Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Sel -Made Algeb aic Semi-Magic Squa es o O de 11
Inde J. Taneja1
The whole wo k as pd iles is a ailable a au ho ’s si es:
h ps://numbe s-magic.com/?p=16767
This wo k is wi hou use o any kind o p og amming language
Abs ac
This wo k b ings semi-magic squa es o o de 11 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 en ies
depending on he ypes o magic squa es. Diffe en kind o magic squa es a e used o b ing hese sel -made magic squa es.
These a e o ype, block-wise,co ne ed,single-digi bo de ed,double-digi bo de ed, e c. In some cases, he idea o
magic ec angles is also applied. In each case, he magic ec angles a e conside ed wi h equal wid h and leng h. This
wo k o he sel -made algeb aic magic squa es o de 11. This wo k is a ailable online a abo e gi en link. Fo simila kind o
wo k o diffe en o de s he eade s a e sugges ed o au ho ’s wo k gi en in e e ences [30]-[45].
1Fo me ly, P o esso o Ma hema ics, Fede al Uni e si y o San a Ca a ina, Flo ianópolis, 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; Ins ag am: @c azynumbe s.
1
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Con en s
1 In oduc ion 2
2 Magic Squa es o O de 11 3
2.1 Sel -MadeSemi-MagicSqua eso o de 11 .................................................. 3
3 Au ho ’s Con ibu ion o Magic Squa es and Rec ea ion Numbe s 74
1 In oduc ion
This wo k b ings sel -made algeb aic semi-magic squa es o o de 11. By sel -made o 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 s o en ies, whe e he magic squa e is comple e
in i sel . Pu ing any in ege alues o hese less en ies, we shall ge always a magic squa e. 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/o nega i e numbe s. In some cases, hese may
be decimal o ac ional alues depending on he way o choosing he en ies. Some ime o a oid decimal o ac ional en ies we
apply ce ain condi ions. These condi ions depends on he ypes o magic squa es. The name sel -made is no known in he li e a u e o
magic squa es. I is being in oduced o he i s ime. The wo k is based on diffe en ypes o magic squa es, such as, pandiagonal,
block-wise, co ne ed, single-digi bo de ed, double-digi bo de ed, e c. I is no necessa y, bu we wo ked wi h magic ec angles
ha ing equal wid h and leng h o he same ca ego y wi hin a magic squa e. I we elax his condi ion, i.e., by conside ing only equali y
o wid h, s ill we ha e good esul s.
This wo k o he o de 11 b ings magic,semi-magic and pandiagonal magic squa es. I is di ided in h ee pa s. The i s pa is
on magic squa es, he second pa on semi-magic squa es and he hi d pa on pandiagonal magic squa es o o de 11. This pa is
on semi-magic squa es o o de 11. Fo he i s pa on magic squa es e e [42]. Fo mo e de ails on hese kind o wo k e e au ho ’s
p e ious wo ks [30]-[45]. The able below gi e he quan i ies and e e ences o each o de .
O de Magic Squa es Semi-Magic Squa es Pandiagonal Magic Squa es To al
31 1 0 2[38]
42 0 2 4[38]
52 1 3 6[38]
65 1 6 12 [38]
75 3 8 16 [38]
88 5 13 26 [39]
910 9 18 28 [40]
10 15 14 29 58 [41]
11 25 23 o be done 48 [42, 43]
12 28 25 o be done 53 [44, 45]
2
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
The au ho [30, 31, 32, 33, 34, 35] also wo ked simila kind o wo k bu om diffe en poin o iew. This wo k is o he magic squa es
o o de s 3 o 12 o he da es and days o he yea 2025, whe e he da es a e ew en ies and days a e he sums o he magic squa es.
2 Magic Squa es o O de 11
Below a e h ee diffe en examples o magic squa es o o de 11 o sequen ial en ies om 1 o 121.
The i s example is known by co ne ed magic squa es. The second example is amous as single-digi bo de ed magic squa e. The
hi d example is known as double-digi bo e ed magic squa e. Fo mo e de ails on hese kind o magic squa es e e au ho ’s wo k
[24, 25, 26, 27].
