NEGATIVE FOUR CORNER MAGIC SQUARES OF ORDER SIX WITH a BETWEEN 1 AND 5

NEGATIVE FOUR CORNER MAGIC SQUARES OF ORDER SIX WITH a BETWEEN 1 AND 5 S. Al-Ashhab Depratement of Mathematics Al-Albayt University Mafraq Jordan Email: ahhab@aabu.edu.jo 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. Keywords Magic 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 same number. This number is called the magic constant. A natural magic square of order n is a matrix of size n n such that its entries consist of all integers from one to n². The n( n 2 1) magic constant in this case is. A symmetric magic square is a natural magic square of order 2 n such that the sum of all opposite entries equals n+1. For example, Table 1 a 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 magic square 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 of the entries 39,12, 46, 22, 20, 23, 13 is 175, which is 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 4x4 magic squares is 880, where he considered a magic square with all its reflections and rotations one square. Hire listed them all in a table in the year 1693. Recently we can use the computer to check that there are magic squares of order 4. 880*8 = 7040 In 1973 the number of all natural magic squares of order five became known. Schoeppel computed it using a PDF-10 machine. It is 64 826 306*32=2 202 441 792 where we multiply with 32 due the existence of type preserving transformations. According to [5] there exists natural nested magic squares of order six. 736 347 893 760 It is well-known that there are pandiagonal magic squares and symmetric squares of order five. But, there are neither pandiagonal magic squares nor symmetric squares of order six. The number of natural magic squares of order six is actually till now unknown. Trump made using statistical methods (Monte Carlo Backtracking) the following interval estimation for this number

(1.7712 e19, 1.7796 e19) 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 Brée s (cf. [10]) combinatorial count ranks as a major achievement, draw attention to another type which 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 Brée (cf. [10]) have a patent for using most-perfect magic squares for cryptography, and Besslich (cf. [7] and [8]) has proposed using pandiagonal magic squares as dither matrices for image processing. A pandiagonal and symmetric magic square is called ultramagic. According to [14] the number of ultramagic squares of order five is 16 and number of ultramagic squares of order seven 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 each row or column yields the magic constant. According to Trump (cf. [14]) the number of semi magic squares of order four is 68 688, and the number of semi magic squares of order five is 2 Four corner magic square 579 043 051 200. This concept was first introduced in [1]. Alashhab studied this type there in very simple cases. In [1] Al-ashhab considered the type called nested four corner magic square with a pandiagonal magic square. We continue here the study of this type. We focus in this paper on the following kind of magic squares: magic squares of order six with magic constant 3s such that a + a + ij (i 3)(j 3) i(j 3) a + a (i = 2s 3)j

holds for each i=1,2,3 and j=1,2,3 and a + a 33 44 + a 34 + a 43 = 2s. We call such squares four corner magic square of order 6. The entries of a four corner magic square of order 6 satisfy a + a 14 25 + a 36 + a 41 + a 52 + a 63 =3s, a + a 13 22 + a 31 + a 61 + a 55 + a 64 = 3s These two conditions represent the sum of the entries 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 a + a 11 44 + a 14 + a 41 = 2s, a + a 22 55 + a 25 + a 52 =2s, a + a 33 66 + a 36 + a 63 =2s holds. Adding up these equations and subtracting from them the following equation yields the desired equation. a + a 11 22 + a 33 + a 44 + a 55 + a 66 = 3s A four corner magic square of order 6 can be written as Table 3 a symbolic four corner magic squares x f G t M G z h N j q N w E e a m D A k 2s a b e b H R 2s j o z p d o 2s p q h T B F W J L Y where A= 2s b t x, B = j + o + t + b s w, D = d + g + n + x a p q,

