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.

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:

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

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

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.

• Siamese method
• Strachey method for magic squares

• Erickson, Martin (2009), Aha! Solutions, MAA Spectrum, Mathematical Association of America, p. 98, ISBN 9780883858295.