### Khajuraho method

Use the famous Khajuraho 4x4 panmagic square to construct larger magic squares which are a multiple of 4 (= 8x8, 12x12, 16x16, 20x20, … magic square).

Rewrite the Khajuraho magic square as follows:

Khajuraho magic square                Basic magic square

 7 12 1 14 7 h-4 1 h-2 2 13 8 11 2 h-3 8 h-5 16 3 10 5 h 3 h-6 5 9 6 15 4 h-7 6 h-1 4

To construct an 8x8 panmagic square, you need the basic square and 3 extending magic squares:

 7 h-4 1 h-2 +8 -8 +8 -8 2 h-3 8 h-5 +8 -8 +8 -8 h 3 h-6 5 -8 +8 -8 +8 h-7 6 h-1 4 -8 +8 -8 +8 +16 -16 +16 -16 +24 -24 +24 -24 +16 -16 +16 -16 +24 -24 +24 -24 -16 +16 -16 +16 -24 +24 -24 +24 -16 +16 -16 +16 -24 +24 -24 +24

The highest number in the 8x8 square is 64. Fill in 64 for h and calculate all the numbers. You get the following 8x8 panmagic square.

Panmagic 8x8 magic square

 7 60 1 62 15 52 9 54 2 61 8 59 10 53 16 51 64 3 58 5 56 11 50 13 57 6 63 4 49 14 55 12 23 44 17 46 31 36 25 38 18 45 24 43 26 37 32 35 48 19 42 21 40 27 34 29 41 22 47 20 33 30 39 28

This magic square is almost Franklin panmagic. Only four 2x2 sub-squares in the middle two columns give not 1/2 of the magic sum (1/2 x 260 = 130). If you swap the colours you get the following Franklin panmagic 8x8 square:

Franklin panmagic 8x8 square

 15 60 1 54 7 52 9 62 2 53 16 59 10 61 8 51 64 11 50 5 56 3 58 13 49 6 63 12 57 14 55 4 31 44 17 38 23 36 25 46 18 37 32 43 26 45 24 35 48 27 34 21 40 19 42 29 33 22 47 28 41 30 39 20

Use the Khajuraho method to construct magic squares of order is multiple of 4 from 8x8 to infinity. See 8x8, 12x12, 16x16, 20x20, 24x24, 28x28 and 32x32

8x8, Khajuraho method original.xls

It is possible to use each 4x4 panmagic square to construct a 8x8 Franklin panmagic square.

See above how to construct the almost perfect 8x8 Franklin panmagic square (replace the numbers 9 up to 16 of the 4x4 panmagic square by 57 up to 64 to create the first 4x4 sub-square and add each time 8 to the eight low numbers and -/- 8 to the eight high numbers to create the three other 4x4 sub-squares).

You must swap half of the numbers to get a perfect 8x8 Franklin panmagic square. Which numbers you must swap and how to swap the numbers, depends on the place of the 1 and the 8 in the 4x4 panmagic square.

 1 2 3 4

If the 1 and the 8 are in the same column, than you must swap half of the numbers of sub-square 1 with 2 and 3 with 4 (= horizontally).

If the 1 and the 8 are in the same row, than you must swap half of the numbers of sub-square 1 with 3 and 2 with 4 (= vertically).

Correction sheet 1                                Correction sheet 2

If the 1 and the 8 are in position 1 & 2 or 3 & 4 of the row/column, than you must use correction sheet 1.

If the 1 and the 8 are in position 2 & 3 or 1 & 4 of the row/column, than you must use correction sheet 2.

8x8, Khajuraho method.xls