Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
Double-Digi Cyclic-Type Bo de ed Reduced En ies
Algeb aic Magic Squa es o O de s 7 o 20
The whole wo k is also is a ailable a au ho ’s si es:
h ps://numbe s-magic.com/?p=17009
Inde J. Taneja1
Abs ac
This wo k b ings double-digi cyclic- ype algeb aic magic squa es o o de s 7 o 20 o educed en ies. By educed o
less en ies, we unde s and ha ins ead o no mal n2en ies o a magic squa e o de n, we a e using less numbe o en ies.
Mo eo e , in hese si ua ions he en ies a e no mo e sequen ial numbe s. These en ies a e non-sequen ial posi i e and nega i e
numbe s. Some imes, we call hese kind o magic squa es as sel -made. I means ha hese a e comple e in hemsel es. Jus
pu he alues o en ies and choose he magic sum, we ge a magic squa e. In some cases, he e maybe decimal o ac ional
alues o he en ies depending on he ypes o magic squa es. The idea o double-digi [30] is applied o b ing hese o magic
squa es. Mo eo e , we ha e conside ed he magic ec angles in a cyclic way, i.e, all he ou side in each case a e o equal sums
in wid hs and leng hs. Fo simila kind o wo k o diffe en o de s in diffe en s yles and ways, he eade s a e sugges ed o
see au ho ’s wo k [13, 14, 15, 16, 17, 18, 19, 20, 21].
1Fo me ly, P o esso o Ma hema ics, Uni e sidade Fede al de San a Ca a ina, Flo ian´
opolis, SC, B azil (1978-2012).
E-mail: [email p o ec ed];
Web-si es: h p://inde j aneja.wo dp ess.com; h p://numbe s-magic.com;
Twi e : @IJTANEJA
1
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
Con en s
1 In oduc ion 3
2 Double-Digi Algeb aic magic squa es 3
2.1 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.8 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.9 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.10Double-Digi Bo de ed Algeb aic Magic Squa e o O de 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.11Double-Digi Bo de ed Algeb aic Magic Squa e o O de 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.12Double-Digi Bo de ed Algeb aic Magic Squa e o O de 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.13Double-Digi Bo de ed Algeb aic Magic Squa e o O de 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.14Double-Digi Bo de ed Algeb aic Magic Squa e o O de 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Au ho ’s Con ibu ion o Magic Squa es and Rec ea ion o Numbe s 34
2
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
1 In oduc ion
This wo k b ings double-digi o double-laye algeb aic magic squa es o o de s 7 o 20 o educed en ies. Some imes, hese ypes
o magic squa es, we call as sel -made, beacause hey a e comple e in hemsel es. Jus choose he en ies and magic sum, we always ge
a magic squa e.
We know ha magic sum o a magic squa e o o de n ha ing 1 o n2numbe o en ies is gi en by
Sn×n:= n×(1 + n2)
2
In his wo k he en ies a e w i en as a iables and hei combina ions. The wo k is based on he ou equal sums magic ec angles
in each bo de wi h wid h as 2. The size o he leng h he magic ec angle depends on he o de s o he magic squa es. Fo simplici y,
hese ypes o magic squa es we call as cyclic- ype. Simila kind o s udy o he diffe en s yles and o de s e e au ho ’s wo k [13, 14,
15, 16, 17, 18, 19, 20, 21]. Fo double-digi wo k o sequen ial en ies e e [24, 25, 26, 27, 28, 29, 30].
2 Double-Digi Algeb aic magic squa es
The sec ion b ing esul s and examples o educed en ies algeb aic magic squa es comple e in i sel o he o de s 7 o 20. These a e
based on equal sums magic ec angles o equal wid h. Fo ou s udy we shall make use o educed en ies algeb aic magic squa es o
o de s 3, 4, 5 and 6. These a e gi en as ollows.