E = 3s a e m w D, F = 3s f h k p E, G = 2s + e + w (j + o + p + q + t), H = e + g + s + w + x j k o p q, J=3s j b 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 = a + b + j + o + p + q + t g 2s w, T= h + j + q + z d s, W=a + b + s d g n, Y=s + p + q b e x. We see that it has seventeen independent variables, which are represented by the small letters. In the code these variables will be assigned to loops. We have for example Table 4 a natural four corner magic squares 6 23 11 13 33 25 19 28 36 3 7 18 2 29 1 17 27 35 21 8 22 34 10 16 32 9 15 20 30 5 31 14 26 24 4 12 2.1 Four corner magic square with negative center. We introduce now the main concept in our work. We call a four corner magic squares such that a * a 33 44 a * a 34 43 < 0

a four corner magic square of order six with negative center. This means that the 2 by 2 square in the center has negative determinant. The number of all different possible values for a, b and e by computing the number of four corner magic squares is 3429. Hence, there are 3429 possible centers of the natural four corner magic squares. The number of squares with positive center (the sign < is replaced by > in the last inequality) is 232. Hence, there are 3197 possible centers of the negative four corner magic squares. These squares include the squares with symmetric centers (cf. [2]) and semi symmetric centers (cf. [3]). The number of their centers is 459. We are here interested in 2738 centers. 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 π/2, π, (3π)/2 and transpose operation. Now, a four corner magic squares with negative center can be transformed as follows into another one of the same kind: we make these interchanges simultaneously: interchange a (res. a ) with a (res. a ), 12 62 15 65 interchange a (res. a ) with a (res. a ), interchange a (res. a ) with a (res. a ), 21 26 51 56 22 55 25 52 interchange a (res. a ) with a (res. a ), interchange a (res. a ) with a (res. a ). 23 24 53 54 32 42 35 45 It is obvious that the center remains unchanged by this transformation. We can use this transformation to reduce the number of computed natural magic squares. In order to eliminate the effect of the previous transformations we compute all natural four corner magic squares with negative center for which the following conditions hold: p < q, a < e < b, a < 2s a b e. This means that we compute first the number of all natural squares satisfying these conditions. We multiply then the number with sixteen in order to get the number of squares. 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 with negative center we fix by each run of the code two specific values for a, b and e, which satisfy the following conditions a < e < b, a < 2s a b e, b*e-a*(2s a b e) < 0, 0 < a < 6.

We exclude the squares with symmetric centers and semi symmetric centers in the next list. We list the number for all squares with respect to different values of a, b and e in the following tables: Table 5 a list of the number of four corner magic squares with a = 1 b e number b e number b e number 20 18 131022698 22 17 134145080 23 17 136644763 20 19 138775961 22 18 130604069 23 18 137869426 21 17 140218160 22 19 133615994 23 19 138183367 21 18 131534372 22 20 133012116 23 20 133747793 21 19 138959709 22 21 131305331 23 21 137623014 21 20 135417961 23 15 142265782 23 22 134121525 22 16 135819458 23 16 138000220 * * * b=24 e number e number e number e number 14 141560325 17 137907213 19 135435276 21 134603089 15 136219542 18 133286816 20 135682553 22 138687369 16 138302549 * * * * 23 142088500 b=25 e number e number e number e number 13 142713486 16 137569348 18 134502101 20 135810466 14 144822987 17 140304129 19 142888748 21 143478320 15 146985749 * * * * 22 136855015 b=26 e number e number e number e number e number 12 140721021 15 138290554 17 135733078 20 137800262 23 139683145 13 135450348 16 147094027 18 134376506 22 137808915 24 143478700 14 139373569 * * 19 138190635 * * 25 139145267 b=27 e number e number e number e number e number 11 149563956 14 140911703 17 145372240 20 139722400 24 140978175 12 143540310 15 146532171 18 149173141 21 143216037 25 145027496 13 150184348 16 142662660 * * 22 139766220 26 143002581 b=28 e number e number e number e number e number 10 141220103 13 144068850 16 143594872 21 136582416 25 143252037 11 138251848 14 143135768 18 143963285 22 139031423 26 144833202 12 142819907 15 145154107 19 139067086 23 138421213 27 156154873 * * * * 20 140422132 24 140420584 * *

