special magic squares of order six and eight

8/12/2019 SPECIAL MAGIC SQUARES OF ORDER SIX AND EIGHT

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Special Magic Squares of Order Six and Eight

S. Al-Ashhab

Assistant Professor

Department of Mathematics, Al-albayt University,

Visiting ProfessorUm Alqura university (KSA)

Email: [email protected]

Abstract:

In this paper we introduce and study special

types of magic squares of order six. We list

some enumerations of these squares. We

present a parallelizable code. This code is

based on the principles of genetic algorithms.

KeywordsMagic Squares, Four Corner Property, Parallel

Computing, Search Algorithms, Nested loops.

1 Introduction

A magic square is a square matrix, where

the sum of all entries in each row or column

and both main diagonals yields the samenumber. This number is called the magic

constant. A natural magic square of order n is

a matrix of size nn such that its entries

consists of all integers from one to n. The

magic constant in this case is2

1)n(n + . A

symmetric magic square is a natural magic

square of order n such that the sum of all

opposite entries equals n+1. For example,

Table 1a natural symmetric magic square

15 14 1 18 17

19 16 3 21 6

2 22 13 4 24

20 5 23 10 7

9 8 25 12 11

A pandiagonal magic square is a magicsquare such that the sum of all entries in all

broken diagonals equals the magic constant.

For example, we note in table 2 that the sum ofthe entries 39,12,46,22,20,23,13 is 175, whichis the magic sum. These entries represent the

first right broken diagonal.

Table 2 a natural pandiagonal and symmetric magic

square of order seven

1 39 34 21 35 8 37

27 9 12 36 24 19 48

40 30 17 46 7 32 3

45 6 28 25 22 44 5

47 18 43 4 33 20 10

2 31 26 14 38 41 23

13 42 15 29 16 11 49

In the seventeenth century Frenicle de

Bessy claimed that the number of the 4x4magic squares is 880, where he considered a

magic square with all its reflections and

rotations one square. Hire listed them all in atable in the year 1693. Recently we can use thecomputer to check that there are

880*8 = 7040

magic squares of order 4.

In 1973 the number of all natural magic

squares of order five became known.

Schoeppel computed it using a PDF-10

machine. It is64 826 306*32=2 202 441 792

where we multiply with 32 due the existenceof type preserving transformations. According

to [5] there exists

736 347 893 760

natural nested magic squares of order six.

It is well-known that there are pandiagonalmagic squares and symmetric squares of order

five. But, there are neither pandiagonal magic

squares nor symmetric squares of order six.

International Journal of Digital Information and Wireless Communications (IJDIWC) 1(4): 733-745The Society of Digital Information and Wireless Communications, 2011(ISSN 2225-658X)

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The number of natural magic squares of order

six is actually till now unknown. Trump madeusing statistical methods (Monte Carlo

Backtracking) the following intervalestimation for this number

(1.7712e19, 1.7796e19)

with a probability of 99%.We give here the number of a subset of

such squares. We define here classes of magic

squares of order six, which satisfy some of the

conditions for both types.

The most-perfect pandiagonal magic

squares of McClintock (cf. [11]) for which

Ollerenshaw and Bres (cf. [10])

combinatorial count ranks as a majorachievement, draw attention to another typewhich have the same sum for all 2 by 2

subsquares (or quartets). The number of

complete magic squares of order four is 48,

and the number of complete magic squares of

order eight (cf. [10]) is

368 640.

Ollerenshaw and Bre (cf. [10]) have a patent

for using most-perfect magic squares forcryptography, and Besslich (cf. [7] and [8])

has proposed using pandiagonal magic squaresas dither matrices for image processing.

A pandiagonal and symmetric magic square

is called ultramagic. According to [14] thenumber of ultramagic squares of order five is

16 and number of ultramagic squares of orderseven is

20 190 684.

The weakest property of a square is being

semi magic. By this concept we mean a

matrix, where the sum of all entries in eachrow or column yields the magic constant.

According to Trump (cf. [14]) the number of

semi magic squares of order four is68 688,

and the number of semi magic squares of order

five is

579 043 051 200.