3
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
2.1 Sel -Made Semi-Magic Squa es o o de 11
Below a e 23 examples o sel -made semi-magic squa es o educed en ies. In o de o b ing hem as magic squa es we shall use
some condions. These condi ions a e gi en below:
R:= 11
9
×L(1)
L:= 9
7
×T(2)
T:= 7
5
×S(3)
S:= 5
3
×M(4)
whe e he le e s M, S, T, L and R ep esen s he magic o semi-magic squa es o o de s 3,5,7,9and 11.
4
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.1. Le ’s conside ollowing sel -made semi-magic squa e o o de 11 wi h educed en ies:
•De ails:
I is a double-digi bo de ed magic squa e o o de 11 embedded wi h a single-digi bo de ed magic squa e o o de 7
ha ing pandiagonal magic squa e o o de 5 in he middle. The magic ec angles o o de s 2×7a e o equal wid h and leng h.
The le e s S, T and R ep esen s he magic squa es o o de s 5, 7 and 11 espec i ely. The diffe ence be ween R and T should
be mul iple o 4 o a oid decimal en ies. I is a semi-magic squa e a one diagonal. I becomes magic squa e by applying he
condi ion gi en in (3), i.e., T:= 7
5×S. See below wo examples. One is semi-magic squa e, and he second one is magic squa e.
5
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.1. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.1:
The semi-magic squa e gi en in Example 2.1 includes he ollowing magic and semi-magic squa es:
6
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.2. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.1:
I includes he ollowing magic squa es:
7
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.2. Le ’s conside ollowing sel -made semi-magic squa e o o de 11 wi h educed en ies:
•De ails:
I is a double-digi bo de ed magic squa e o o de 11 embedded wi h a single-digi bo de ed magic squa e o o de 7.
I again con ains a co ne ed magic squa e o o de 3 a he uppe -le co ne . The magic ec angles o o de s 2×3and 2×7
a e o equal wid h and leng h in each case. The le e s M, S, T and R ep esen s he magic squa es o o de s 3, 5, 7 and 11
espec i ely. I is a semi-magic squa e a one diagonal. I becomes magic squa e by applying he condi ion gi en in (3), i.e.,
T:= 7
5×S. Mo eo e he magic squa e o o de 3 should be mul iple o 3 o a oid decimal en ies. Also he diffe ence be ween
R and T should be mul iple o 4. See below wo examples. One is semi-magic squa e, and he second one is magic squa e.
8
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.3. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.2:
The semi-magic squa e gi en in Example 2.3 includes he ollowing magic and semi-magic squa es:
9
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.8. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.4:
The abo e Example 2.8 is ob ained by applying he condi ion R:= 11
9×L. I includes he ollowing magic squa es:
16
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.5. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de 11 embedded wi h a double-digi bo de ed magic squa es o o de 9
ha ing a co ne ed magic squa e o o de 5 in he middle. I con ains magic squa e o o de 3 a he uppe -le co ne . The
magic ec angles o o de 2×3and 2×5a e o equal wid h and leng h in each case. The le e s M, S, L and R ep esen s he
magic squa es o o de s 3, 5, 9 and 11 espec i ely. I is a semi-magic squa e a one diagonal. I becomes magic squa e by
applying he condi ion gi en in (1), i.e., R:= 11
9×L. To a oid decimal en ies he diffe ence be ween S and L should be mul iple
o 4. Mo eo e , he magic squa e o o de 3 should be mul iple o 3. See below wo examples. One is semi-magic squa e, and
he second one is magic squa e.
17
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.9. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.5:
The magic squa e gi en in Example 2.9 includes he ollowing magic and semi-magic squa es:
18
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.10. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.5:
The abo e Example 2.10 is ob ained by applying he condi ion R:= 11
9×L. I includes he ollowing magic squa es:
19
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.6. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de 11 embedded wi h a double-digi bo de ed magic squa es o o de 9
ha ing again a single-digi bo de ed magic squa e o o de 5, whe e magic squa e o o de 3 is in he middle. I con ains
magic squa e o o de 3 a he uppe -le co ne . The magic ec angles o o de 2×5a e o equal wid h and leng h. The le e s
M, S, L and R ep esen s he magic squa es o o de s 3, 5, 9 and 11 espec i ely. I is a semi-magic squa e a one diagonal. I
becomes magic squa e by applying he condi ions gi en in (1) and (4), i.e., R:= 11
9×Land S:= 5
3×M. To a oid decimal en ies
he diffe ence be ween S and L should be mul iple o 4. Mo eo e , he magic squa e o o de 3 should be mul iple o 3. See
below wo examples. One is semi-magic squa e, and he second one is magic squa e.