b=29 e number e number e number e number e number e number 9 152944098 12 149617095 16 145423524 19 147012552 23 144666619 26 143855893 10 149062066 13 151153974 17 144693264 20 139302692 24 144407022 27 157731720 11 148645882 14 149240219 18 139351400 21 143962741 25 151356839 28 155569340 b=30 8 147682181 12 148879623 17 137608303 22 144446182 26 151879426 9 143091351 14 146926872 18 139940935 23 147651730 27 155154999 10 154348850 15 145302440 19 143241868 24 148982167 28 163126976 11 141700950 16 143805250 20 147308911 25 147116549 29 166116865 * * * * 21 142612859 * * * * b=31 7 154995334 12 151575571 17 148907122 23 152324861 27 163620548 8 151231411 13 149813747 18 141646497 24 148055832 28 164321701 9 153757900 14 148163849 19 147061694 25 153196655 29 172742265 10 151881687 15 149512821 20 145527012 26 166292157 30 172828814 * * * * 22 145951640 * * * * b=32 7 147653350 12 148067738 18 151711935 22 150004248 27 159146949 8 153710531 13 150783376 19 150226056 23 149654418 28 173746359 10 149959760 15 145670244 20 154230283 24 155848899 29 171308655 14 143142790 16 147533316 21 151137258 25 156755631 30 178363735 11 143529129 17 146479277 * * 26 158827137 31 184374964 b=33 5 168168958 11 152975902 15 158683431 19 161205077 24 161463708 28 170327643 6 154187048 12 154661315 16 152343356 21 159294692 25 168209790 29 181864053 8 154744187 13 156989953 17 152710044 22 148820123 26 165521466 30 189916806 9 157792763 14 152817654 18 146426820 23 157716003 27 174567256 31 191124197 10 162439769 * * * * * * * * 32 197126038 b=34 4 166385994 11 140812446 17 149087413 22 154833350 28 172485021 6 145337495 12 147280551 18 148393535 23 173467680 29 181953194 7 148949886 13 146839250 19 163330582 24 148061821 30 184628002 8 150887436 14 154412429 20 162083451 25 169038003 31 191482378 9 146106132 15 148383014 21 150367084 26 162959057 32 202146147 10 151441375 16 150586413 * * 27 169952518 33 206733283

b=35 4 172963666 10 153523754 16 157738214 23 164038715 29 190583123 5 161824291 11 155826604 17 158845589 24 169505590 30 193408521 6 153920660 12 159813044 18 153968470 25 170625275 31 202990677 7 157946900 13 154142329 20 151134961 26 173771036 32 199149110 8 154427653 14 160854778 21 161873798 27 181641254 33 225048341 9 158458900 15 144184056 22 158040813 28 181003002 34 240782997 The total number of the squares with a = 1 is 38523022675. Table 5 a list of the number of four corner magic squares with a = 2 b e number b e number b e number 19 17 131022698 21 18 125765276 22 16 137404253 20 16 140218160 21 19 126046913 22 17 130111045 20 18 133210140 21 20 130279869 22 18 132080655 20 19 128935500 22 14 142265782 22 19 126485028 21 15 135819458 * * * 22 20 130564302 21 17 131879063 * * * 22 21 128022232 b=23 13 141560325 16 135109297 18 127643826 19 129607428 21 137809464 15 133049722 17 140356000 * * 20 141274205 22 131797562 b=24 12 142713486 15 137871625 17 133991049 20 135108324 22 135246322 14 138361972 16 143773682 18 134834873 21 136013377 23 141617171 * * * * 19 129286147 * * * * b=25 11 140721021 14 135897367 16 134485908 18 127643145 20 132397335 23 140458558 13 139588939 15 138980499 17 140868624 19 132200077 21 134760196 24 138679259 b=26 10 149563956 14 144984171 16 144847545 19 136432511 22 136316609 12 143393496 15 139218947 17 150732635 21 135099351 24 139228056 13 149876695 * * 18 139701915 * * 25 136652741 b=27 9 141220103 13 142972817 15 143067910 21 139912609 22 134366843 24 140132722 10 187515974 14 141196329 16 134488912 19 138669868 23 138141530 25 137980825 11 139928843 * * 17 138342068 20 135005688 * * 26 142403698 b=28 8 152944098 11 149892037 14 152800164 18 139164148 21 144148808 25 140866305 9 188437984 12 149082943 15 144389905 19 136711784 23 145865748 26 150772851 10 149823919 13 150902338 17 139151292 20 138664151 24 141822477 27 146484868