Bi-magic squares are magic squares that when

the entries are squared, it also forms a magicsquare. Here is a bi-magic square of order six

Table 3a bi-magic square

17 36 55 124 62 114

58 40 129 50 111 20

108 135 34 44 38 49

87 98 92 102 1 28

116 25 86 7 96 78

22 74 12 81 100 119

We focus in this paper on the following

kind of magic squares: magic squares of order

6 )(a ij with magic constant 3s such that

ija +

3)3)(j(ia

+++

3)i(ja

++

3)j(ia

+= 2s

holds for each i=1,2,3 and j=1,2,3 and

33a +

44a +

34a +

43a = 2s.

We call such squares four corner magic square

of order 6. The entries of a four corner magic

square of order 6 satisfy

14a +

25a +

36a +

41a +

52a +

63a =3s,

13a +

22a +

31a +

46a +

55a +

64a = 3s

These two conditions represent the sum of theentries of two broken diagonals. If the magic

square is pandiagonal, then we have to

consider all broken diagonals. To see the

validity of the first equation we know from the

definition that

11a +

44a +

14a +

41a = 2s,

22a +

55a +

25a +

52a =2s,

33a +

66a +

36a +

63a =2s

holds. Adding up these equations andsubtracting from them the following equation

11a +

22a +

33a +

44a +

55a +

66a = 3s

yields the desired equation.

We introduce now the main concept in our

work. We call a four corner magic squares

)(aij

such that

33a +

44a = s and

34a +

43a = s.


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a four corner magic square of order 6 with

symmetric center. This means that the 2 by 2square in the center is symmetric. A four

corner magic squares with symmetric center isa four corner magic square of order 6 can be

written as

Table 4 a symbolic four corner magic squares with

symmetric center

x f g t G M

z h n j q N

w E e a m D

A k sa se H R

2sj

oz

p D o 2sp

qh

T

B F W J L p+qx

whereA=e+s t x,

B = j+o+t e w,

D=d+g+n+x a p q,

E=3s a e m w D,

F=3s f h k p E,

G = j+o+p+q+s e f g w x,

H=e+g+s+w+x j k o p q,

J=2s+e j o a t,M=3s f g t x G,

N=3s h j n q z,

L=f+h+k+p m s,R=s+a+e k A H,

T=h+j+q+z d s,

W=a+2s d e g n.

Table 5 a natural four corner magic squares with

symmetric center

6 23 11 13 33 25

19 28 36 3 7 182 29 1 17 27 35

21 8 22 34 10 16

32 9 15 20 30 5

31 14 26 24 4 12

We see that it has seventeen independent

variables. We can consider a special class of

the class of four corner magic squares with

symmetric center. The squares, which can bewritten in the following form, will be called

four corner magic squares with double

symmetric center.

Table 6 a symbolic four corner magic squares withdouble symmetric center

x f g t I s+w+ej

ot

Z h s o j q o+2sj

hqz

w sR e a m g+s+xajo

A sm s a se R a+j+o+t

egw

2sj

oz

sq sj o sh h+2j+q+

z 2sB K a+j+o

eg

J Q sx

where

A=e+s t x,

B = j+o+t e w,I = j+o+2s e f g w x,

J = 2s+e j o a t,

K = e+g+2m+q+w+x f h j o s,

Q = f+h+s 2m q,

R = e+g +m+w+x j o s,

2 Four corner magic squareThis concept was first introduced in [1].

Alashhab considered there the type called

nested four corner magic square with a

pandiagonal magic square. By this kind of

squares we mean matrices having the

following structure

YpdcZb37fA14A13A12A11f37XA24A23A22A21X37WA34A33A32A31W37VA44A43A42A41V37b37p37d37c37Z37Y

where


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V=37 +afj,

W=37+gbd,X=111 a b c g h,

Y=111 b c d h j,Z= h+j p,

and the inside square (matrix A) is

74ghjjhgh+jaa+gj74agha37h37gg+h+j3737j

a+g+h2s37a37+ahj37+jag

We note that the inside 4x4 square is apandiagonal square. The number of the naturalmagic squares of this kind is

32*79118 = 2531776

Here the number 32 refers to the 32 type

preserving transformations, which are any

mixture of the following transformations:

1) Rotation with angle2 counter

clockwise,

2) Transpose of the matrix,3) Exchange the second and fifth entry of

the first and last row,

4) Exchange the second and fifth entry ofthe first and last column.

