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:

:<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

  • .