Editing Conway's LUX method for magic squares

{{Short description|Algorithm for creating magic squares}}
'''Conway's LUX method for magic squares''' is an [[algorithm]] by [[John Horton Conway]] for creating [[magic square]]s of order 4''n''+2, where ''n'' is a [[natural number]].

==Method==
Start by creating a (2''n''+1)-by-(2''n''+1) square array consisting of 
* ''n''+1 rows of '''L'''s,
* 1 row of '''U'''s, and
* ''n''-1 rows of '''X'''s,
and then exchange the '''U''' in the middle with the '''L''' above it.

Each letter represents a 2x2 block of numbers in the finished square.

Now replace each letter by four consecutive numbers, starting with 1, 2, 3, 4 in the centre square of the top row, and moving from block to block in the manner of the [[Siamese method]]: move up and right, wrapping around the edges, and move down whenever you are obstructed. Fill each 2x2 block according to the order prescribed by the letter:

:<math>\mathrm{L}: \quad \begin{smallmatrix}4&&1\\&\swarrow&\\2&\rightarrow&3\end{smallmatrix} \qquad \mathrm{U}: \quad \begin{smallmatrix}1&&4\\\downarrow&&\uparrow\\2&\rightarrow&3\end{smallmatrix} \qquad \mathrm{X}:\quad \begin{smallmatrix}1&&4\\&\searrow\!\!\!\!\!\!\nearrow&\\3&&2\end{smallmatrix}</math>

==Example==
Let ''n'' = 2, so that the array is 5x5 and the final square is 10x10.
:{| class="wikitable" style="text-align: center"
|L||L||L||L||L
|-
|L||L||L||L||L
|-
|L||L||U||L||L
|-
|U||U||L||U||U
|-
|X||X||X||X||X
|}

Start with the L in the middle of the top row, move to the 4th X in the bottom row, then to the U at the end of the 4th row, then the L at the beginning of the 3rd row, etc.

:{| class="wikitable" style="text-align: center"
| width="10%" | 68 
| width="10%" | 65 
| width="10%" | 96 
| width="10%" | 93 
| width="10%" |  4 
| width="10%" |  1 
| width="10%" | 32 
| width="10%" | 29 
| width="10%" | 60 
| width="10%" | 57
|-
| 66 || 67 || 94 || 95 ||  2 ||  3 || 30 || 31 || 58 || 59
|-
| 92 || 89 || 20 || 17 || 28 || 25 || 56 || 53 || 64 || 61
|-
| 90 || 91 || 18 || 19 || 26 || 27 || 54 || 55 || 62 || 63
|-
| 16 || 13 || 24 || 21 || 49 || 52 || 80 || 77 || 88 || 85
|-
| 14 || 15 || 22 || 23 || 50 || 51 || 78 || 79 || 86 || 87
|-
| 37 || 40 || 45 || 48 || 76 || 73 || 81 || 84 ||  9 || 12
|-
| 38 || 39 || 46 || 47 || 74 || 75 || 82 || 83 || 10 || 11
|-
| 41 || 44 || 69 || 72 || 97 ||100 ||  5 ||  8 || 33 || 36
|-
| 43 || 42 || 71 || 70 || 99 || 98 ||  7 ||  6 || 35 || 34
|}

==See also==
*[[Siamese method]]
*[[Strachey method for magic squares]]

==References==
*{{citation|title=Aha! Solutions|series=MAA Spectrum|first=Martin|last=Erickson|publisher=[[Mathematical Association of America]]|year=2009|isbn=9780883858295|page=98|url=https://books.google.com/books?id=ywKyQz7_4-MC&pg=PA98}}.

[[Category:Magic squares]]