2.1. Property preserving transformations

There are seven classical transformations,which take a magic square into another magic

square. They are the combinations of the

rotations with angles /2, , (3)/2 and

transpose operation. Now, a four corner magic

squares with symmetric center can be

transformed as follows into another one of the

same kind: we make these interchanges

simultaneously: interchange12

a (res.62

a )

with15

a (res.65

a ), interchange21

a (res.26

a )

with51

a (res.56

a ), interchange22

a (res.55

a )

with25

a (res.52

a ), interchange23

a (res.

24a ) with

53a (res.

54a ), interchange

32a (res.

42a ) with

35a (res.

45a ).

We can use this transformation to reducethe number of computed natural magicsquares. In order to eliminate the effect of the

previous transformations we compute all

natural four corner magic squares with

symmetric center for which the following

conditions hold:

p


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We consider the problem of counting the

natural four corner magic squares with doublesymmetric center. Regarding these squares we

eliminate the effect of the seven classicaltransformations by requiring

q< h , 1 q 17, h 18 ..... (2.1)

When we hold the entries q and h, then we

can transform the square into another square

of the same kind. This is possible by

interchanging the first row (res. column) withthe last row (res. row) and simultaneously

interchanging the two middle rows (res.

columns). Hence, the number of naturalsquares by fixed values of q and h is divisible

by four. Actually, there is a third propertypreserving transformation. If we take the dual

of the square, and reflect about the main right

diagonal followed by a reflection about the

main left diagonal, then we obtain a naturalfour corner magic squares with double

symmetric satisfying (2.1). For example, the

following square is a natural four corner magic

squares with double symmetric satisfying the

condition (2.1):

15 24 9 16 33 14

32 7 27 19 1 25

20 26 2 3 31 29

8 6 34 35 11 17

13 36 18 10 30 4

23 12 21 28 5 22

This square will be transformed according to

the last transformation into the followingsquare

15 32 9 16 25 14

33 7 27 19 1 24

20 26 2 3 31 29

8 6 34 35 11 17

12 36 18 10 30 5

23 4 21 28 13 22

Hence, the number of natural four corner

magic squares with double symmetric centerby fixed values of q and h is divisible by eight.

2.2. Number of squares

We used computers to count several types

of magic squares. The algorithm is constructed

in such a way that we take specific values at

the beginning. In the case of four corner magic

squares we fix by each run of the code two

specific values for a and e, which satisfy the

following conditions

a < e,1 a 17,

e 18.

The algorithm uses then nested for-loops

representing the independent variables (the

small letters) in order to assign all possible

values for these variables between 1 and 36.

When we make a specific assignment for the

independent variables, we substitute in the

formulas, which are written in the definition.

This determines a numerical matrix, which isthen examined to be a possible magic square

or not, i. e. the computed value for being in the

range from 1 to 36 and for being differentfrom other existing values.

We used Pentium IV computers core 2 duo

CPU (3 GHz) to count the four-corner magicsquares with semi-symmetric center. It took

about two months to finish. The C code is

presented in the appendix. The code is

parallelizable since we can fix the value of the

outer for-loop before running the code. By thisway we can split the task into 36 tasks, which

can run in parallel.

We list the number for all squares withrespect to different values of a and e in the

following tables:


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Table 7 a list of the number of four corner magic

