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Different Types and Aspects of Magic Squares of Order 16

Author: Inder J. Taneja
Publisher: Zenodo
DOI: 10.5281/zenodo.17543834
Source: https://zenodo.org/records/17543834/files/BW-BB-New-16.pdf
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Diffe en Types and Aspec s o
Magic Squa es o O de 16
The whole wo k as pd iles is a ailable a au ho ’s si es:
h ps://numbe s-magic.com/?p=16911
Inde J. Taneja1
Abs ac
This wo k summa izes au ho ’s p e ious wo ks on magic squ es o o de 16. I b ings diffe en ypes o
magic squa e such as block-wise, single-digi bo de ed, double-digi bo de ed, co ne ed, designs,
s yles, e c. In e ms os aspec s, we ha e conside ed he idea o Py hago ean iples, upside-down,
mi o -looking, wa e e lec ion, pe ec squa e sum en ies, La in squa e dis ibu ions, e c.
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
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Con en s
1 In oduc ion 3
2 Diffe en Types o Magic Squa es 4
2.1 Block-WiseMagicSqua es ................................................ 5
2.2 Block-WiseBimagicSqua e................................................ 7
2.3 Single-Digi Bo de edMagicSqua es.......................................... 8
2.4 Embedded Single-Digi Bo de ed Magic Squa es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Co ne edMagicSqua e.................................................. 13
2.6 Double-Digi Bo de edMagicSqua es ......................................... 14
2.7 S ipedMagicSqua eso O de 16........................................... 17
2.8 Diffe en S yleso MagicSqua es............................................ 21
2.9 Fou -Digi sBo de edSqua es .............................................. 25
3 Diffe en Aspec s o Magic Squa es 29
3.1 La inSqua eDis ibu ions ................................................ 29
3.2 Palind omic wi h Equal Sum Blocks o O de 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Palind omicBimagicSqua e ............................................... 33
3.4 Pe ec Squa eEn iesSum................................................ 34
3.4.1 Uni o mi yP ope y ................................................ 34
3.4.2 Py hago eanT iple ................................................. 36
3.4.3 Minimum Pe ec Squa e En ies Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Upside-Down, Mi o -Looking and Wa e Re lec ion 39
4.1 Upside-DownandMi o Looking............................................ 40
4.2 Upside-Down and Mi o Looking wi h Magic Rec angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Wa e Re lec ionMagicSqua es............................................. 54
5 Au ho ’s Con ibu ion o Magic Squa es and Rec ea ion o Numbe s 61
2
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
1 In oduc ion
In he p e ious wo ks [21, 22, 23, 24, 25, 26, 27], he au ho wo ked wi h block-wise cons uc ions o magic squa es.
The wo k is om he o de s 8 o 45. In each case, all he possibili ies a e conside ed. These possibili ies a e based
on di isions o magic squa es. The magic sums o o de no consecu i e numbe s om 1 o n2is gi en by
Sn×n:= n×(1 + n2)
2, n ≥3.(1)
This o mula is applied o all o de magic squa es. Based on his o mula we shall b ink blocks o he block-wise
magic squa es. Tha is, whene e is possible, we shall y o b ing blocks o equal sum magic squa es. In some cases,
hey a e magic,semi-magic,pandiagonal, e c. When he ques ion come o bimagic squa es, in some cases, we
ha e semi-bimagic squa es.
On he o he hand he idea o bo de ed magic squa es is well explained in he wo k by H.Whi e [6, 7]. Few esul s
in his di ec ion can be seen in au ho ’s wo k [28, 29, 30, 31, 32, 33]. In some case, he magic ec angles a e also used
o w i e i in diffe en s yles. Mos o he au ho ’s wo k on magic squa es is summa ized below in de ails acco ding
o opic:
