# Block-Wise Construction of Bimagic Squares: Multiples of Orders 8 and 16

#### ByInder J. Taneja

Dec 12, 2023

Whole the work is done manually on excel sheets.

Below are bimagic squares written in blocks multiples of orders 8 and 16. Most of the work is done manually by author in 2011. Few of them are with the help of Aale de Winkel <aaledewinkel@magichypercubes.com>, Magic Encyclopedia magic square, magic cube, magic tesseract, magic hypercube. See the reference below:

Before proceeding further below are the basic formulas to check the sums of magic and bimagic squares

• Magic Sum
• Bimagic Sum

where n is the order the magic square.

### Bimagic Square of Order 8

In this case, we have

S8×8:=260; Sb8×8:= 11180

2×4 blocks are of equal sums as of magic square, i.e., 260. See below the bimagic square of order 8.

The construction of bimagic square of order 8 is well known in the history, and is done by G. Pfeffermann in 1891. In this case, we have a pandiagonal bimagic square of order 8, where the blocks of order 2×4 are of same sum as of magic square of order 8.

### Bimagic Square of Order 16

In this case, we have

S16×16:=2056; Sb16×16:= 351576, S4×4:=(1/4) S16×16=514.

See below the bimagic square of order 16.

### Semi-Bimagic Square of Order 24: Blocks of Order 8

In this case, we have

S24×24:=6924; Sb24×24:= 2661124 (semi), S8×8:=2308.

Magic squares of order 8 are of equal sums. These are either bimagic or semi-bimagic. See semi-bimagic square of order 24

### Bimagic Square of Order 32: Blocks of Order 8

In this case, we have

S32×32:=16400; Sb32×32:= 11201200, S8×8:=4100.

Magic squares of order 8 are of equal sums. These are either bimagic or semi-bimagic. See bimagic square of order 32. It is also pandiagonal.

### Bimagic Square of Order 40: Blocks of Order 8

In this case, we have

S40×40:=32020; Sb40×40:= 34165340, S8×8:=6404.

Magic squares of order 8 are of equal sums. These are either bimagic or semi-bimagic. See bimagic square of order 40. It is also pandiagonal.

### Semi-Bimagic Square of Order 48: Blocks of Order 16

In this case, we have

S48×48:=55320; Sb48×48:= 84989960 (semi-bimagic); S16×16:=18440.

Magic squares of order 16 are of equal sums. These are all bimagic. See below semi-bimagic square of order 48.

### Bimagic Square of Order 56: Blocks of Order 8

In this case, we have

S56×56:=87836; Sb56×56:= 183665076; S8×8:=12548.

Magic squares of order 8 are of equal sums. These are either bimagic or semi-bimagic squares of order 8. See below bimagic square of order 56.

### Bimagic Squares of Order 64: Blocks of Orders 8 and 16

Below are two types of bimagic squares of order 64:

#### First Type: Blocks of Order 8

In this case, we have

S64×64:=131104; Sb64×64:= 358045024; S8×8:=16338.

See below bimagic square of order 64 with blocks of order 8. These blocks of order 8 are either bimagic or semi-bimagic with different sums, but the magic sum of all order magic squares is same.

#### Second Type: Blocks of Order 16

In this case, we have

S64×64:=131104; Sb64×64:= 358045024; S16×16:=32776 and S4×4:==8194.

Magic squares of order 16 are of equal sums. These are all bimagic. See below bimagic square of order 64 with blocks of order 16. These blocks of order 16 are bimagic squares with different sums, but the magic sum of all order 16 magic squares is same.

### Bimagic Square of Order 72: Blocks of Order 8

In this case, we have

S72×72:=186660; Sb72×72:= 645159180; S8×8:=20740.

Magic squares of order 8 are of equal sums. These are either bimagic ou semi-bimagic squares of different sums. See the excel sheet given at the end.

### Bimagic Squares of Order 80: Blocks of Orders 8 and 16

Similar to order 64, here also we have two ways to write bimagic square of order 80.

#### First Type: Blocks of Order 8

In this case, we have

S80×80:=256040; Sb80×80:= 1092522680; S8×8:=25604.

These blocks of order 8 are either bimagic or semi-bimagic with different sums, but the magic sum of all order magic squares is same. See the magic square of order 80 in excel sheet given at the end.

#### Second Type: Blocks of Order 16

In this case, we have

S80×80:=256040; Sb80×80:= 1092522680; S16×16:=51208 and S4×4:=12802.