b=29 7 147682181 12 141632641 17 152655065 21 140024665 25 142888313 9 142932719 13 148615081 18 134921100 22 138865215 26 155212458 10 141787698 15 145708818 19 134766649 23 155215569 27 151248380 11 156145769 16 139079926 20 150469020 24 141049562 28 158660795 b=30 6 154995334 11 145560084 15 143927688 19 137876624 23 141326689 27 155487317 8 149428123 13 147971264 16 148166778 20 141479893 24 146170044 28 168747314 9 147374279 14 154300490 17 145891388 22 140454402 25 150466663 29 164171149 10 149058722 * * 18 138136820 * * 26 159461791 * * b=31 5 148610466 11 146734890 15 143228937 19 140157023 23 145727906 27 160373086 7 146735788 12 142595769 16 141954069 20 144275332 24 144479018 28 169784706 8 153698333 13 152103600 17 142366004 21 147823673 25 150010838 29 166262378 9 142275478 14 142864054 18 136386198 22 141780381 26 155608970 30 166239573 b=32 4 168168958 11 158483913 16 157091633 22 149151130 27 163206706 6 154737421 12 152832470 17 153162632 23 152695310 28 176426211 7 148817832 13 151327969 18 148164591 24 162222044 29 181400940 9 147565846 14 156348877 19 143793615 25 158079652 30 181124525 10 153705235 15 148804837 21 143988138 26 166070261 31 178003382 b=33 3 166385994 10 146472147 14 145207989 19 151539859 23 155069971 29 178203175 5 151849825 11 149296432 15 147039405 20 151070318 24 157612401 30 180573483 7 146825910 12 143630891 16 144825350 21 144650131 26 171526925 31 193135050 8 152201319 13 159423592 17 148555360 22 146131117 27 163527872 32 189033416 9 150762100 * * 18 138337790 * * 28 167954143 * * b=34 5 156564171 10 151108284 15 146448836 20 150968634 24 158229465 29 176155537 6 146635847 11 144990802 16 149420333 21 147648296 25 160672435 30 182215443 7 148000383 12 149121335 17 146616542 22 153158013 26 170532608 31 185515595 8 151178944 13 151502087 18 148171293 23 152285271 27 164385210 32 197114762 9 144198830 14 148932312 * * * * 28 180265730 33 204478536

b=36 3 172963666 9 153523754 15 157738214 22 193204904 28 190583123 4 161824291 10 155826604 16 158845589 23 169505590 29 193408521 5 153920660 11 159813044 17 153968470 24 170625275 30 188457639 6 157946900 12 154142329 19 151134961 25 173771036 31 207305164 7 142591089 13 160854778 20 161873798 26 181641254 32 199057819 8 158458900 14 158144276 21 158040813 27 181003002 33 240782997 The total number of the squares with a = 2 is 40662919383. Table 5 a list of the number of four corner magic squares with a = 3 b e number b e number b e Number 18 17 138775961 21 19 128760410 22 14 133049722 19 16 131534372 21 20 127679332 22 16 131182601 19 17 133210140 22 13 136219542 22 17 130380694 21 14 138000220 20 15 134145080 22 18 124580957 21 15 137404253 20 16 131879063 22 19 126747584 21 17 131571966 20 18 122900180 22 20 128384420 21 18 125939493 20 19 128297404 22 21 128899433 b=23 12 144822987 15 136667297 17 133335601 19 132363823 21 131031874 13 138361972 16 140471492 18 128114514 20 131063343 22 131447556 b=24 11 135450348 14 140441790 16 134644322 19 127926756 21 131115942 12 139588939 15 133197578 17 133550283 20 140330105 22 134264186 * * * * 18 129499341 * * * * b=25 10 143540310 13 143018053 15 146935974 17 142521759 19 135608506 22 136755260 11 143393496 14 134604239 16 143584771 18 133747474 20 132847711 24 135424627 b=26 9 138251848 13 134316359 16 135622044 20 131879763 23 134200506 10 139928843 14 134901275 17 133867159 21 134564403 24 136351628 12 136781870 15 122503013 18 134995272 22 134816670 25 138147354 b=27 8 149062066 13 147896321 18 136950958 23 139747717 9 149823919 14 142163907 19 138164882 24 144667060 11 147175408 15 143940250 20 146606427 25 142540300 12 143838412 16 138917234 21 136784165 26 143303870