1. Digi al Fon s: Upside-down and Mi o Looking.
2. Two Digi s Uni e sal Magic Squa es.
3. Diffe en Digi s Magic Squa es.
4. Py hago ean T iples Magic Squa es.
5. Block-Wise Magic Squa es.
6. Sel ie and Palind omic Type Magic Squa es.
7. Block Bo de ed Magic Squa es.
8. Block-Wise Bo de ed Magic Squa es.
9. Magic C osses, Le e s and Numbe s.
10. Co ne ed Magic Squa es.
11. Single-Digi Bo de ed Magic Squa es.
12. Double-Digi Bo de ed Magic Squa es.
13. Mul iple-Digi Bo de ed Magic Squa es.
14. S iped Magic Squa es.
3
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
15. Upside-down and Mi o Looking.
16. Wa e e lec ion.
The aim o his wo k is o w i e magic squa es o o de 16 in diffe en ways and s yles using he aspec s o blocks,
bo de ed,block-bo de ed and magic ec angles,co ne - ype,single-digi ,double-digi , e c.
Mo e de ails on hese wo ks on magic squa es can be seen in au ho ’s web-si 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/
2 Diffe en Types o Magic Squa es
Acco ding o Equa ion (1), he magic sum o o de 16 is gi en by
S16×16 := 16 ×(1 + 162)
2= 2056.
We can w i e, 16 := 42= 4 ×4. This gi es he possibili y o blocks o o de 4 and 8. Le ’s see di ision o 2056 by 4
and 2:
(i)2056
4= 514 =⇒equal blocks o o de 4;
(ii)2056
2= 1024 =⇒equal blocks o o de 8.
Below a e ew examples o diffe en ypes o magic squa es o o de 16.
4
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
2.1 Block-Wise Magic Squa es
Example 2.1.
I is a block-wise pandiagonal magic squa e o o de 16 wi h equal sums pandiagonal magic squa es o o de 4.
5

Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 2.2.
I is a block-wise magic squa e o o de 16, whe e he blocks a e single-digi bo de ed magic squa es o o de 8.
Fo mo e de ails e e au ho ’s wo k [22, 24].
6
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
2.2 Block-Wise Bimagic Squa e
Example 2.3.
I is a bimagic squa e o o de 16. The blocks o o de 4 a e equal sums magic squa e o o de 4.
Fo mo e de ails e e au ho ’s wo k [13]
7
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
2.3 Single-Digi Bo de ed Magic Squa es
Example 2.4.
I is a single-digi bo de ed magic squa e o o de 16. Remo ing he highe bo de s s ill we a e le wi h magic
squa es o lowe o de s, such as o o de s 14, 12, 10,... e c. The in e nal block is a magic squa e o o de 4.
Fo mo e de ails e e au ho ’s wo k [24, 26]
8
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
2.4 Embedded Single-Digi Bo de ed Magic Squa es
Example 2.5.
I is a single-digi bo de ed magic squa e o o de 16 embedded wi h a magic squa e o o de 12 o med by equal
sums semi-magic squa es o o de 3.
9
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 2.12.
I is a alsodouble-digi bo de ed magic squa e o o de 16 w i en in li le diffe en way. In each blocks he magic
ec angles a e o equal wid h and leng h. We call hese ypes as cyclic magic ec angles. Remo ing he highe bo de s
s ill we a e le wi h magic squa es o lowe o de s, such as o o de s 12, 8 and 4. The in e nal block is a pandiagonal
magic squa e o o de 4.
Fo mo e de ails e e au ho ’s wo k [35, 36]
16

Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
2.7 S iped Magic Squa es o O de 16
Example 2.13.
I is a also double-digi bo de ed magic squa e o o de 16 o med by magic ec angle s ips o equal wid h.
The only diffe ence is in he leng h o each magic ec angle. These ypes i magic squa es we call as s iped magic
squa es.
17
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 2.14.
I is a also double-digi bo de ed magic squa e o o de 16 o med by magic ec angle s ips o equal wid h.
The only diffe ence is in he leng h o each magic ec angle. These ypes i magic squa es we call as s iped magic
squa es.
18
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 2.15.
I is a also double-digi co ne ype magic squa e o o de 16 o med by magic ec angle s ips o equal wid h.
The only diffe ence is in he leng h o each magic ec angle. These ypes i magic squa es we call as s iped magic
squa es.
19
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 2.16.
I is a also s iped magic squa e o o de 16 as i is o med by equal wid h and leng h o magic ec angles, i.e., 2×4.