Magic squares of order 16 are of equal sums. See below bimagic square of order 80 with blocks of order 16. These blocks of order 16 are bimagic squares with different bimagic sums, but the magic sum of all order 16 magic squares is same. See the magic square of order 80 in excel sheet given at the end.

### Bimagic Square of Order 88: Blocks of Order 8

In this case, we have

S88×88:=340780; Sb88×88:= 1759447140; S8×8:=30980.

Magic squares of order 8 are of equal sums. These are either bimagic ou semi-bimagic squares of different sums. See the excel sheet given at the end.

### Bimagic Square of Order 96: Blocks of Order 8

In this case, we have

S96×96:=442416; Sb96×96:= 2718351376; S8×8:=36868.

Magic squares of order 8 are of equal sums. These are either bimagic ou semi-bimagic squares of different sums. See the excel sheet given at the end.

Even though 96 is divisible by 16, still, we don’t have bimagic square of order 96 with blocks of order 16.

### Bimagic Square of Order 104: Blocks of Order 8

In this case, we have

S104×104:=562484; Sb104×104:=4056072124; S8×8:=43268.

Magic squares of order 8 are of equal sums. These are either bimagic ou semi-bimagic squares of different sums. See the excel sheet given at the end.

### Bimagic Squares of Order 112: Blocks of Orders 8 and 16

Similar to orders 64 and 80, here also we have two ways to writing bimagic square of order 112.

#### First Type: Blocks of Order 8

In this case, we have

S112×112:=702520; Sb112×112:= 5875174760; S8×8:=50180.

These blocks of order 8 are either bimagic or semi-bimagic with different sums, but the magic sum of all order magic squares is same. See the magic square of order 112 in excel sheet given at the end.

#### Second Type: Blocks of Order 16

In this case, we have

S112×112:=702520; Sb112×112:= 5875174760; S16×16:=100360 and S4×4:=25090.

See below bimagic square of order 80 with blocks of order 16. These blocks of order 16 are bimagic squares with different bimagic sums, but the magic sums of all order 16, and order 4 are of equal sums.

### Bimagic Square of Order 120: Blocks of Order 8

In this case, we have

S120×120:=864060; Sb120×120:=8295264020; S8×8:=57604.

Magic squares of order 8 are of equal sums. These are either bimagic ou semi-bimagic squares of different sums. See the excel sheet given at the end.

### Bimagic Square of Order 128: Blocks of Orders 8 and 16

Similar to orders 64, 80 and 112, here also we have two ways to writing bimagic square of order 128.

#### First Type: Blocks of Order 8

In this case, we have

S128×128:=1048640; Sb128×128:=11454294720; S8×8:=65540.

These blocks of order 8 are either bimagic or semi-bimagic with different sums, but the magic sum of all order magic squares is same. See the magic square of order 128 in excel sheet given at the end.

#### Second Type: Blocks of Order 16

In this case, we have

S128×128:=1048640; Sb128×128:=11454294720; S16×16:=131080 and S4×4:=32770.

See below bimagic square of order 128 with blocks of order 16. These blocks of order 16 are bimagic squares with different bimagic sums, but the magic sums of all order 16, and order 4 are of equal sums.

### Bimagic Square of Order 136: Blocks of Order 8

In this case, we have

S136×136:=1257796; Sb136×136:=15509882476; S8×8:=73988.

Magic squares of order 8 are of equal sums. These are either bimagic ou semi-bimagic squares of different sums. See the excel sheet given at the end.

### Bimagic Square of Order 144: Blocks of Order 16

In this case, we have

S144×144:=1493064; Sb144×144:=20640614424; S16×16:=165896 and S4×4:=41474.

See below bimagic square of order 128 with blocks of order 16. These blocks of order 16 are bimagic squares with different bimagic sums, but the magic sums of all order 16, and order 4 are of equal sums.

### Bimagic Square of Order 152: Blocks of Order 8

In this case, we have

S152×152:=1755980; Sb152×152:=27047359940; S8×8:=92420.

Magic squares of order 8 are of equal sums. These are either bimagic ou semi-bimagic squares of different sums. See the excel sheet given at the end.

### Bimagic Square of Order 160: Blocks of Order 16

In this case, we have

S144×144:=2048080; Sb144×144:=34954581360; S16×16:=204808 and S4×4:=51202.

See below bimagic square of order 128 with blocks of order 16. These blocks of order 16 are bimagic squares with different bimagic sums, but the magic sums of all order 16, and order 4 are of equal sums.