org/web/20171114111409/http://home.cc.umanitoba.ca/~loly/MathGaz.pdf | url-status = dead }}</ref>

; Magic tori
Cross-referenced to the above sequence, a new classification enumerates the magic tori that display these magic squares. The number of magic tori of order ''n'' from 1 to 5, is: 
: 1, 0, 1, 255, 251449712 {{OEIS|id=A270876}}.

; Higher-order squares and tori
[[File:Journal.pone.0125062.g001.PNG|thumb|right|Semi-log plot of Pn, the probability of magic squares of dimension n]]
The number of distinct normal magic squares rapidly increases for higher orders.<ref>[http://www.trump.de/magic-squares/howmany.html How many magic squares are there?] by Walter Trump, Nürnberg, January 11, 2001</ref>

The 880 magic squares of order 4 are displayed on 255 magic tori of order 4 and the 275,305,224 squares of order 5 are displayed on 251,449,712 magic tori of order 5. The numbers of magic tori and distinct normal squares are not yet known for orders beyond 5 and 6, respectively.<ref name=sudoku>[https://plus.maths.org/content/anything-square-magic-squares-sudoku Anything but square: from magic squares to Sudoku] by Hardeep Aiden, [[Plus Magazine]], March 1, 2006</ref>{{Citation needed|reason=The claim in this sentence is not in this reference.|date=August 2023}}

Algorithms tend to only generate magic squares of a certain type or classification, making counting all possible magic squares quite difficult. Since traditional counting methods have proven unsuccessful, statistical analysis using the [[Monte Carlo method]] has been applied. The basic principle applied to magic squares is to randomly generate ''n'' × ''n'' matrices of elements 1 to ''n''<sup>2</sup> and check if the result is a magic square. The probability that a randomly generated matrix of numbers is a magic square is then used to approximate the number of magic squares.<ref>{{cite journal |last1=Kitajima |first1=Akimasa |last2=Kikuchi |first2=Macoto |last3=Altmann |first3=Eduardo G. |title=Numerous but Rare: An Exploration of Magic Squares |journal=PLOS ONE |date=14 May 2015 |volume=10 |issue=5 |pages=e0125062 |doi=10.1371/journal.pone.0125062 |pmc=4431883 |pmid=25973764|bibcode=2015PLoSO..1025062K |doi-access=free }}</ref>

More intricate versions of the Monte Carlo method, such as the exchange Monte Carlo, and Monte Carlo backtracking have produced even more accurate estimations. Using these methods it has been shown that the probability of magic squares decreases rapidly as n increases. Using fitting functions give the curves seen to the right.