b=28 7 143091351 12 150668098 17 134512950 20 142039752 24 137997817 8 142932719 13 134990744 18 132699010 21 137390717 25 143463063 10 142777307 14 139990551 19 134159098 22 139595231 26 144187692 11 134757646 16 139581464 * * 23 140537625 27 149214700 b=29 6 151231411 11 140187814 16 140011089 20 131714933 25 151104816 7 149428123 12 141213764 17 143220643 22 135603578 26 143842943 9 144308025 14 140105235 18 135703344 23 139606576 27 155148177 10 143005910 15 142191849 19 137437360 24 140015144 28 154560529 b=30 5 147653350 12 142095599 16 146315135 21 138205111 25 155193992 6 146735788 13 136525756 17 139262896 22 140316994 26 157174767 8 149428039 14 139326049 18 132882313 23 140086473 27 155496602 9 141596589 15 141237131 19 137184826 24 146720764 28 154606221 10 142830575 * * 20 139623352 * * 29 158329145 b=31 4 154187048 10 148500474 14 144128093 18 135853968 23 151124879 27 169043704 5 154737421 11 146441757 15 151524988 19 143425932 24 148871650 28 174948383 7 147292870 12 157807245 16 143924052 21 147396017 25 161119372 29 164760928 8 145839616 13 151880679 17 145604578 22 140676672 26 155773693 30 168928501 b=32 4 151849825 11 133332053 15 141716446 19 144673900 23 148010248 27 162821495 6 149198817 12 141523735 16 148729216 20 146784425 24 150780190 28 160736429 8 140078872 13 141490766 17 138998901 21 137410150 25 150563201 29 166508557 9 143138108 14 135957317 18 136263171 22 143929397 26 156710496 30 176015705 10 144166155 * * * * * * * * 31 173094520 b=33 6 158236584 11 148550796 15 146040375 20 140450343 24 161223882 28 167360316 7 146421224 12 146584509 16 142633119 21 145803942 25 160787355 29 174570017 8 149302137 13 148021070 17 144060771 22 147890097 26 162237461 30 179710972 9 148090824 14 151267863 18 135924074 23 152927917 27 169587055 31 181550099 10 147496198 * * * * * * * * 32 191916900 b=35 4 156564171 11 149121335 19 150968634 26 164385210 5 146635847 12 151502087 20 147648296 27 180265730 6 148000383 13 148932312 21 153158013 28 176155537 7 151178944 14 146448836 22 152285271 29 182215443 8 144198830 15 149420333 23 158229465 30 185515595 9 151108284 16 146616542 24 160672435 31 197114762 10 144990802 17 148171293 25 170532608 32 204478536

b=36 4 145337495 11 146839250 18 162083451 25 169952518 5 148949886 12 154412429 19 150367084 26 172485021 6 150887436 13 148383014 20 154833350 27 181953194 7 146106132 14 150586413 21 159658120 28 184628002 8 151441375 15 149087413 22 161871381 29 191482378 9 140812446 16 148393535 23 169038003 30 202146147 10 147280551 17 163330582 24 168376244 31 206733283 The total number of the squares with a = 3 is 39628193947. Table 5 a list of the number of four corner magic squares with a = 4 b e number b e number b e number 18 16 138959709 20 15 130111045 21 14 135109297 18 17 128935500 20 16 131571966 21 15 131182601 19 15 130604069 20 18 125191689 21 17 127375107 19 16 125765276 20 19 122508696 21 18 124109386 19 17 122900180 21 13 138302549 21 19 123549439 20 14 136644763 * * * 21 20 128980534 b=22 12 146985749 14 136667297 17 126745385 18 128000888 20 121418695 13 137871625 16 136020554 * * 19 126146079 21 130665900 b=23 11 139373569 13 140441790 16 128625558 19 130230377 21 129648211 12 135897367 15 138436974 17 132530709 20 124066378 22 128764559 * * * * 18 124662988 * * * * b=24 10 150184348 12 143018053 15 141840168 18 129942329 20 124754709 11 149876695 14 135458748 16 135657171 19 127023594 21 131439570 * * * * 17 132563331 * * * * b=25 9 142819907 13 135110087 16 139970728 18 126840267 22 131550882 10 138698795 14 132164809 17 136645500 19 133569930 23 133492696 11 136781870 15 135707151 * * 21 133627920 24 131248649 b=26 8 148645882 12 145397176 15 133406533 19 135241642 23 132950667 9 149892037 13 136479088 16 135320277 20 125920989 24 129815439 10 147175408 14 134951971 17 134093384 21 135777936 25 141023114 b=27 7 143211547 11 141189870 14 130550812 18 130762412 21 138300646 24 132304052 8 141787698 12 132865415 15 137244633 19 132180675 22 136476148 25 134971056 9 142777307 13 140290057 17 136276139 20 129776191 23 136383758 26 139984910