squares with a=1,2

a e number a e number

1 2 80012582 2 3 1387041061 3 93587366 2 4 138566816

1 4 137572494 2 5 166212466

1 5 130446558 2 6 156797758

1 6 161682674 2 7 172088726

1 7 194448126 2 8 186805792

1 8 175127312 2 9 188437984

1 9 177194810 2 10 187515974

1 10 193502584 2 11 203101826

1 11 185469236 2 12 192563748

1 12 196104980 2 13 198537572

1 13 194270982 2 14 200999970

1 14 195467492 2 15 194713394

1 15 187447864 2 16 191759218

1 16 195084338 2 17 203881432

1 17 184936940 2 18 185508218

1 18 190538808

Table 8a list of the number of four corner magic

squares with a=3, 4


3 4 166984902 4 5 176458428

3 5 157057934 4 6 177214506

3 6 178288550 4 7 192022722

3 7 174733174 4 8 187140692

3 8 185501038 4 9 196003756

3 9 190481524 4 10 203009482

3 10 197666168 4 11 197765138

3 11 189738168 4 12 200327396


squares with a=3, 4

a e number a e number3 12 207005744 4 13 207983258

3 13 198104476 4 14 197878610

3 14 197186976 4 15 201703132

3 15 194821376 4 16 195874772

3 16 195261598 4 17 184054844

3 17 193169344 4 18 187361040

3 18 182197140


squares with a=5,6


5 6 182674758 6 7 1920227225 7 185076984 6 8 190067620

5 8 202246296 6 9 200239806

5 9 186078138 6 10 192906394

5 10 199812094 6 11 197264336

5 11 211074178 6 12 204338134

5 12 204847206 6 13 203936564

5 13 201808012 6 14 202150964

5 14 228209842 6 15 204245156

5 15 197562226 6 16 188724932

5 16 195399258 6 17 185672352

5 17 195513800 6 18 185229452

5 18 186946600


squares with a=7,8


7 8 198867408 8 9 201833290

7 9 198535068 8 10 200137238

7 10 202469588 8 11 218984602

7 11 192498024 8 12 194348924

7 12 205310982 8 13 196333868

7 13 198048566 8 14 202921386

7 14 201259028 8 15 192255112

7 15 201184468 8 16 190172216

7 16 192323906 8 17 208098890

7 17 189866824 8 18 189151120

7 18 192803750


squares with a=9, 10

a e number a e number9 10 210139916 10 11 205746730

9 11 197103700 10 12 202161512

9 12 208482002 10 13 205937482

9 13 198459704 10 14 190702480

9 14 196245850 10 15 186280692

9 15 187397670 10 16 186260912

9 16 186039290 10 17 186234712

9 17 188258346 10 18 185507292

9 18 193867030


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squares with a=11,12


11 12 201527484 12 13 19861796211 13 195908880 12 14 190414138

11 14 197655756 12 15 190263702

11 15 185552014 12 16 192317526

11 16 201069568 12 17 187546684

11 17 186852956 12 18 188181614

11 18 180859498


squares with a=13,14


13 14 198697560 14 15 18846315213 15 194493502 14 16 197238686

13 16 191413396 14 17 229808362

13 17 190031104 14 18 195395024

13 18 197288494


squares with a=15, 16, 17


15 16 205305156 16 17 193594738

15 17 199979596 16 18 202198792

15 18 199645518 17 18 226528028

The total number of the squares is

28 634 584 244.Hence, there are

28 634 584 244*2*8=458 153 347 904

different natural squares. The number of pairs

(a,e) is 153. Each pair determines uniquely a

center of the square. The average of squares

per center is

2.995e9153

044581533479=

There are 3429 possible centers of the natural

four corner magic squares. Based on the

information about the considered 153 centers

of the natural four corner magic squares we

estimate their total number to be

13e103429*2.995e9 13 =

Also, we present the count of the natural

four corner magic squares with doublesymmetric center. We list the number with

respect to all values of q in the following

table:

Table 16 a list of the number of natural four corner

magic squares with double symmetric center

q number q number

1 976800 10 391968

2 996096 11 376064

3 894816 12 263216

4 808608 13 230736

5 7977472 14 217616

6 640592 15 1399687 595024 16 85200

8 515344 17 78096

9 458784

The total number of the squares is 15646400.

Hence, there are15 646 400*8=125 171 200

different natural squares.

3 Semi pandiagonal magic squares

We introduce a generalization of the

concept of four corner magic square. This

square will be called semi pandiagonal magic

square. It is a magic square with the threeadditional conditions: the middle right and left

broken diagonal sum up to the magic constant.

The sum of the elements in the center 2x2

square is two-third the magic constant. Theformula for such squares is

Table 17a symbolic semi pandiagonal magic squarea D c d f G

H 2semo k l m H

A r u v J K

Q p z 2suvz y L

N o i x e M

B E Q R F N

where


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A=d +l+m+o+p+q+x+ycs2uvz,

B=c+4s+2u+v+zxyadhlmnop2q,D=4s+2u+2v +2z2a2dfhlnp2qxy,

E=2a+2d+e+f+h+l+m+n+2q+x+yr3s2u2v2z,

F=k+l+p+r+i+xefms,

G=a+d+h+l+n+p+2q+x+ycs2u2v2z,

H=e+o+sklh,

J=4siklprxy,

K=c+k+i+u+zdmoq,

L=s+u+vqpy,

M=3seinox,N=m+o+v+zas,

Q=3scikuz,

R=s+u+zdlx.