Fo mo e de ails e e au ho ’s wo k [35, 43, 47].
20
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
2.8 Diffe en S yles o Magic Squa es
Example 2.17.
I is a magic squa e o o de 16 o med by magic, bo de ed magic and bo de ed magic ec angles.
21

Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 2.18.
I is a magic squa e o o de 16 o med by magic squa es and magic ec angles.
22
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 2.19.
I is a magic squa e o o de 16 o med by magic ec angles.
23
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 2.20.
I is a magic squa e o o de 16 o med by magic ec angles.
Fo mo de ails e e au ho ’s wo k [8, 9, 10, 12]
24
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
2.9 Fou -Digi s Bo de ed Squa es
Example 2.21.
I is a ou -digi bo de ed magic squa e o o de 16. Fou digi s means bo de o med by magic squa es o o de
4. In his case case easily eplace he inne pa o he magic squa e oming magic squa e o o de 8.
25
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
3.2 Palind omic wi h Equal Sum Blocks o O de 4
Example 3.2. Apandiagonal magic squa e o o de 16 wi h equal sum pandiagonal magic squa es o o de 4 is
gi en by
In his case, he magic sum is S16×16 = 479960. All he 4×4blocks a e pandiagonal magic squa es o o de 4 wi h
equal magic sums, S4×4= 119990.
32

Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
3.3 Palind omic Bimagic Squa e
Example 3.3. Abimagic squa e o o de 16 is gi en by
In his case, he magic and bimagic sums a e S16×16 := 479960 and Sb16×16 := 16484528520 espec i ely. De ails o his
bimagic squa e can be seen in au ho ’s wo k [?]. The blocks o o de 4 a e also magic squa es wi h equal magic sums
S4×4:= 119990
Fo mo e de ails e e au ho ’s wo k [14, 15]
33
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
3.4 Pe ec Squa e En ies Sum
Below a e i e diffe en magic squa es o o de 16 esul ing in uni o mi y,Py hago ean iples and minimum
pe ec squa e sum p ope ies. Ou o hese i e, h ee o hem a e wi h ac ion numbe s en ies.
3.4.1 Uni o mi y P ope y
Example 3.4. Ablock-wise pandiagonal magic squa e o o de 16 o consecu i e odd numbe s en ies {1,3,5, ..., 509,511}
is gi en by
34
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 3.5. Ablock-wise pandiagonal magic squa e o o de 16 o consecu i e ac ion numbe s en ies
{257/2,259/2,...,763/2,767/2}is gi en by
Bo h he Examples 3.4 and 3.5 a e wi h equal magic sums. The blocks o o de 4 a e pandiagonal magic squa es
wi h equal magic sums. See he de ails below:
S16×16 = 4096 = 163;T256 := 16 ×4096 = 65536 = 2562= 164
S4×4:= 1024; T16 := 4 ×1024 = 4096 = 642.
The Examples 3.4 and 3.5 also sa is y he uni o mi y p ope y, i.e., D16,162,163,164E.
35
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
3.4.2 Py hago ean T iple
Example 3.6. Ablock-wise pandiagonal squa e o o de 16 o consecu i e odd numbe s en ies {69,71,...,577,579}
is gi en by
36
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 3.7. Ablock-wise pandiagonal squa e o o de 16 o consecu i e ac ion numbe s en ies {393/2,395/2,...,901/2,903/2}
is gi en by
Bo h he Examples 3.6 and 3.7 a e wi h equal magic sums. The blocks o o de 4 a e pandiagonal magic squa es
wi h equal magic sums. See he de ails below:
S16×16 = 5184; T256 := 16 ×5184 = 82944 = 2882;
S4×4= 1296; T16 := 4 ×1296 = 5184 = 722.
Bo h he Examples 3.6 and 3.7 a e gene a ed by Py hago ean iple (34, 288, 290), i.e., 342+ 2882= 2902wi h
leas possible en ies esul ing in pe ec squa e en ies sum.