b=28 6 153757900 11 136375446 16 139434980 19 135218481 24 139405882 7 147374279 12 138418197 17 135693912 20 129499607 25 145101120 8 144308025 13 137756276 18 130829365 22 131322834 26 144462604 10 146167982 15 135686259 * * 23 136789515 27 146603808 b=29 5 153710531 10 143896759 15 137145979 18 131169404 21 142718694 25 155350736 6 153698333 11 140975397 16 147360520 19 138455065 22 140227489 26 143750852 7 149428039 13 141073928 17 139629640 20 135783997 23 135638568 27 147459143 9 142024009 14 130631241 * * * * 24 146513807 28 154000655 b=30 5 148817832 11 139986164 15 136896127 18 133083796 22 139913236 26 152780567 6 147292870 12 142255243 16 138843458 19 133348340 23 145999133 27 157308042 8 139032704 13 145081708 17 140935863 21 137005584 24 146099505 28 156358183 9 142272721 14 138028817 * * * * 25 147200727 29 161896729 b=31 5 149198817 11 135417793 15 146493520 19 139804938 23 143040280 27 152390239 7 142967065 12 135267668 16 134874012 20 137451319 24 155319326 28 153330055 9 141644356 13 139489295 17 137503147 21 138011289 25 147813290 29 170330796 10 144243215 14 133987059 18 131093083 22 139221713 26 154116063 30 164200853 b=32 7 149820579 11 140958904 15 148685882 20 139239632 24 152090614 28 169536431 8 139804478 12 152018675 16 143565743 21 139561406 25 151396105 29 165608031 9 143870157 13 142070217 17 135839807 22 147131538 26 164518539 30 169153656 10 149448475 14 141515054 18 138231371 23 148825102 27 157761704 31 177140294 b=34 5 158236584 10 148550796 14 146040375 19 140450343 23 161223882 27 167360316 6 146421224 11 146584509 15 142633119 20 145803942 24 160787355 28 174570017 7 149302137 12 148021070 16 144060771 21 147890097 25 162237461 29 179710972 8 148090824 13 151267863 17 135924074 22 152927917 26 169587055 30 181550099 9 147496198 * * * * * * * * 31 191916900 b=35 5 146825910 10 143630891 14 144825350 18 151070318 22 157612401 26 167954143 6 152201319 11 159423592 15 148555360 19 144650131 23 156670961 27 178203175 7 150762100 12 145207989 16 138337790 20 146131117 24 171526925 28 180573483 8 146472147 13 147039405 17 151539859 21 155069971 25 163527872 29 193135050 9 149296432 * * * * * * * * 30 189033416