We note here that this square is a magicsquare with the magic constant is 3s. Further,

the sum of both entries d, m ,K, q, o, Q and c,

2some, A, L, e, R is equal to 3s. We can

easily check that any four corner magic squarewith symmetric center satisfy the conditions

imposed on the semi pandiagonal magic

square.

We have now twenty independentvariables. We note that the inside 4 by 4

square is given in the following table:

2some k l m

r u v J

p z 2suvz y

o i x e

The variable J can be determined by variables,which do not appear in the outer frame.

Further, the elements of the outer frame of thissquare do not depend on the variables of the

center.

The semi pandiagonal magic square will

be transformed into another one of the same

type by applying the eight classicaltransformations. It has another property

preserving transformation. In fact, the square

in table 17 can be transformed into the

following one

a f c d D G

N e i x o M

A J u v r K

Q y z 2s-u-v-z p L

H m k l 2s-o-m-e H

B F Q R E N

without losing its magic properties.

4 Franklin squares

We are here interested in Franklin squares of

order eight, which are semi magic squares (cf.

[2], [3] and [4]). They have some similarities

to four corner magic squares. As explained by

Franklin, each row and column of the square

have the common sum 260. Also, he notedthat half of each row or column sums to half of260. In addition, each of the ''bent rows'' (as

Franklin called them) have the sum 260. The

total number of natural Franklin squares is

1 105 920.

The general form of a Franklin square with

magic constant 2s is an eight by eight matrix

consisting of 16 blocks arranged as follows:

B1 B2 B3 B4

B5 B6 B7 B8

B9 B10 B11 B12

B13 B14 B15 B16

The blocks are defined in the following way:

Table 18Block B1

-j+f+2b-q+x-p s+j-f-2b-x

j-2b-x+m+q+p -j+2b+x-m

Table 19Block B2

-s+f+k+q+x+p s-k-x-f

s+m-k-q-x-p -m+k+x


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Table 20Block B3

b-q+x s-b-p-x

p+q-b b

Table 21Block B4

j +k+2p+q+xb s b+sjkpx

b+skjpq j+kb

Table 22Block B5

-f+b-k-q-p+s f-b+k

-m-b+k+q+p m+b-k

Table 23Block B6

j-f-b+q+p f+b-j

s-j-m+b-q-p j+m-b

Table 24Block B7

-j+2b-k-2p-q+s j-2b+k+p

j-2b+k-x+q+p -j+2b-k+x

Table 25Block B8

q p

s-q-p-x x

Table 26 Block B9

a+b+x+fj sbxfa+mbx+j m+b+x

Table 27Block B10

a+b+f+k+xs s+jbf kx

baxk+m+s b+k+xjm

Table 28Block B11

p+xa sjxp

A j

Table 29 Block B12

p+a+x+j+ks skpxsajk k

Table 30 Block B13

sa fk fj+k

km+a jk+m

Table 31Block B14

j+af f

sajm m

Table 32Block B15

b+s a j k p pb+k

j+axb+k x+bk

Table 33Block B16

a+bp j+pb

sabx b+xj

We notice that the sum of all entries in each

block is s.

Using Maple we computed the

characteristic polynomial of this 8 by 8 matrix.

It is5678 )(4 EDxb +++

where

222 2kj+2b+qxpxkx2jx

km+jm+kq2kpjq+2bx+axjk2bs

bqbp2ap2bk+3bjakajab=F

F]+f2q)2pkj2b(

q)s2p2kjfa(2b8[+4s=D 2

222 2kj+2b+qxpxkx2jxkm+jm

+kq2kpjq+2bx+axjk2bmbqbp2ap2bk+3bjakajab=L

L]s+f2q)2pkj2b16[(+s=E 2

We see that the eigenvalue zero has

algebraic multiplicity five. Hence, the rank of

the Franklin square is at most three in general.