37

Inde J. Taneja
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Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
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3.4.3 Minimum Pe ec Squa e En ies Sum
Example 3.8. Ablock-wise pandiagonal magic squa e o o de 16 o consecu i e ac ion numbe s en ies
{33/2,35/2,...,541/2,543/2}is gi en by
The en ies sum is minimum pe ec squa e. The blocks o o de 4 a e pandiagonal magic squa es wi h equal
magic sums. See below he de ails:
S16×16 = 2304; T256 := 16 ×2304 = 36864 = 1922
S4×4= 576; T16 := 4 ×576 = 2304 = 482.
Fo mo e de ails e e au ho ’s wo k [16, 17, 18, 19, 20].
38
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
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4 Upside-Down, Mi o -Looking and Wa e Re lec ion
Le ’s conside he ollowing image:
Sou ce: h ps://www.ma hsis un.com/de ini ions/ e ical- lip.h ml
F om he abo e image we unde s and ha ho izon al lip is same as mi o looking image and he e ical lip
is same as wa e e lec ion image. The same e ms a e gi en in pain b ush o mic oso . Le ’s see how i wo ks
on numbe s.
Le ’s conside ollowing 9 digi s w i en in digi al o m:
•180oRo a ion
In his case, he eadable numbe s a e 0, 1, 2, 5, 6, 8 and 9 whe e he numbe 6 becomes 9 and 9 as 6. Thus he
su i al numbe s a e 180o o a ion a e 0, 1, 2, 5, 6, 8 and 9. Some imes we call hem as upside-down numbe s.
•Mi o Looking
I is same as ho izon al lip as desc ibe abo e. In his case, we ha e
39
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
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In his case, he eadable numbe s a e 0, 1, 2, 5 and 8, whe e 2 becomes 5 and 5 as 2. Thus he su i al numbe s
a e ho izon al lip a e 0, 1, 2, 5 and 8. Some imes we call hem as mi o -looking numbe s.
•Wa e Re lec ion
I is same as e ical lip as desc ibe abo e. In his case, we ha e
In his case, he eadable numbe s a e 0, 1, 2, 3, 5 and 8, whe e 2 becomes 5 and 5 as 2. Thus he su i al
numbe s a e e ical lip a e 0, 1, 2, 3, 5 and 8. Fo he i s ime we call hese numbe s as wa e e lec ion o
wa e e lexi e numbe s.
We obse e ha he numbe s 0, 1, 2, 5 and 8 a e in all he h ee si ua ions, i.e., hese a e upside-down,mi o -
looking and wa e e lexi e. We call hem as uni e sal numbe s p o ided hey a e w i en in digi al o m. The e
le only one numbe 3. I is only wa e e lexi e. While he numbe s 6 and 9 a e only upside-down.
The e is a lo o wo k by au ho on upside-down and mi o -looking numbe s. This wo k is concen a ed only
owa ds magic squa es ha ing wa e e lexi e numbe s, i.e., 0, 1, 2, 3, 5 and 8. The numbe s 0, 1, 2, 5 and 8 a e
al eady s udied p e iously. This wo k b ings magic squa es o o de 14 o 16 specially in numbe 3 along wi h 0, 1,
2, 5 and 8.
4.1 Upside-Down and Mi o Looking
This sec ion b ings magic and bimagic squa es o o de 16 w i en in such way ha , i we make 180o o a ion and/o
see in he mi o , s ill we a e able o ead he en ies again esul ing in magic squa es. I is possible only wi h he
digi s 0, 1, 2, 5, 6, 8 and 9 w i en in digi al o ms.
Example 4.1. The magic squa es o o de 16 o he digi s (1,2,5,8) is gi en by
40
Inde J. Taneja
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Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
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The abo e magic squa e o he digi s (1,2,5,8) is upside-down and mi o -looking, i.e., i is a uni e sal magic
squa es o o de 16 wi h magic sum S16×16(1,2,5,8) := 71104. The blocks o o de 4 a e magic squa es wi h diffe en
magic sums.
41
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Below a e h ee examples o bimagic squa es o o de 16 o he digi s (1,8),(2,5) and (6,9).