b=36 5 154744187 9 154661315 13 152343356 18 159294692 22 168209790 26 181864053 6 157792763 10 156989953 14 152710044 19 148820123 23 165521466 27 189916806 7 162439769 11 152817654 15 146426820 20 157716003 24 174567256 28 191124197 8 152975902 12 158683431 16 161205077 21 161463708 25 170327643 29 197126038 The total number of the squares with a = 4 is 40866368479. Table 5 a list of the number of four corner magic squares with a = 5 b e number b e number b e number 17 16 135417961 20 16 127375107 21 13 143773682 18 15 133615994 20 18 122884440 21 14 140471492 18 16 126046913 20 19 119320724 21 15 136020554 18 17 128297404 21 12 137569348 21 17 122254286 20 13 137907213 19 14 137869426 21 18 122438456 20 14 140356000 19 15 132080655 21 19 125982545 20 15 130380694 19 16 125939493 21 20 121120361 * * * 19 17 125191689 * * * b=22 11 138290554 13 133197578 16 129435767 18 120369921 20 126281250 12 138980499 14 138436974 17 123081010 19 119459984 21 121839807 b=23 10 140911703 12 134604239 15 132640534 17 131906636 19 125843075 21 126599940 11 144984171 13 135458748 16 131407287 18 126767780 20 135611220 22 130498831 b=24 9 144068850 12 135110087 15 131506079 18 123451990 20 126275986 10 142972817 14 141533466 16 135917747 19 122768564 22 130130936 11 134316359 * * 17 130317255 * * 23 134020039 b=25 8 149617095 11 145397176 14 141380028 16 131887930 18 136352497 21 129252970 9 149082943 13 140687218 15 135740761 17 132626465 20 127799492 23 133908073 10 143838412 * * * * * * * * 24 129448376

b=26 7 141700950 12 135245917 15 133571878 19 125992751 22 129171902 8 156145769 13 128809853 16 142075092 20 144373363 23 136115552 9 134757646 14 143270620 18 135893351 21 128140980 24 138759848 10 141189870 * * * * * * 25 143686572 b=27 7 149058722 12 132683677 16 144683623 19 127496228 23 134239948 8 143005910 13 139231388 17 134195093 20 130647016 24 129974954 9 146167982 14 143761353 18 127323270 22 132106893 25 140158737 11 136907402 * * * * * * 26 149285802 b=28 7 141596589 12 132542832 17 134300418 20 131369647 24 142422589 8 142024009 14 141540613 18 126405309 21 127993820 25 140125945 10 134468605 15 134646698 19 127901959 22 132006986 26 142915847 11 134799950 16 122342784 * * 23 133206738 27 146679031 b=29 6 145839616 12 136872080 16 138445641 21 139707991 25 147066447 7 138464019 13 140708204 17 139392139 22 129734036 26 152798202 9 135615715 14 148546649 18 126488067 23 144867062 27 153998306 10 139363210 15 138173480 19 137583464 24 143107636 28 151076886 b=30 6 142967065 12 133868512 15 133270407 19 130560165 23 138993391 26 149467047 8 148835837 13 133204041 16 133982017 20 137177939 24 143928039 27 144269696 10 139439502 14 142096749 17 128746577 21 133592738 25 147756468 28 154614786 11 136638984 * * 18 130368146 22 135891835 * * 29 159279125 b=31 8 151527930 12 138753829 15 135266258 20 131313336 24 151650896 27 151793820 9 136862992 13 144254317 16 138888100 21 135603911 25 157136203 28 155126965 10 138846161 14 134349520 17 132194757 22 139394728 26 152609089 29 159842747 11 140250177 * * 18 128536456 23 145192656 * * 30 172109148

b=33 6 149820579 10 140958904 14 148685882 19 139239632 23 152090614 27 169536431 7 139804478 11 152018675 15 143565743 20 139561406 24 151396105 28 165608031 8 143870157 12 138086695 16 135839807 21 147131538 25 164518539 29 169153656 9 149448475 13 141515054 17 138231371 22 148825102 26 157761704 30 177140294 b=34 6 140078872 10 141523735 14 148729216 18 146784425 22 150780190 26 160736429 7 143138108 11 141490766 15 138998901 19 137410150 23 150563201 27 166508557 8 144166155 12 135957317 16 136263171 20 143929397 24 156710496 28 176015705 9 133332053 13 141716446 17 144673900 21 148010248 25 162821495 29 173094520 b=35 6 150269529 10 151327969 13 157091633 18 143988138 22 158079652 25 176426211 7 153705235 11 156348877 14 153162632 19 149151130 23 166070261 26 181400940 8 157102148 12 148804837 15 148164591 20 152695310 24 163206706 27 195201865 9 152832470 * * 16 143793651 21 150137639 * * 28 178003382 b=36 6 149959760 10 143142790 13 146479277 17 151137258 21 159057117 24 173746359 7 143529129 11 145670244 14 151711935 18 150004248 22 158827137 25 171308655 8 148067738 12 147533316 15 150226056 19 149654418 23 159146949 26 178363735 9 150783376 * * 16 154230283 20 * * 27 184374964 The total number of the squares with a = 5 is 39666915624. The total number of the squares with is Hence, there are a = 1, 2, 3, 4, 5 199 347 420 108. 199 347 420 108*16=3 189 558 721 728 different four corner magic squares of order six with a = 1, 2, 3, 4, 5, which does not have neither a symmetric center nor a semi symmetric center.