The investigation of the natural Franklinsquares shows that the rank is always three.

5 Applications of magic squares

The main area of the application of magic

squares to music is in rhythm, rather than

notes. This is because for rhythm, consecutive

numbers 1 to 2n are not used to fill the cells of

the nn magic square. This relationship is:


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The total sum of the magic squares numbers

is equal to central number x 9

This is important to music as it shows the sizeof the magic square, which is how many

pulses or sub-divisions there are in the

sequence, this will indicate how and where to

apply it.Table 34 Magic square for rhythm

3 5 7

5 8 11

7 11 15

Using table 34 as an example, 8x9=72 gives

the size of the magic square. This can

therefore be applied to a piece of music with

18 crotchet beats since 18x4=72. Rests can

also be added between the first and second or

second and third rows to create a feeling of the

music building towards a cadence. By

choosing different values for the rests, the

same magic square can create many differentmusical passages. Finally, we mention that

semi magic squares have applications todigital halftoning.

6 Conclusions

We have introduced several types of magic

squares. The problem of counting these

squares is not completely solved yet. We can

find some numbers and estimations in [14].

The development of computers can help bythis task. In this paper we presented some

counting and ideas how to count. In the futurethis research can be extended to include more

types and give counting for the introduced

types. The code, which we presented, is based

on the idea of search over all possibilities insuch a way that we continue the search at each

dead end from the nearest exit.

7 PROGRAM CODE (The C-code)#include

#include

#include

#include

#include

#include #include

const int N = 6; const int NN =

N*N;const int Sum2 = NN - 1;

const int Sum4 = Sum2 + Sum2; const

int Msum = Sum2 + Sum4;

struct bools {bool used[NN];};

struct bools allFree;

#define Uint unsigned int

void writeSquare(int *p, FILE *wfp)

{char squareString[120], *s =

squareString; int cells = 0;

{int i; for (i = 0; i < NN; ++i)

{int x = p[i] + 1;

if (x < 10) { *s++ = ' '; *s++ = '0'+ x; }

else if (x < 20) { *s++ = '1';

*s++ = '0' - 10 + x; }

else if (x < 30) { *s++ = '2';

*s++ = '0' - 20 + x; }

else { *s++ = '3'; *s++ = '0' -

30 + x; }

if (++cells == N) { *s++ = '\n';

cells = 0; } else *s++ = ' ';}}

*s++ = '\n'; *s++ =

'\0';fputs(squareString, wfp);}

Uint makeSquares(int a, int b, int e,

int J, FILE *wfp){Uint count = 0, pcount = 0; bools v

= allFree; int Z[NN];

Z[20]=J;v.used[e] = true; v.used[a] =

true; v.used[b] =

true;v.used[J] = true;

{int t; for (t = 1; t < 2 ; ++t) if

(!v.used[t]) {v.used[t] = true;

{int x; for (x = 21; x < 22; ++x) if

(!v.used[x]) {Z[18] = Sum4-b-t-x;

if ((Z[18] < 0) || (Z[18] >= NN) ||

v.used[Z[18]] || (Z[18] == x))

continue;

v.used[x] = true; v.used[Z[18]] =

true;{int j; for (j = 0; j < NN; ++j)if (!v.used[j]){v.used[j] = true;

{int o; for (o = 0; o < NN; ++o) if

(!v.used[o]) { Z[33]=Msum-j-b-o-a-t;

if ((Z[33] < 0) || (Z[33] >= NN) ||

v.used[Z[33]] || (Z[33] == o))

continue;v.used[o] = true;

v.used[Z[33]] = true;

{int z; for (z = 0; z < NN; ++z) if

(!v.used[z]) {Z[24]= Sum4-j-z-o;

if ((Z[24] < 0) || (Z[24] >= NN) ||

v.used[Z[24]] || (Z[24] == z))

continue; v.used[z] = true;


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{int w; for (w = 0; w < NN; ++w)

if (!v.used[w]){Z[30]=b+j+o+t-w-Sum2;

if ((Z[30] < 0) || (Z[30] >= NN) ||v.used[Z[30]] || (Z[30] == w))

continue;

v.used[w] = true; v.used[Z[30]] =

true;