Resul 2.1. A magic squa e o o de 3 wi h educed en ies algeb aic is gi en by
•De ails
Knowing only wo en ies A1 and A2 and he magic sum M, we can cons uc a magic squa e o o de 3. To a oid decimal
en ies, we mus always conside he magic sum as a mul iple o 3. This magic squa e can seen in he web-si e o F. Gaspalou
[3].
Resul 2.2. Le ’s conside he ollowing magic squa e o o de 4
3
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
•De ails
I is a magic squa e o o de 4 o educed en ies. The le e S ep esen s he magic sum o o de 4. This magic squa e is also
due o F. Gaspalou [3].
Resul 2.3. Le ’s conside ollowing educed en ies magic squa e o o de 5:
•De ails
I is an algeb aic pandiagonal magic squa e o o de 5 wi h educed en ies. The le e M ep esen s he magic sum o o de
5. This magic squa e can seen in F. Gaspalou [3] web-si e. See below wo examples.
Resul 2.4. Le ’s conside a ollowing magic squa e o o de 6 wi h educed en ies:
•De ails
I is an algeb aic magic squa e o o de 6 o educed en ies. I cons uc ed based on ou equal sums magic squa es o de
3. The le e S ep esen s he magic sum o o de 3.
4
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
We shall equen ly use hese o magic squa e in ou wo k based on magic ec angles. These ou magic squa es shall also be in he
middle o he magic squa e. Magic squa es o o de s 3, 4 and 5 can be seen in F. Gaspalou’s si e [3]. The wo k on o de 6 is gi en by he
au ho [14, 15].
2.1 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 7
Resul 2.5. Le ’s conside a ollowing magic squa e o o de 7 wi h educed en ies:
•De ails
I is a composed o ou equal sums magic ec angles o o de 2×5embedded wi h a magic squa e o o de 3. Since he magic
squa e o o de 3 equi es magic sum as mul iple o 3, o he wise we ha e decimal en ies, hen he magic sum o o de 7 is also
mul iple o 3. The magic sum o o de 7 is gi en as S7×7:= 7S
3, whe e Sis he magic sum o o de 3.
Below a e wo examples based on he Resul 2.5.
Example 2.1. Le ’s conside ollowing wo examples based on he Resul 2.5:
5
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
The magic sums a e:
•Fi s example: S3×3:= 21 and S7×7:= 49.
•Second example: S3×3:= 24 and S7×7:= 56.
2.2 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 8
Resul 2.6. Le ’s conside a ollowing magic squa e o o de 8 wi h educed en ies:
•De ails
I is a composed o ou equal sums magic ec angles o o de 2×6embedded wi h a magic squa e o o de 4. Since he magic
squa e o o de 4 equi es magic sum as mul iple o 2, o he wise we ha e decimal en ies, hen he magic sum o o de 8 is also
6
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
mul iple o 2. The magic sum o o de 8 is gi en as S8×8:= 2 S, whe e Sis he magic sum o o de 4.
Below a e wo examples based on he Resul 2.6.
Example 2.2. Le ’s conside ollowing wo examples based on he Resul 2.6:
The magic sums a e:
•Fi s example: S4×4:= 26 and S8×8:= 52.
•Second example: S4×4:= 34 and S8×8:= 68.
2.3 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 9
Resul 2.7. Le ’s conside a ollowing magic squa e o o de 9 wi h educed en ies:
7
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
•De ails
I is a composed o ou equal sums magic ec angles o o de 2×7embedded wi h a pandiagonal magic squa e o o de 5.
He e he magic sum o o de 5 don’ ha e any condi ion, bu he magic sum o o de 9 depends on numbe 5, i.e., S9×9:= 9S
5,
whe e Sis he magic sum o o de 5. This equi es he magic sum o o de 9 should be mul iple o 5, o he wise we may ha e
decimal en ies.
Below a e wo examples based on he Resul 2.7.
Example 2.3. Le ’s conside ollowing wo examples based on he Resul 2.7:
8
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
The magic sums a e:
•Fi s example: S5×5:= 45 and S9×9:= 81.
•Second example: S5×5:= 50 and S9×9:= 90.