4 The number of four corner magic squares of order 6 The number of all different possible values for a, b and e by computing the number of four corner magic squares is 3429. Hence, there are 3429 possible centers of the natural four corner magic squares. The number of squares with positive center is (as illustrated) 232. The remaining squares include the squares with symmetric centers (cf. [2]) and semi symmetric centers (cf. [3]). There are 153 possible symmetric centers of the natural four corner magic squares. According to [2] there are 28 634 584 244*16 = 458 153 347 904 different natural four corner magic squares with symmetric center. There are 306 possible semi symmetric centers of the natural four corner magic squares. According to [3] there are 101 425 060 998*16 = 1 622 800 975 968 different natural four corner magic squares with semi symmetric center. Based on the information about the computed natural four corner magic squares we will estimate the whole number. We have considered 153+306+1366 = 1825 centers. The total number of squares associated with them is 5 270 513 045 600 We want here to estimate the number of four corner magic squares of order 6. By computing the average number of squares per center 5 270 513 045 600 180 497 022 *16 1825 Hence, we estimate the number of the four corner magic squares to be 12 618 924 288 438*16 9.9 *10. 5 Parallelization and grid computing The problem itself is split into several part problems, since counting squares for each center is a separate problem. The code is constructed so that the input is the center of the square. This is the first step by splitting the job of counting into many smaller jobs, which run in parallel. We can fix the value of the outer for-loop before running the code. By this way we can split the task into 36 tasks, which can run in parallel.

The used code is parallelizable. The code (algorithm) uses nested for loops representing the independent variables (the small letters). The first loop is for the variable t. When we fix one center we also run the code for an interval of the values of the variable t. This interval is a part of the input. We have the freedom to choose any subinterval of [0, 36]. By choosing smaller intervals we run smaller jobs since they do not involve much more computations. The loops are used 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 is then 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 different from other existing values. The computations were done with the aid of the EUMED GRID system. The jobs were submitted to the system, which distributes the jobs on the connected computers. 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 by this task. In this paper we presented some counting and ideas how to count. In the future this 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 in such a way that we continue the search at each dead end from the nearest exit. 8 References [1] Al-Ashhab, S.: Magic Squares 5x5, the international journal of applied science and computations, Vol. 15, No.1, pages 53-64 (2008). [2] Al-Ashhab, S.: Even-order Magic Squares with Special Properties, International Journal of Open Problems in Mathematics and Computer Science, Vol. 5, No. 2 (2012). [3] Al-Ashhab, S.: Special Magic Squares of Order Six and Eight, International Journal of Digital Information and Wireless Communications (IJDIWC) 1(4): 769-781 (2012). [4] Ahmed, M.: Algebraic Combinatorics of Magic Squares, Ph.D. Thesis, University Of California (2004). [5] Ahmed, M.: How Many Squares Are There, Mr. Franklin?: Constructing and Enumerating Franklin Squares, American Mathematical Monthly 111, pages 394 410 (2004). [6] Amela, M.: Structured 8 x 8 Franklin Squares, http://www.region.com.ar/amela/franklinsquares/

[7] Bellew, J.: Counting the Number of Compound and Nasik Magic Squares, Mathematics Today, pages 111-118 August (1997). [8] Besslich, Ph. W.: Comments on Electronic Techniques for Pictorial Image Reproduction, IEEE Transactions on Communications 31, pages 846 847 (1983). [9] 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). [10] Ollerenshaw, K., Brée, D. S.: Most-perfect Pandiagonal 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 99 120 (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