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??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!<drs/downrev.xmlDn0DV8E)&j+ &#@R;;3oU8V:@WNn/K!"kLJ]16"pQAK,ӭvbLoxM涗oY ng`64꯳rN})4~~4{nF"&%0?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!<drs/downrev.xmlPK]  fff̙___PPT10i.Qk+D=' = @B +"  (  m  f` ??Rectangle 1 q A typical Sudoku puzzle r 4 cg && g 8N '(PR   A??æA typical Sudoku puzzle grid, with nine rows and nine columns that intersect at square spacesPicture 2A typical Sudoku puzzle grid, with nine rows and nine columns that intersect at square spaces. Some of the spaces are pre-filled with one number each; others are blank spaces for a solver to fill with a number.0V2  *  A??äThe previous puzzle, solved with additional numbers that each fill a blank spacePicture 3The previous puzzle, solved with additional numbers that each fill a blank space.}9  , fff̙___PPT10i. f+D=' = @B +"    (     f@"??Rectangle 3  ! N"g 3f&3f& 3fc 1   A??æA typical Sudoku puzzle grid, with nine rows and nine columns that intersect at square spacesPicture 4A typical Sudoku puzzle grid, with nine rows and nine columns that intersect at square spaces. Some of the spaces are pre-filled with one number each; others are blank spaces for a solver to fill with a number.jP%  3  D ^?eW[eJX 1   yExample of Sudoku     fff̙___PPT10i.;{+D=' = @B +"   0 $m(  $ $ ^"??Text Box 4iw jTwo Latin squares and of order n are orthogonal if 0<n0 2: *l   ~ $ BA 4?Object 5m K 4 $ l"??Rectangle 8-- R   ~ $ BA 5?Object 73 . 5 $ n R??Rectangle 10LL R    $ n E??Rectangle 12LL R    $ DA 6?Object 11 6  $ `E??Text Box 137  'Two orthogonal Latin squares of order 4( 2((    $ rA ??latin03Picture 14latin03S ^ fff̙___PPT10i.@X+D=' = @B +" @(o(  ( (  @0e0e33f8cԔ    ?? jL 1"9 /d \If you have two fingers, then you can prove that the above two Latin squares are orthogonal.]]$\  S (  l 0e0e? oRjL 26 9  0d How? 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P2 p0r   fff̙___PPT10i.й+D=' = @B +"     ` 0 (  0 0 ^`y??Text Box 47Y P are mutually orthogonal Latin squares of order n (MOLS(n)), if for 0 :b0 283f#_    0 l??Rectangle 6(( R   ~ 0 BA 7?Object 5d" 7 0 lu??Rectangle 8%% R   ~ 0 BA 8?Object 7l m 8 0 nZR??Rectangle 1011 R   ~ 0 BA 9?Object 9jo 9  0 rA ??latin04Picture 11latin04R   0 `??Text Box 12` jMOLS(4)  2    N  0 `??Text Box 13 7  Theorem Let n be a prime power and n`"2 . Then there exists n-1 mutually orthogonal Latin squares (best possible!) of order n. z 23fW   fff̙___PPT10i.0RN+D=' = @B +"  p4l(  4< 4 ^??Text Box 4U  p( ) NpvN PcR _N/fcR GR vQ-N . (**) There do not exist two orthogonal Latin squares of order 2 and 6. 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Euler s conjecture (1782) There does not exist a pair of orthogonal Latin squares of order n where n 2 (mod 4).