Frncis Gslou Second edition of Ferury 10, 2012 (First edition on Jnury 28, 2012) HOW MANY SQUARES ARE THERE, Mr TARRY? ABSTRACT In this er, I enumerte ll the 8x8 imgic sures given y the Trry s ttern. This ttern is 8x8 Greco-Ltin mgic ttern nd I show tht in fct we cn find 14,784 different 8x8 Greco-Ltin mgic tterns of similr tye. Among these 14,784 tterns, only some ones give imgic sures nd I clculte the totl numer of 8x8 imgic sures generted with this method. THE TARRY S PATTERN In his rticle ref. [1], Trry gives the ttern of 8x8 ndigonl mgic sure which hs ll the rows nd ll the columns imgic: +r +s -r +d -c +s -c +s +d -r +s +r +d +r -c +s -r +s +s -r +s -c +d +r +s +r +d +s -c -r -r -c +d +s +r +s +s +d -c +r +s -r -r +s -c +r +d +s The 8 numers,, -c,,,,, +d hve to e chosen mong (1,2,3,,8) nd the 8 numers, +r, -r,, +s,,, +s mong (0,8, 16,,56). The two min digonls re lso imgic if the sulementry condition r(-)=c(-) is stisfied. For exmle if: =1, =4, c=1, d=4, =0, =24, r=8, s=32.
2 Trry writes little ridly there is n infinity of sures which cn e constructed with this ttern. In fct, it is wrong. The exct numer cn e esily clculted y tultion, i.e. when giving to the rmeters ll the ossile vlues stisfying the conditions indicted y Trry. I found there re 2,304 semi-imgic sures 320 imgic sures (I cll semi-imgic the mgic sures which hve ll the rows nd ll the columns imgic). I studied rticulrly the 320 imgic solutions. There re only 80 uniue sures ecuse the grou (I, V, H, R2) is working on this set (see my site ref [4] for the nottions). All these imgic sures re Greco-Ltin, ndigonl of tye comlete nd with the 2 semidigonls imgic (the semi-digonls re A4, B3, C2,, H5 nd A5, B6, C7,, H4). Ech elementry sure is Ltin digonl, self-orthogonl (orthogonl to its trnsosed) nd ndigonl (ut not Ltin ndigonl ; the two semi-digonls re nevertheless Ltin). I hve rinted ll these sures. For exmle, here is the first imgic one, which is mde with the ove-mentioned rmeters of Trry: 9 51 8 62 44 18 37 31 4 58 13 55 33 27 48 22 46 24 35 25 15 53 2 60 39 29 42 20 6 64 11 49 21 47 28 34 56 14 57 3 32 38 17 43 61 7 52 10 50 12 63 5 19 41 30 40 59 1 54 16 26 36 23 45 The clssicl sis with the numers (1, 2, 3,, 8) nd (0, 8, 16,, 56) ws here used y Trry. But, other sis re ossile: 20 sis exctly (see my site ref [4]). I rememer the 10 first sis: 1, 2, 3, 4, 5, 6, 7, 8 ; 0, 8,16,24,32,40,48,56 1, 2, 3, 4, 9,10,11,12 ; 0, 4,16,20,32,36,48,52 1, 2, 3, 4,17,18,19,20 ; 0, 4, 8,12,32,36,40,44 1, 2, 3, 4,33,34,35,36 ; 0, 4, 8,12,16,20,24,28 1, 2, 5, 6, 9,10,13,14 ; 0, 2,16,18,32,34,48,50 1, 2, 5, 6,17,18,21,22 ; 0, 2, 8,10,32,34,40,42 1, 2, 5, 6,33,34,37,38 ; 0, 2, 8,10,16,18,24,26 1, 2, 9,10,17,18,25,26 ; 0, 2, 4, 6,32,34,36,38 1, 2, 9,10,33,34,41,42 ; 0, 2, 4, 6,16,18,20,22 1, 2,17,18,33,34,49,50 ; 0, 2, 4, 6, 8,10,12,14 I found tht the Trry s ttern with ll the sis gives: 46,080 semi-imgic sures, i.e. 2,304 for ech sis from 1 to 20, 3,584 imgic sures, distriuted s follows:
3 Bsis N of s. 1 320 2 192 3 128 4 128 5 128 6 320 7 128 8 192 9 128 10 128 11 128 12 128 13 192 14 128 15 320 16 128 17 128 18 128 19 192 20 320 Totl 3,584 Among the 3,584 imgic sures, there re 3,584/4 = 896 uniue sures. All re Greco- Ltin (with the indicted sis) nd ndigonl of tye comlete. I hve rinted ll these sures. For exmle, here is the first imgic sure with the second sis: 17 39 12 62 24 34 13 59 4 54 25 47 5 51 32 42 30 44 7 49 27 45 2 56 15 57 22 36 10 64 19 37 41 31 52 6 48 26 53 3 60 14 33 23 61 11 40 18 38 20 63 9 35 21 58 16 55 1 46 28 50 8 43 29 The two semi-digonls of this sure re mgic ut not imgic. There is n isomorhism etween the 320 sures of the first sis nd the 320 sures of the 20 th sis, nd more generlly etween the sures of one sis nd the sures of the corresonding sis. The exlntion lies in the fct tht we ss from one sis to the corresonding sis y exchnge etween the low nd the high comonent sure (see my site) nd tht there is here etween these two elementry sures - the geometric trnsformtion (16745238) ll *G, with convenient ermuttion fter (see next rgrh). The geometric trnsformtion (16745238) ll *G lies then the sures of one sis on the sures of the corresonding sis.