2.4 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 10
Resul 2.8. Le ’s conside a ollowing magic squa e o o de 10 wi h educed en ies:
•De ails
I is a composed o ou equal sums magic ec angles o o de 2×8embedded wi h a magic squa e o o de 6. This magic squa e
o o de 6 is again composed o ou equal sums magic squa es o o de 3. Since he magic squa e o o de 3 equi es magic
sum as mul iple o 3, o he wise we ha e decimal en ies, hen he magic sum o o de 10 is also a mul iple o 3. The magic
sum o o de 10 is gi en as S10×10 := 10 S
3, whe e Sis he magic sum o o de 3. In his case he magic sum o o de 6 is gi en as
S6×6:= 2 S.
Below a e wo examples based on he Resul 2.8.
Example 2.4. Le ’s conside ollowing wo examples based on he Resul 2.8:
9
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
The magic sums a e:
•Fi s example: S5×5:= 65,S9×9:= 117 and S13×13 := 169
•Second example: S5×5:= 70,S9×9:= 126 and S13×13 := 182.
2.8 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 14
Resul 2.12. Le ’s conside a ollowing magic squa e o o de 14 wi h educed en ies:
16
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
•De ails
I is a composed o ou equal sums magic ec angles o o de s 2×8and 2×12 (equal in each case) embedded wi h a magic
squa e o o de 6. This magic squa e o o de 6 is again composed o ou equal sums magic squa es o o de 3. Since he magic
squa e o o de 3 equi es magic sum as mul iple o 3, o he wise we may ha e decimal en ies. The magic sums o o de s 10
and 14 a e also mul iple o 3. The magic sum o o de s 10 and 14 a e gi en as S10×10 := 10 S
3and S14×14 := 14 S
3, whe e Sis he magic
sum o o de 3. In his case he magic sum o o de 6 is gi en as S6×6:= 2 S.
Below a e wo examples based on he Resul 2.12
Example 2.8. Le ’s conside ollowing wo examples based on he Resul 2.12:
17
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
The magic sums a e:
•Fi s example: S3×3:= 66,S6×6:= 132,S10×10 := 220 and S14×14 := 108.
•Second example: S3×3:= 75,S6×6:= 150,S10×10 := 250 and S14×14 := 350.
2.9 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 15
Resul 2.13. Le ’s conside a ollowing magic squa e o o de 15 wi h educed en ies:
18
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
•De ails
I is a composed o ou equal sums magic ec angles o o de s 2×8and 2×12 (equal in each case) embedded wi h a magic
squa e o o de 6. This magic squa e o o de 6 is again composed o ou equal sums magic squa es o o de 3. Since he magic
squa e o o de 3 equi es magic sum as mul iple o 3, o he wise we may ha e decimal en ies. The magic sums o o de s 10
and 14 a e also mul iple o 3. The magic sum o o de s 10 and 14 a e gi en as S10×10 := 10 S
3and S14×14 := 14 S
3, whe e Sis he magic
sum o o de 3. In his case he magic sum o o de 6 is gi en as S6×6:= 2 S.
Below a e wo examples based on he Resul 2.13
Example 2.9. Le ’s conside ollowing wo examples based on he Resul 2.13:
19
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
The magic sums a e:
•Fi s example: S3×3:= 63,S7×7:= 147,S11×11 := 231 and S15×15 := 315.
•Second example: S3×3:= 72,S7×7:= 168,S11×11 := 264 and S15×15 := 360.
2.10 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 16
Resul 2.14. Le ’s conside a ollowing magic squa e o o de 16 wi h educed en ies:
20
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
•De ails
I is a composed o ou equal sums magic ec angles o o de s 2×6,2×10 and 2×14 (equali y in each case) embedded wi h a
magic squa e o o de 4. Since he magic squa e o o de 4 equi es magic sum as mul iple o 2, o he wise we ha e decimal
en ies, hen he magic sum o o de s, 8, 12 and 16 a e also mul iple o 2. The magic sum o o de s 16, 12 and 8 a es gi en as
S16×16 := 4 S,S16×16 := 3 Sand S16×16 := 2 S, whe e Sis he magic sum o o de 4.
