Conway's LUX method for magic squares

Conway's LUX method for magic squares

Conway's LUX method for magic squares is an algorithm by John Horton Conway for creating magic squares of order 4n+2, where n is a natural number.

Method

Start by creating a (2n+1)-by-(2n+1) square array consisting of

  • n+1 rows of Ls,
  • 1 row of Us, and
  • n-1 rows of Xs,

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:

\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}

Example

Let n = 2, so that the array is 5x5 and the final square is 10x10.

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.

68 65 96 93 4 1 32 29 60 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


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