LUX method

The LUX method is a method of creating magic squares . It comes from the English mathematician John Horton Conway .

Procedure

The LUX method is used to generate magic squares of order , where is a positive natural number . First, consider a square matrix with rows and columns, which is filled with letters as follows: ${\ displaystyle 4n + 2}$ ${\ displaystyle n}$ ${\ displaystyle 2n + 1}$

The first lines are completely filled with . ${\ displaystyle n + 1}$ ${\ displaystyle \ mathrm {L}}$
It is followed by a line with . ${\ displaystyle \ mathrm {U}}$
The remaining lines are also described. ${\ displaystyle n-1}$ ${\ displaystyle \ mathrm {X}}$
Finally, the middle is swapped with the one above. ${\ displaystyle \ mathrm {U}}$ ${\ displaystyle \ mathrm {L}}$

Now, using the Siamese method, a magic square of order is created that lies on top of the letter. Then in turn all the numbers of this magic square as follows by the four consecutive numbers , , and replaced in accordance with the provision of the related letter: ${\ displaystyle 2n + 1}$ ${\ displaystyle i}$ ${\ displaystyle (i-1) \ cdot 4 + a}$ ${\ displaystyle (i-1) \ cdot 4 + b}$ ${\ displaystyle (i-1) \ cdot 4 + c}$ ${\ displaystyle (i-1) \ cdot 4 + d}$

{\ displaystyle {\ begin {bmatrix} a & b \\ c & d \ end {bmatrix}} \ quad with \ quad \ mathrm {L} \ colon {\ begin {bmatrix} 4 & 1 \\ 2 & 3 \ end {bmatrix}} \ quad \ mathrm {U} \ colon {\ begin {bmatrix} 1 & 4 \\ 2 & 3 \ end {bmatrix}} \ quad \ mathrm {X} \ colon {\ begin {bmatrix} 1 & 4 \\ 3 & 2 \ end {bmatrix}}}

One imagines drawing the letters with a pen (hence the name LUX method).

example

For the letter matrix has the form ${\ displaystyle n = 2}$

{\ displaystyle {\ begin {bmatrix} \ mathrm {L} & \ mathrm {L} & \ mathrm {L} & \ mathrm {L} & \ mathrm {L} \\\ mathrm {L} & \ mathrm {L } & \ mathrm {L} & \ mathrm {L} & \ mathrm {L} \\\ mathrm {L} & \ mathrm {L} & \ mathrm {U} & \ mathrm {L} & \ mathrm {L} \\\ mathrm {U} & \ mathrm {U} & \ mathrm {L} & \ mathrm {U} & \ mathrm {U} \\\ mathrm {X} & \ mathrm {X} & \ mathrm {X} & \ mathrm {X} & \ mathrm {X} \ end {bmatrix}}}

Using the Siamese method, the result is the following magic square:

{\ displaystyle {\ begin {bmatrix} \ mathrm {L} = 17 & \ mathrm {L} = 24 & \ mathrm {L} = 1 & \ mathrm {L} = 8 & \ mathrm {L} = 15 \\\ mathrm {L } = 23 & \ mathrm {L} = 5 & \ mathrm {L} = 7 & \ mathrm {L} = 14 & \ mathrm {L} = 16 \\\ mathrm {L} = 4 & \ mathrm {L} = 6 & \ mathrm { U} = 13 & \ mathrm {L} = 20 & \ mathrm {L} = 22 \\\ mathrm {U} = 10 & \ mathrm {U} = 12 & \ mathrm {L} = 19 & \ mathrm {U} = 21 & \ mathrm {U} = 3 \\\ mathrm {X} = 11 & \ mathrm {X} = 18 & \ mathrm {X} = 25 & \ mathrm {X} = 2 & \ mathrm {X} = 9 \ end {bmatrix}}}

Now you start at the very top in the middle and replace the number 1 with the numbers 1 to 4 according to the letter . It follows on the last line, where the number 2 is replaced by the numbers 5 to 8 according to the letter . The next field is then and so on. The end result is the following magic square: ${\ displaystyle \ mathrm {L} = 1}$ ${\ displaystyle \ mathrm {L}}$ ${\ displaystyle \ mathrm {X} = 2}$ ${\ displaystyle \ mathrm {X}}$ ${\ displaystyle \ mathrm {U} = 3}$