Example 4.8. The bimagic squa es o o de 16 o he digi s (1,8) is gi en by
The magic and bimagic sums o abo e magic squa e a e S16×16(1,8) := 799999992 and Sb16×16(1,8) := 59797978997979800.
The blocks o o de 4 a e magic squa es wi h equal magic sums. I is upside-down and mi o -looking, i.e., uni-
e sal magic squa e.
48

Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 4.9. The bimagic squa es o o de 16 o he digi s (2,5) is gi en by
The magic and bimagic sums o abo e magic squa e a e S16×16(2,5) := 622222216 and Sb16×16(2,5) := 27833894016610552.
The blocks o o de 4 a e magic squa es wi h equal magic sums. I is upside-down and mi o -looking, i.e., uni-
e sal magic squa e.
49
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 4.10. The bimagic squa es o o de 16 o he digi s (6,9) is gi en by
The magic and bimagic sums o abo e magic squa e a e S16×16(6,9) := 1333333320 and Sb16×16(6,9) := 114747472525252536.
The blocks o o de 4 a e magic squa es wi h equal magic sums. I is only upside-down.
50
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
4.2 Upside-Down and Mi o Looking wi h Magic Rec angles
Example 4.11. The magic squa es o o de 16 o med by magic ec angles o o de 2×4 o he digi s (1,8) is gi en
by
The magic sums o abo e magic squa e is S16×16(1,8) := 799999992. The blocks o o de 4 a e pandiagonal magic
squa es wi h equal magic sums. I is upside-down and mi o -looking, i.e., uni e sal magic squa e . I is also
pandiagonal.
51
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 4.12. The magic squa es o o de 16 o med by magic ec angles o o de 2×4 o he digi s (2,5) is gi en
by
The magic sums o abo e magic squa e is S16×16(2,5) := 622222216. The blocks o o de 4 a e pandiagonal magic
squa es wi h equal magic sums. I is upside-down and mi o -looking, i.e., uni e sal magic squa e . I is also
pandiagonal.
52
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 4.13. The magic squa es o o de 16 o med by magic ec angles o o de 2×4 o he digi s (6,9) is gi en
by
The magic sums o abo e magic squa e is S16×16(6,9) := 1333333320. The blocks o o de 4 a e pandiagonal magic
squa es wi h equal magic sums. I is only only upside-down, I is also pandiagonal.
53

Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
4.3 Wa e Re lec ion Magic Squa es
Below a e ew examples o magic squa e wi h wa e e lec ion. These examples a e o 4-digi s: (2,3,5,8), (1,2,3,5),
(0,2,3,5) and (0,1,3,8) wi h wa e e lec ion p ope y.
Example 4.14. Le ’s conside a magic squa e o o de 16 wi h 4-digi s (2,3,5,8) gi en by
The abo e magic squa es is bimagic. The magic sums a e S16×16(2,3,5,8) = 79992 and Sb16×16(2,3,5,8) = 484768488. The
blocks o o de 4 a e magic squa es wi h equal magic sums, i.e., S4×4(2,3,5,8) = 19998.
54
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
•Wa e Re lec ion
Applying he e ical lip o e he abo e magic squa e, we ge
Thus, he abo e magic squa e o o de 16 wi h digi s is (2,3,5,8) is wa e e lexi e and bimagic. The magic sums
a e S16×16(2,3,5,8) = 79992 and Sb16×16(2,3,5,8) = 484768488. The blocks o o de 4 a e magic squa es wi h equal magic
sums, i.e., S4×4(2,3,5,8) = 19998. Le ’s see below ew mo e examples.
55
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
Example 4.15. Le ’s eplace 8 by 1 in Example 4.14, we ge
I is bimagic squa e. The magic sums a e S16×16(1,2,3,5) = 48884 and Sb16×16(1,2,3,5) = 184706376. The blocks o o de
4 a e magic squa es wi h equal magic sums, i.e., S4×4(1,2,3,5) = 12221.
56
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
•Wa e Re lec ion
Applying he e ical lip o e he abo e magic squa e, we ge
I is bimagic squa e. The magic sums a e S16×16(1,2,3,5) = 48884 and Sb16×16(1,2,3,5) = 184706376. The blocks o o de
4 a e magic squa es wi h equal magic sums, i.e., S4×4(1,2,3,5) = 12221.