20
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.11. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.6:
The magic squa e gi en in Example 2.11 includes he ollowing magic and semi-magic squa es:
21
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.12. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.6:
The abo e Example 2.12 is ob ained by applying he condi ions R:= 11
9×Land S:= 5
3×M. I includes he ollowing magic squa es:
22
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.7. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de 11 embedded wi h a co ne ed magic squa es o o de 9 ha ing a double-
digi bo de ed magic squa e o o de 7 a he uppe -le co ne . I con ains magic squa e o o de 3 in he middle. The magic
ec angles o o de s 2×3and 2×7a e o equal wid h and leng h. The le e s M, T, L and R ep esen s he magic squa es o o de s
3, 7, 9 and 11 espec i ely. I is a semi-magic squa e a one diagonal. I becomes magic squa e by applying he condi ion gi en
in (1), i.e., R:= 11
9×L. To a oid decimal en ies he diffe ence be ween T and S should be mul iple o 4. Mo eo e , he magic
squa e o o de 3 should be mul iple o 3. See below wo examples. One is semi-magic squa e, and he second one is magic
squa e.
23
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.13. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.7:
The magic squa e gi en in Example 2.13 includes he ollowing magic and semi-magic squa es:
24
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.14. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.7:
The abo e Example 2.14 is ob ained by applying he condi ion R:= 11
9×L. I includes he ollowing magic squa es:
25
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.10. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de 11 embedded wi h a co ne ed magic squa es o o de 9 and 7 ha ing a
single-digi bo de ed magic squa e o o de 5 a he uppe -le co ne . I con ains a magic squa e o o de 3 in he middle.
The magic ec angles o o de s 2×5and 2×7a e o equal wid h and leng h in each case. The le e s M, S, T, L and R ep esen s
he magic squa es o o de s 3, 5, 7, 9 and 11 espec i ely. I is a semi-magic squa e a one diagonal. I becomes magic squa e
by applying he condi ion gi en in (1), i.e., R:= 11
9×L. To a oid decimal en ies he pai s (M, S), (S, T) and (T, L) should be
mul iple o 2. Mo eo e , he magic sum o o de 3 should be mul iple o 3. See below wo examples. One is semi-magic squa e,
and he second one is magic squa e.
32
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.19. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.10:
The magic squa e gi en in Example 2.19 includes he ollowing magic and semi-magic squa es:
33
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.20. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.10:
The abo e Example 2.20 is ob ained by applying he condi ion R:= 11
9×L. E en hough i is no necesa y, bu an ex a condi ion
S:= 5
3×Mis applied o b ing a semi-magic suua e o o de 5 as magic squa e. See below he magic squa es appea ing in second
example o Example 2.19.
34
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.11. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de 11 embedded wi h a co ne ed magic squa es o o de 9 ha ing a magic
squa e o o de 7 a he uppe -le co ne . The magic ec angles o o de 2×7a e o equal wid h and leng h. The le e s T, L
and R ep esen s he magic squa es o o de s 7, 9 and 11 espec i ely. I is a semi-magic squa e a one diagonal. I becomes
magic squa e by applying he condi ion gi en in (1), i.e., R:= 11
9×L. To a oid decimal en ies he pai s (T, L) should be mul iple
o 2. See below wo examples. One is semi-magic squa e, and he second one is magic squa e.
35
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.21. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.11:
The magic squa e gi en in Example 2.21 includes he ollowing magic and semi-magic squa es:
36
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.22. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.11:
The abo e Example 2.22 is ob ained by applying he condi ion R:= 11
9×L. I includes he ollowing magic squa es:
37
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.12. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de 11 embedded wi h a co ne ed magic squa es o o de 9 ha ing a single-
digi bo de ed magic squa e o o de 7 a he uppe -le co ne embedded wi h a pandiagonal magic squa e o o de 5 in
he middle. The magic ec angles o o de 2×7a e o equal wid h and leng h. The le e s S, T, L and R ep esen s he magic
squa es o o de s 5, 7, 9 and 11 espec i ely. I is a semi-magic squa e a one diagonal. I becomes magic squa e by applying
he condi ion gi en in (1), i.e., R:= 11
9×L. To a oid decimal en ies he pai s (T, L) should be mul iple o 2. See below wo
examples. One is semi-magic squa e, and he second one is magic squa e.