4 GENERALIZATION I insected the 320 imgic sures coming from the Trry s ttern with the first sis. All these sures hve the sme structure: - the low comonent sure is vrition of the s. # 454 1 2 3 4 5 6 7 8 5 6 7 8 1 2 3 4 4 3 2 1 8 7 6 5 8 7 6 5 4 3 2 1 7 8 5 6 3 4 1 2 3 4 1 2 7 8 5 6 6 5 8 7 2 1 4 3 2 1 4 3 6 5 8 7 - the high comonent sure is vrition of the s # 49 1 2 3 4 5 6 7 8 3 4 1 2 7 8 5 6 5 6 7 8 1 2 3 4 7 8 5 6 3 4 1 2 6 5 8 7 2 1 4 3 8 7 6 5 4 3 2 1 2 1 4 3 6 5 8 7 4 3 2 1 8 7 6 5 The sures # 454 nd # 49 elong to the list of the 1,152 digonl Ltin sures of order 8 which re self-orthogonl nd which hve 1,2,3,4,5,6,7,8 on the first row. Cf my enumertion of some 8x8 Greco-Ltin mgic sures, ref [3]. A Greco-Ltin sure cn then e mde with the s. # 454 nd # 49 which re orthogonl. These two sures elong to the fmily of the s #2 (I rememer I showed tht ll the 1,152 sures cn e generted from 8 sic elementry sures #1, #2, #3, #4, #6, #8, #10 nd #13 y lying the 1,536 geometric trnsformtions of G 1,536 nd the 40,320 ermuttions on the 8 first numers). The fmily of the s #2 is mde of 96 sures (out of 1152). We cn then try to generlize the Trry s ttern y serching other coules of digonl Ltin sures for uilding Greco-Ltin sure. Ech Greco-Ltin ttern genertes 20*(8!)*(8!) sures. We cn serch fter, mong ll these sures, those which re imgic. I mde this tsk on the self-orthogonl digonl Ltin sures. Among the 1,152 elementry digonl Ltin sures which re self-orthogonl, I found there re 14,784 coules of sures which re orthogonl or Greco-Ltin: there re 14,784 different digonl Greco-Ltin tterns mde from 8x8 selforthogonl digonl Ltin sures
5 (ech elementry sure hs severl orthogonl sures in the list ; mong these 14,784 coules, we hve nturlly the 1,152 coules mde with one sure nd its trnsosed ; these 1,152 coules never give imgic solutions). But imgic solutions re only found with coules mde with 2 sures coming from the 96 sures of the fmily of the s #2 (there re 1,152 orthogonl such coules, nd finlly only 864 orthogonl coules give imgic solutions. More exctly, ech sure out of 96 hs 12 orthogonl sures, ut ech sure hs only 9 imgic orthogonl sures, nd 9*96=864). there re 864 different imgic digonl Greco-Ltin tterns mde from 8x8 selforthogonl digonl Ltin sures In my enumertion of coules, I count for 2 solutions the coules x-y nd y-x. I hve file with the 14,784 coules nd the 14,784 digonl Greco-Ltin tterns. I hve defined stndrd osition of Greco-Ltin sure y mgic sure with the first sis nd with 1,10,19,, 64 on the first row. I found 552,960 imgic sures with the first sis. The grou G 1,536 is working on this set nd then we cn reduce the numer to 552,960/1,536 = 360 elementry (or essentilly different) sures. I hve rinted these 552,960 nd 360 solutions. For ll the sis, I found 2,016 elementry imgic solutions y reduced rogrm/1536 with 10 runs: Bsis Numer of elementry s. 1 360 2 216 3 144 4 144 5 144 6 360 7 144 8 216 9 144 10 144 Totl 2,016 We hve similr distriution (ccording to the sis) s for the ove-mentioned Trry s sures tultion: the sme numer for sis #1 nd 6, for sis #2 nd #8, nd for sis #3, 4, 5, 7, 9, 10. And nturlly the sme numer for sis nd for its corresonding sis. But the sures of one sis re here identicl to the sures of the corresonding sis, ecuse there is trnsformtion of G 1,536 etween the 2 sures of ech coule of Ltin sures, nd when we exchnge these 2 sures, we find the sme resulting elementry sure. I hve rinted ll these 2,016 solutions in 10 files. Unfortuntely, ll these 2,016 sures re not different: they re different in ech file, ut sme sure cn er in 3 different files out of 10.