Thus, he abo e magic squa e is wa e e lexi e wi h equal magic sums. The blocks o o de 4 a e magic squa es
wi h equal magic sums.
57
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
•Block-Wise, Bo de ed and Block-Bo de ed Magic Squa es
[21] Inde J. Taneja, Block-Wise Cons uc ions o Magic and Bimagic Squa es o O de s 8 o 108, May 15, 2019, pp.
1-43, Zenodo,h ps://doi.o g/10.5281/zenodo.2843326.
[22] Inde J. Taneja, Block-Wise Equal Sums Pandiagonal Magic Squa es o O de 4k, Zenodo, Janua y 31, 2019,
pp. 1-17, h ps://doi.o g/10.5281/zenodo.2554288.
[23] Inde J. Taneja, Magic Rec angles in Cons uc ion o Block-Wise Pandiagonal Magic Squa es, Zenodo, Janua y
31, 2019, pp. 1-49, h ps://doi.o g/10.5281/zenodo.2554520.
[24] Inde J. Taneja, Block-Wise Magic and Bimagic Squa es o O de s 12 o 36, Zenodo, Feb ua y 1, 2019, pp. 1-53,
h ps://doi.o g/10.5281/zenodo.2555343.
[25] Inde J. Taneja, Block-Wise Magic and Bimagic Squa es o O de s 39 o 45, Zenodo, Feb ua y 2, 2019, pp. 1-73,
h p://doi.o g/10.5281/zenodo.2555889.
[26] Inde J. Taneja, Block-Wise and Block-Bo de ed Magic and Bimagic Squa es o O de s 10 o 47. Zenodo, Janua y
14, 2021, pp. 1-185, h ps://doi.o g/10.5281/zenodo.4437783.
[27] Inde J. Taneja, Bo de ed Magic Squa es Wi h O de Squa e Magic Sums, Zenodo, Janua y 20, 2020, pp. 1-26,
h p://doi.o g/10.5281/zenodo.3613690.
[28] Inde J. Taneja, Magic Squa es wi h Pe ec Squa e Sum o En ies: O de s 3 o 47. Zenodo. Augus 16, 2021,
pp. 1-317, h ps://doi.o g/10.5281/zenodo.5205214.
[29] Inde J. Taneja, Symme ic P ope ies o Nes ed Magic Squa es, Zenodo, June 29, 2019, pp. 1-55,
h ps://doi.o g/10.5281/zenodo.3262170.
[30] Inde J. Taneja, Bo de ed and Block-Wise Bo de ed Magic Squa es: E en O de Mul iples, Zenodo, Feb ua y
10, 2021, pp. 1-96, h ps://doi.o g/10.5281/zenodo.4527746.
[31] Inde J. Taneja, Nes ed Magic Squa es wi h Pe ec Squa e Sums, Py hago ean T iples, and Bo de s Diffe ences,
Zenodo, June 14, 2019, pp. 1-59, h ps://doi.o g/10.5281/zenodo.3246586.
[32] Inde J. Taneja, Magic and Semi-Magic Squa es wi h Blocks o Magic Rec angles, May 28, 2022, pp. 1-27,
Zenodo,h ps://doi.o g/10.5281/zenodo.6590637.
64

Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
[33] Inde J. Taneja, Magic Rec angles in Cons uc ion o Magic and Block Bo de ed Magic Squa es, June 03, 2022,
pp. 1-70, Zenodo,h ps://doi.o g/10.5281/zenodo.6621071.
•Mul iple O de s Bo de ed Magic Squa es
[34] Inde J. Taneja, Block-Wise Bo de ed and Pandiagonal Magic Squa es Mul iples o 4, Zenodo, Augus 31, 2021,
pp. 1-148, h ps://doi.o g/10.5281/zenodo.5347897.
•Double Digi s and Co ne ed Magic Squa es
[35] Inde J. Taneja, Two Digi s Bo de ed Magic Squa es Mul iples o 4: O de s 8 o 24, Zenodo, Ap il, 26, 2023,
pp. 1-43, h ps://doi.o g/10.5281/zenodo.7866956.