Below a e wo examples based on he Resul 2.14
Example 2.10. Le ’s conside ollowing wo examples based on he Resul 2.14:
21
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
The magic sums a e:
•Fi s example: S4×4:= 102,S8×8:= 204,S12×12 := 312 and S16×16 := 408.
•Second example: S4×4:= 110,S8×8:= 220,S12×12 := 330 and S16×16 := 440.
2.11 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 17
Resul 2.15. Le ’s conside a ollowing magic squa e o o de 17 wi h educed en ies:
22
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
•De ails
I is a composed o ou equal sums magic ec angles o o de s 2×7,2×11 and 2×15 (equali y in each case) embedded wi h a
pandiagonal magic squa e o o de 5. The magic squa e o o de 5 don’ equi es any condi ion bu he magic sums o o de s
17, 13 and 9 depends on i e. Thus, we mus ha e hese magic sums as mul iples o 5 o a oid decimal en ies. The magic sum
o o de s 17, 13 and 9 a es gi en as S17×17 := 17 S
5,S13×13 := 13 S
5, and S9×9:= 9S
5, whe e Sis he magic sum o o de 5.
Below a e wo examples based on he Resul 2.15
Example 2.11. Le ’s conside ollowing wo examples based on he Resul 2.15:
23
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
and
24
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
The magic sums a e:
•Fi s example: S5×5:= 85,S9×9:= 153,S13×13 := 221 and S17×17 := 289.
•Second example: S5×5:= 105,S9×9:= 189,S13×13 := 273 and S17×17 := 357.
2.12 Double-Digi Bo de ed Algeb aic Magic Squa e o O de 18
Resul 2.16. Le ’s conside a ollowing magic squa e o o de 18 wi h educed en ies:
25
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
The in e nal pa is he exac ly he same as gi en in he Resul 2.14. The comple e s uc u e can be seen in a web-si e link: h ps://numbe s-
magic.com/?p=17009.
•De ails
I is a composed o ou equal sums magic ec angles o o de s 2×6,2×10,2×14 and 2×18 (equali y in each case) embedded wi h
a magic squa e o o de 4. Since he magic squa e o o de 4 equi es magic sum as mul iple o 2, o he wise we ha e decimal
en ies, hen he magic sum o o de s, 8, 12 and 16 a e also mul iple o 2. The magic sum o o de s 20, 16, 12 and 8 a es gi en
as S20×20 := 5 S,S16×16 := 4 S,S12×12 := 3 Sand S8×8:= 2 S, whe e Sis he magic sum o o de 4.
Below a e wo examples based on he Resul 2.18
32
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
Example 2.14. Le ’s conside ollowing wo examples based on he Resul 2.18:
and
33
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
The magic sums a e:
•Fi s example: S4×4:= 102,S8×8:= 204,S12×12 := 312,S16×16 := 408 and S20×20 := 510.
•Second example: S4×4:= 152,S8×8:= 304,S12×12 := 456,S16×16 := 608 and S20×20 := 760.
3 Au ho ’s Con ibu ion o Magic Squa es and Rec ea ion o Numbe s
Fo au ho ’s con ibu ion o magic squa es and ec ea ion o numbe s please see he links below:
•Inde J. Taneja, Magic Squa es,
(i) h ps://numbe s-magic.com/?p=668
(ii) h ps://inde j aneja.wo dp ess.com/2019/06/27/publica ions-magic-squa es/
34
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
Zenodo, No embe 21, 2025, pp. 1-37, h ps://doi.o g/10.5281/zenodo.17675032
•Inde J. Taneja, Rec ea ion o Numbe s,
(i) h ps://numbe s-magic.com/?p=671
(ii) h ps://inde j aneja.wo dp ess.com/2019/06/27/publica ions- ec ea ion-o -numbe s/
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h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
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36
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Double-Digi Cyclic-Type Bo de ed Reduced En ies Algeb aic Magic Squa es o O de s 7 o 20,
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