### Binary grids of 4x4 panmagic square

See below the binary grids of the 3 basic 4x4 panmagic squares:

1x digit            +    2x digit            +    4x digit            +    8x digit +1      =    panmagic 4x4

 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 8 13 12 0 1 0 1 1 0 1 0 1 0 0 1 1 1 0 0 15 10 3 6 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 4 5 16 9 1 0 1 0 0 1 0 1 1 0 0 1 1 1 0 0 14 11 2 7

1x digit            +    2x digit               4x digit           +    8x digit +1         panmagic 4x4

 0 1 0 1 0 1 1 0 0 1 0 1 0 0 1 1 1 8 11 14 0 1 0 1 1 0 0 1 1 0 1 0 1 1 0 0 15 10 5 4 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 6 3 16 9 1 0 1 0 1 0 0 1 0 1 0 1 1 1 0 0 12 13 2 7

1x digit            +    2x digit            +    4x digit            +   8x digit +1       =    panmagic 4x4

 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 1 1 8 10 15 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 0 14 11 5 4 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 1 7 2 16 9 1 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 12 13 3 6

See below the binary grids to construct all 4x4 panmagic squares:

 H1a H1b 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 H2a H2b 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 V1a V1b 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 V2a V2b 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0

Construct a 4x4 panmagic square in the following 3 steps:

[1] Choose H1a ór H1b and H2a ór H2b and V1a ór V1b and V2a ór V2b (for example choose H1b, H2b, V1b and V2a; see below). There are 2x2x2x2 = 16 possibilities.

[2] Choose a sequence H1H2V1V2 ór H1H2V2V1 ór H1V1H2V2 ór H1V1V2H2 ór H1V2H2V1 ór H1V2V1H2 ór H2H1V1V2 ór H2H1V2V1 ór H2V1H1V2 ór H2V1V2H1 ór H2V2H1V1 ór H2V2V1H1 ór V1H1H2V2 ór V1H1V2H2 ór V1H2H1V2 ór V1H2V2H1 ór V1V2H1H2 ór V1V2H2H1 ór V2H1H2V1 ór V2H1V1H2 ór V2H2H1V1 ór V2H2V1H1 ór V2V1H1H2 ór V2V1H2H1 (for example choose H1H2V2V1; see below). There are (4x3x2x1=) 24 possibilities.

[3] Construct the 4x4 panmagic square:

1x digit (H1b)  +  2x digit (H2b)   +  4x digit (V2a)  8x digit (V1b) +1 =   panmagic 4x4

 1 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 12 6 9 7 0 0 1 1 0 1 1 0 1 0 1 0 1 0 1 0 13 3 16 2 1 1 0 0 1 0 0 1 1 0 1 0 0 1 0 1 8 10 5 11 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 15 4 14

You can construct all 16 (step 1) x 24 (step 2) = 384 panmagic 4x4 squares (including rotating and or mirroring).

See also the binary grids of the (most perfect) Franklin panmagic 8x8 square.

4x4, binairy grids.xls