{int p; for (p = 0; p < (NN-1); ++p)

if (!v.used[p]) {v.used[p] = true;

{int q; for (q = p+1; q < NN; ++q) if

(!v.used[q])

{Z[5]=Sum4-o-p-q-j-t+w+e;

if ((Z[5] < 0) || (Z[5] >= NN) ||

v.used[Z[5]] || (Z[5] == q))

continue;

Z[35]=Sum2-b+p+q-x-e;

if ((Z[35] < 0) || (Z[35] >= NN) ||v.used[Z[35]] ||(Z[35] == q) ||

(Z[35] == Z[5])) continue;

v.used[q] = true; v.used[Z[5]] =

true; v.used[Z[35]] = true;

{int h; for (h = 2; h < 3; ++h) if

(!v.used[h]) {Z[28]= Sum4-p-q-h;

if ((Z[28] < 0) || (Z[28] >= NN) ||

v.used[Z[28]] || (Z[28] == h))

continue;

v.used[h] = true; v.used[Z[28]] =

true;

{int n; for (n = 0; n < NN; ++n) if

(!v.used[n]) {Z[11]= Msum-j-z-n-q-h;if ((Z[11] < 0) || (Z[11] >= NN) ||

v.used[Z[11]] || (Z[11] == n))

continue; v.used[n] = true;


{int d; for (d = 0; d < NN; ++d) if

(!v.used[d]) {

Z[29]=Msum-Z[24]-p-d-o-Z[28];

if ((Z[29] < 0) || (Z[29] >= NN) ||

v.used[Z[29]] || (Z[29] == d))

continue; v.used[d] = true;


{int g; for (g = 0; g < NN; ++g) if

(!v.used[g]) {Z[32]= a+b-d-g-n+Sum2;

if ((Z[32] < 0) || (Z[32] >= NN) ||v.used[Z[32]] || (Z[32] == g))

continue; Z[17]=d-a+g+n-p-q+x;

if ((Z[17] < 0) || (Z[17] >= NN) ||

v.used[Z[17]] || (Z[17] == g) ||

(Z[17] == Z[32])) continue;

Z[23]=a+b-g+j+o+p+q+t-w-Sum4;

if ((Z[23] < 0) || (Z[23] >= NN) ||

v.used[Z[23]] ||(Z[23] == g) ||

(Z[23] == Z[32]) || (Z[23] == Z[17]))

continue;

v.used[g] = true; v.used[Z[32]] =

true; v.used[Z[17]] = true;


{int f; for (f = 34; f < 35; ++f) if

(!v.used[f]) {Z[4]=Msum-x-g-t-f-Z[5];

if ((Z[4] < 0) || (Z[4] >= NN) ||v.used[Z[4]] ||(Z[4] == f))

continue; v.used[f] = true;


{int m; for (m = 0; m = NN) ||

v.used[Z[13]] || (Z[13] == m))

continue; v.used[m] = true;


{int k; for (k = 0; k < NN; ++k) if

(!v.used[k]) {

v.used[k] = true; Z[22]=Msum-k-b-

Z[18]-Z[23]-Z[20];if ((Z[22] >= 0) && (Z[22] < NN) &&

!v.used[Z[22]]) {


Z[34]=Msum-m-q-Z[4]-Z[22]-Z[28];

if ((Z[34] >= 0) && (Z[34] < NN) &&

!v.used[Z[34]]){v.used[Z[34]] = true;

Z[31]=Msum-f-h-k-p-Z[13];

if ((Z[31] >= 0) && (Z[31] < NN) &&

!v.used[Z[31]]) {

Z[0]=x; Z[1]=f; Z[2]=g; Z[3]=t;

Z[6]=z; Z[7]=h; Z[8]=n; Z[9]=j;

Z[10]=q; Z[12]=w; Z[14]=e; Z[15]=a;

Z[16]=m; Z[19]=k; Z[21]=b; Z[25]=p;Z[26]=d; Z[27]=o;

++count; writeSquare(Z, wfp);

if (++pcount == 1000000)