6 I hve filtered ll these 2,016 sures, nd I found 1,344 different sures: 1,008 sures er only 1 time in 1 file nd 336 sures er 3 times in 3 different files (2,016=1,008+3*336=1,344+2*336=3/2*1344). These 1344 sures re rinted in file with the identifiction of ech sure mong the 10 source files The totl numer of imgic sures generted y this method is then 1,344*1,536 = 2,064,384 when counting ll the sures or 1,344*192 = 258,048 uniue imgic sures TYPES OF THE BIMAGIC SOLUTIONS I insected the elementry imgic nd the uniue imgic solutions. There re 97 tyes of sures, distriuted in 5 fmilies defined in n revited form y A1+X=65, X eing one of the 49 coloured cell in the figure herefter (in fct the ttern is lso necessry for defining tye nd there re 2 tyes for given X, excet for X=H8): Here re the 5 fmilies: Red (1 cell)....1 tye (ssocitive) Yellow (6 cells)... 12 tyes (comlete nd isomorhic tyes) Ornge (6 cells)...12 tyes Green (12 cells) 24 tyes Blue (24 cells)..48 tyes Totl (49 cells) 97 tyes I count 2 tyes for 2 cells symmetricl y the first digonl (insted of only one Dudeney s tye). Here is the distriution of the sures into the 5 fmilies:
7 Fmily Totl N of uniues (x192) N of tyes N of uniues y tye Red 112 21,504 1 21,504 Yellow 224 43,008 12 3,584 Ornge 336 64,512 12 5,376 Green 224 43,008 24 1,792 Blue 448 86,016 48 1,792 Totl 1,344 258,048 97 We find 21,504 ssocitive uniue sures out of the totl of 161,472 found y Wlter Trum nd 3,584 comlete uniue sures 29,376 CONCLUSION I enumerted ll the sures coming from the Trry s ttern nd I showed method for finding rticulr kind of imgic sures: the sures coming from Greco-Ltin sures. This method ws lied to the 8x8 digonl Ltin sures which re self-orthogonl. An extension to ll the 8x8 digonl Ltin sures cn e tried, ut the tsk is surely very hrd. Idem for n extension to orders suerior to 8 (I rememer tht the order 8 is the minimum order for finding imgic sures). When crrying through this study, I found n ccessory result: there re 14,784 different digonl Greco-Ltin tterns mde from 8x8 self-orthogonl digonl Ltin sures. And ech ttern genertes 20*(8!)*(8!) sures. ACKNOWLEDGEMENTS Thnk you to Ale de Winkel who ointed out tht my 2,016 elementry imgic sures were not ll different nd who found the numer of 1,344 for the reduced set. He gve me lso the rincile of filtering rogrm. I found the sme numer y rogrm which gives esides the identifiction of ech sure in ech sis. REFERENCES [1] Gston Trry, Comte-rendu de l AFAS, 1903,.141 [2] Jcues Boutelou, Crrés mgiues, crrés ltins et eulériens, Choix 1991 See ges 152-154 for Trry nd ges 114-116 for Greco-Ltin sure. On ge 152, there is formul for the imgic condition on Greco-Ltin sure. [3] Frncis Gslou, enumertion of some 8x8 Greco-Ltin mgic sures (mil of July 18, 2010 nd mil of Octoer 5, 2010) [4] Site htt://www.gslou.fr/mgic-sures/