38
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.23. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.12:
The magic squa e gi en in Example 2.23 includes he ollowing magic and semi-magic squa es:
39
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.24. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.12:
The abo e Example 2.24 is ob ained by applying he condi ion R:= 11
9×L. An ex a condi ion R:= 7
5×Lis applied o b ing a semi-magic
squa e o o de 7 o a magic squa e. I includes he ollowing magic squa es:
40
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.13. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de 11 embedded wi h a co ne ed magic squa es o o de 9 ha ing a single-
digi bo de ed magic squa e o o de 7 a he uppe -le co ne . I is again embedded wi h a co ne ed magic squa e o o de
5 ha ing a magic squa e o o de 3 a he uppe -lel co ne . The magic ec angles o o de s 2×3and 2×7a e o equal wid h
and leng h in each case. The le e s M, S, T, L and R ep esen s he magic squa es o o de s 3, 5, 7, 9 and 11 espec i ely. I
is a semi-magic squa e a one diagonal. I becomes magic squa e by applying he condi ion gi en in (1), i.e., R:= 11
9×L. To
a oid decimal en ies he pai s (M, S) and (T, L) should be mul iple o 2. The magic squa e o o de 3 should also be mul iple o
3. See below wo examples. One is semi-magic squa e, and he second one is magic squa e.
41
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.29. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.15:
The magic squa e gi en in Example 2.7 includes he ollowing magic and semi-magic squa es:
48
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.30. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.15:
The abo e Example 2.30 is ob ained by applying he condi ions R:= 11
9×Land L:= 9
7×T. I includes he ollowing magic squa es:
Ins ead o conside ing semi-magic squa es o o de 3, we can also conside magic squa es o o de 3. In his case we need he
condi ion ha sums o o de 3 should be mul iple o o de 3 o a oid decimal en ies.
49
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.16. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de s 11 and 9 embedded wi h a double-digi bo de ed magic squa es o
o de 7 ha ing a magic squa e o o de 3 in he middle. I is a semi-magic squa e a one diagonal. I becomes magic squa e
by applying he condi ions gi en in (1) and (2), i.e., R:= 11
9×Land L:= 9
7×T. The le e M, T, L and R ep esen s he magic
squa es o o de s 3, 7, 9 and 11 espec i ely. To a oid decimal en ies we mus ha e he pai s (T, L) and (L, R) as mul iple o
2. Also magic squa e o o de 3 should be mul iple o 3. See below wo examples. One is semi-magic squa e, and he second
one is magic squa e.
50
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.31. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.16:
The magic squa e gi en in Example 2.31 includes he ollowing magic and semi-magic squa es:
51
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.32. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.16:
The abo e Example 2.32 is ob ained by applying he condi ions R:= 11
9×Land L:= 9
7×T. I includes he ollowing magic squa es:
52
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.17. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de s 11 and 9 embedded wi h a magic squa es o o de 7. I is a semi-magic
squa e a one diagonal. I becomes magic squa e by applying he condi ions gi en in (1) and (2), i.e., R:= 11
9×Land L:= 9
7×T.
The le e T, L and R ep esen s he magic squa es o o de s 7, 9 and 11 espec i ely. To a oid decimal en ies we mus ha e
he pai s (T, L) and (L, R) as mul iple o 2. See below wo examples. One is semi-magic squa e, and he second one is magic
squa e.