[36] Inde J. Taneja, New Concep s in Magic Squa es: Double Digi s Bo de ed Magic Squa es o O de s 7 o 108,
Zenodo, Augus 09, 2023, pp. 1-30, h ps://doi.o g/10.5281/zenodo.8230214.
[37] Inde J. Taneja, Co ne ed Magic Squa es o O de s 14 o 24, Zenodo, June 03, 2023, pp. 1-39,
h ps://doi.o g/10.5281/zenodo.8000471.
[38] Inde J. Taneja, New Concep s in Magic Squa es: Co ne ed Magic Squa es o O de s 5 o 81, Zenodo, Augus
09, 2023, pp. 1-27, h ps://doi.o g/10.5281/zenodo.8231157.
[39] Inde J. Taneja, Co ne ed Magic Squa es in Cons uc ion o Magic Squa es o O de s 16, 20, 24 and 28, Augus
23, 2023, h ps://numbe s-magic.com/?p=10172.
[40] Inde J. Taneja, New Concep s in Magic Squa es: Co ne ed Magic Squa es o O de s 5 o 108, Zenodo, Janua y
29, 2025, pp. 1-33, h ps://doi.o g/10.5281/zenodo.14759238.
•S iped Magic Squa es
[41] Inde J. Taneja, S iped Magic Squa es o E en O de s 4, 6, 8 and 10, Zenodo, No embe 10, 2023, pp. 1-34,
h ps://doi.o g/10.5281/zenodo.15228903.
65
Inde J. Taneja
h ps://inde j aneja.wo dp ess.com; h ps://numbe s-magic.com
Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo,
No embe 06, 2025, pp. 1-66, h ps://doi.o g/10.5281/zenodo.17543834
[42] Inde J. Taneja, S iped Magic Squa es o 12 – Re ised, Zenodo, Sep embe 07, 2024, pp. 1-30,
h ps://zenodo.o g/ eco ds/13725031.
[43] Inde J. Taneja, 5600+ S iped Magic Squa es o O de 16, Zenodo, Feb ua y 05, 2025, pp. 1-52,
h ps://doi.o g/10.5281/zenodo.14807639.
[44] Inde J. Taneja, S iped Magic Squa es o 18, Zenodo, June 13, 2024, pp. 1-34,
h ps://doi.o g/10.5281/zenodo.11629567.
[45] Inde J. Taneja, 8000+ S iped Magic Squa es o 20, Zenodo, Ma ch 15, 2025, pp. 1-37,
h ps://doi.o g/10.5281/zenodo.15032524.
[46] Inde J. Taneja, S iped and Semi-S iped Double Digi s Bo de ed Magic Squa es: O de s 7 o 50, Zenodo,
Ma ch 13, 2025, pp. 1-30, h ps://doi.o g/10.5281/zenodo.15021581.
[47] Inde J. Taneja, S iped and Semi-S iped Co ne ed Magic Squa es o O de s 6 o 50 – Rec ea ing Numbe s
and Magic Squa es, Ma ch 17, 2025, h ps://numbe s-magic.com/?p=15048
•Upside-down, Mi o Looking and Wa e Re lec ion Magic and Bimagic Squa es
[48] Inde J. Taneja, Upside-Down, Mi o Looking and Wa e Re lec ion Magic Squa es: O de s 14 o 16, Zenodo,
Janua y 15, 2024, pp. 1-140, h ps://doi.o g/10.5281/zenodo.14649519.
[49] Inde J. Taneja, 2-Digi s Uni e sal and Upside-Down Palind omic Magic and Bimagic Squa es: O de s 3 o 16,
Zenodo, Ap il 07, 2020, pp. 1-103, h ps://doi.o g/10.5281/zenodo.3743362.
•Diffe en Types and Aspec s o Magic Squa es
[50] Inde J. Taneja, Diffe en Types and Aspec s o Magic Squa es o O de 16, Zenodo, No embe 06, 2025, pp.
1-66, h ps://doi.o g/10.5281/zenodo.17543834
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