{printf("count %lu\n", count);

pcount = 0; fflush(wfp);}}

v.used[Z[34]] = false;}

v.used[Z[22]] = false;}

v.used[k] = false;}}

v.used[m] = false;

v.used[Z[13]] = false;}}

v.used[f] = false;


v.used[g] = false;

v.used[Z[17]] = false;v.used[Z[23]] = false;


v.used[d] = false;


v.used[n] = false;


v.used[h] = false;


v.used[q] = false;

v.used[Z[5]] = false;


v.used[p] = false;}}


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12/13

v.used[w] = false;


v.used[z] = false;

v.used[Z[24]] = false;}}v.used[o] = false;


v.used[j] = false;}}

v.used[x] = false;


v.used[t] = false;}}

printf("number of squares %d\n",

count);return count;}

void get_rest_of_line(int c) {

if (c != '\n') do { c = getchar(); }

while (c != '\n');}

void get_abe(int *a, int *b, int *e)

{

int unused = scanf("%d %d %d", a, b,e);

int c = getchar();

get_rest_of_line(c);}

void getNumPatterns(int *num)

{int unused = scanf("%d", num);

int c = getchar();

get_rest_of_line(c);}

bool check_abe(int a, int b, int e)

{bool rv = true;

if ((a NN)|| (b NN)||(e NN))

{

printf("\aValue range is 1 to%d.\n\n", NN);rv = false;}

return rv;}

bool checkNum(int num) {

bool rv = true;if (num


13/13

if (wfpc != NULL) { time_t startTime

= time(NULL);

int patterns = getSquares(a, b, e,

num, wfpc);int elapsed_t = (int)(time(NULL) -

startTime);

int hr = elapsed_t/3600; elapsed_t %=

3600;

{ int min = elapsed_t/60, sec =

elapsed_t%60;

{ char *fmt = "\na, b, e patterns: %d

elapsed time: %d:%02d:%02d\n";

printf(fmt, patterns, hr, min, sec);

fprintf(wfpc, fmt, patterns, hr, min,

sec);

fclose(wfpc); } } } }} { int unused =

getchar(); } return 0;}

8 References

[1] Al-Ashhab, S.: Magic Squares 5x5, the internationaljournal of applied science and computations, Vol. 15,

No.1, pages 53-64 (2008).

[2] Ahmed, M.: Algebraic Combinatorics of Magic

Squares , Ph.D. Thesis, University Of California (2004).

[3] Ahmed, M.: How Many Squares Are There, Mr.

Franklin?: Constructing and Enumerating Franklin

Squares, American Mathematical Monthly 111, pages

394410 (2004).

[4] Amela, M.: Structured 8 x 8 Franklin Squares,

http://www.region.com.ar/amela/franklinsquares/

[5] Bellew, J.: Counting the Number of Compound and

Nasik Magic Squares, Mathematics Today, pages 111-

118 August (1997).

[6] Benson, W. H.: O. Jacoby, New Recreations With

Magic Squares, Dover, New York (1976).

[7] Besslich, Ph. W.: Comments on Electronic

Techniques for Pictorial Image Reproduction, IEEE

Transactions on Communications 31, pages 846 847(1983).

[8] Besslich, Ph. W.: A Method for the Generation and

processing of Dyadic Indexed Data, IEEE Transactions

on Computers, C-32(5), pages 487 494 (1983).

[9] Diaconis, P., Gamburd, A.: Random Matrices,

Magic Squares and Matching Polynomials, The

Electronic Journal of Combinatorics, Vol. 11, No. 2,

pp. 1 -- 26 (2004).

[10] Ollerenshaw, K., Bre, D. S., Most-perfectPandiagonal Magic Squares: Their Construction and

Enumeration, The Institute of Mathematics And its

Applications, Southend-on-Sea, U.K., (1998).[11] McClintock, E.: On the Most Perfect Forms of

Magic Squares, with Methods for Their Production,

American Journal of Mathematics 19, pages 99120

(1897).

[12] Kolman, B.: Introductory Linear Algebra with

Applications, 3rd edition (1991).

[13] Van den Essen, A.: Magic squares and linear

algebra, American Mathematical Monthly 97, pp. 60-62

(1990).

[14] Walter Trump, www.trump.de/magic-squares

9 Acknowledgment Thanks are due to Harry

White from Canada, who has made great

contributions by developing the code (email:[email protected]).


745

special magic squares of order six and eight

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