53
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.33. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.17:
The magic squa e gi en in Example 2.33 includes he ollowing magic and semi-magic squa es:
54
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.34. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.17:
The abo e Example 2.20 is ob ained by applying he condi ions R:= 11
9×Land L:= 9
7×T. I includes he ollowing magic squa es:
55
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.18. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de s 11 and 9 embedded wi h a co ne ed magic squa es o o de 7. I con ains
apandiagonal magic squa e o o de 5 a he uppe -le co ne . I is a semi-magic squa e a one diagonal. I becomes magic
squa e by applying he condi ions gi en in (1) and (2), i.e., R:= 11
9×Land L:= 9
7×T. The magic ec angles o o de 2×5a e o
equal wid h and leng h. The le e S, T, L and R ep esen s he magic squa es o o de s 5, 7, 9 and 11 espec i ely. See below
wo examples. One is semi-magic squa e, and he second one is magic squa e.
56
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.35. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.18:
The magic squa e gi en in Example 2.35 includes he ollowing magic and semi-magic squa es:
57
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.40. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.20:
The abo e Example 2.40 is ob ained by applying he condi ions R:= 11
9×Land L:= 9
7×T. An ex a condi ion S:= 5
3×Mis also applied
o b ing block o o de 5 as a magic squa e. I includes he ollowing magic squa es:
64
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.21. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de s 11, 9 and 7 embedded wi h a pandiagonal magic squa es o o de 5.
I is a semi-magic squa e a one diagonal. I becomes magic squa e by applying he condi ions gi en in (1),(1), and (2) and
(3), i.e., R:= 11
9×L,L:= 9
7×Tand T:= 7
5×S. The le e s S, T, L and R ep esen s he magic squa es o o de s 5, 7, 9 and 11
espec i ely. See below wo examples. One is semi-magic squa e, and he second one is magic squa e.
65
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.41. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.21:
The magic squa e gi en in Example 2.41 includes he ollowing magic and semi-magic squa es:
66
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.42. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.21:
The abo e Example 2.42 is ob ained by applying he condi ions R:= 11
9×L,L:= 9
7×Tand T:= 7
5×S. I includes he ollowing magic
squa es:
67
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.22. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de s 11, 9 and 7 embedded wi h a co ne ed magic squa es o o de 5. I
con ains a magic squa e o o de 3 a he uppe - le co ne . I is a semi-magic squa e a one diagonal. I becomes magic
squa e by applying he condi ions gi en in (1),(1), and (2) and (3), i.e., R:= 11
9×L,L:= 9
7×Tand T:= 7
5×S. The le e s S, T, L
and R ep esen s he magic squa es o o de s 5, 7, 9 and 11 espec i ely. To a oid decimal en ies we mus ha e he (S,M) as
mul iple o 2. Mo eo e , he magic squa e o o de 3 should also be mul iple o 3. See below wo examples. One is semi-magic
squa e, and he second one is magic squa e.
68
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.43. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.22:
The magic squa e gi en in Example 2.43 includes he ollowing magic and semi-magic squa es:
69
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.44. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.22:
The abo e Example 2.44 is ob ained by applying he condi ions R:= 11
9×L,L:= 9
7×Tand T:= 7
5×S.. I includes he ollowing magic
squa es:
70
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Resul 2.23. Le ’s conside ollowing magic squa es o o de 11 wi h educed en ies:
•De ails:
I is a single-digi bo de ed magic squa e o o de s 11, 9, 7 and 5 embedded wi h a magic squa e o o de 3. I is a semi-
magic squa e a one diagonal. The le e s M, S, T, L and R ep esen s he magic squa es o o de s 3, 5, 7, 9 and 11 espec i ely.
I becomes magic squa e by applying he condi ions gi en in (1)-(4), i.e., R:= 11
9×L,L:= 9
7×T,T:= 7
5×Sand S:= 5
3×M. See
below wo examples. One is semi-magic squa e, and he second one is magic squa e.
71
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.45. Le ’s conside an example o semi-magic squa e o o de 11 based on he Resul 2.23:
The magic squa e gi en in Example 2.45 includes he ollowing magic and semi-magic squa es:
72
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com;
Sel -Made Algeb aic Magic and Semi-Magic Squa es o O de 11,
Zenodo, Oc obe 12, 2025, pp. 1-77, h ps://doi.o g/10.5281/zenodo.17330822
Example 2.46. Le ’s conside an example o magic squa e o o de 11 based on he Resul 2.23:
The abo e Example 2.46 is ob ained by applying he condi ions R:= 11
9×L,L:= 9
7×T,T:= 7
5×Sand S:= 5
3×M. I includes he ollowing
magic squa es:
73