[<  5  pMp  p  p8  Y   fff̙___PPT10i.[+D=' = @B +"     < (  <0  <  j@0e0e? jL 1"ðPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!7Idrs/shapexml.xmlU͎6;{]'+Y0AHlɒ#g [@ߢ3#hC/z7PIv:T]D•]J"-8SuQ}յFa>UD? b WkKʅa=AGm5f0 +2T!TTb|N.ՠ7lXb,dQxҞ%X]$@R8ec)7 <5(}ƣxj4y͙ZHeIڸqav Ӝ4 b*y@ǟZTM.cj&fjDT?ض;$` $Αvtp\lȵq',>zw.ۀbEDF穭l;؉/P/"N? M½fG-"YTRjyvgװ7a+*g!rhzvM\BŲdv2GhG=1KӓԾ/zmMX׼͎g8o^k6L {!>GBnۦnܧZfTi{!#\X}{$ym,k)[I2TՁTy識]d^ F28W\"Fx պژݛL]KAo@Mu}3+]U@Bu6C6Co^re3uꘇF,u~9z-3,l}[NrUP:4. 7jx)BQ>~蚏ᛀUmD?Eσ3/z`$?c~z,PK!q Kdrs/downrev.xmlDMK@EB nN(;-EhT5fބ1M\r/rf4#k & K ˤ` jg^S )*BhS)}QA?-q 1Rj7&ɣ4Xs|(0|en_K{c)u=  i+QoZu8Zr [4 PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!7I)drs/shapexml.xmlPK-!q Kdrs/downrev.xmlPKA`}j  F>___PPT9  fDisproof   + <  z0e0e?gQ[HrbMn@S 2 `  ]Theorem (B.S.P., 1959) There exists a pair of orthogonal Latin squares of order n for all n which is not 1, 2 or 6. Theorem (?) There do not exist 9 mutually orthogonal Latin squares of order 10.T` G` G   fff̙___PPT10i.}p+D=' = @B +"#  *"@ (  @ @ ^??Text Box 4M  LD00bPSNx n pPrime Power, YdkN(n) NvWvvnbNecxe(max) 6q/fg'YN(n) = n-10 00 00 \e N/fPrime Powervn 0000 ffx0W . [T. Beth, Abh. Math. Sem. Hamburg 53(1983)]c 0 / 6gggg/gg-g                       ,   @ l` ??Rectangle 6 R   ~ @ BA <?Object 5 p  < @ l??Rectangle 8 R   ~ @ BA =?Object 7S   = fff̙___PPT10i.9+D=' = @B +"  D(  D2  D  j c0e0e? jL 1"ðPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!aIdrs/shapexml.xmlU͎6;{]'+Y0AHlɒ#g [@ߢ3#hC/z7PIv:T]D•]J"-8SuQ}յFa>UD? b WkKʅa=AGm5f0 +2T!TTb|N.ՠ7lXb,dQxҞ%X]$@R8ec)7 <5(}ƣxj4y͙ZHeIڸqav Ӝ4 b*y@ǟZTM.cj&fjDT?ض;$` $Αvtp\lȵq',>zw.ۀbEDF穭l;؉/P/"N? M½fG-"YTRjyvgװ7a+*g!rhzvM\BŲdv2GhG=1KӓԾ/zmMX׼͎g8o^k6L {!>GBnۦnܧZfTi{!#\X}{$ym,k)[I2TՁTy識]d^ F28W\"Fx պژݛL]KAo@Mu}3+]U@Bu6C6Co^re3uꘇF,u~9z-3,l}[NrUP:4. 7jx)BQ>~蚏wMPBr .ƌ gfLoXONVwZa$1PK!ypOdrs/downrev.xmlDAK@a؍UNK[-9Z4;Mݰ;Ń=񴷭swW қU/;PQjc#GNNOƔp+ޯR bNHk˚-Łإn%I1T:$m0nƥ:~\Y{)b1 /o ~vJE=!q([2M'?PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!aI)drs/shapexml.xmlPK-!ypOdrs/downrev.xmlPKA`}j  F>___PPT9  h Conjecture     D  z m0e0e?gQ[HrbMn@S 2 `   A collection of n-1 mutually orthogonal Latin squares of order n exists if and only if n is a prime power and n 2. ( ) The proof follows by using the existence of a finite field of order n. (Exercise!),3f9  p0     " D VmG78cT]{_ 3#"@ÆPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!7drs/shapexml.xmlUn1#(ݤem"Q58z4S7np*x &iig/*A\RɔvڔpT^qJoGWJPt =?{NLM4I҉uEMx@\PK=j ,:Dtn JI*9$v=hR)=DB.?~o>!]5f%@!zxA(tՠAkwB,Rzǽ)Z{r$|a C}04qQ km[DE)Z;duLlxZ`v)3c<|${M R Swj9q09WCVB<??gٜ"&8x+|ΫR*7_< NRK',}s$(Rfuayf |[%fR,ln ǸQSً "]$$*;GGD`ϡ//˟)20O6j<29^3IB 9Cej6`6MͺpZ\bvmڋ>ahhfF06kB $Mh$c1|pLgUYD`)]MZ}u;9|vNYg)ԍ!]NqH5%SY-p 55'*势KtC ͥh ';BTWB}DMNGVnid .k^C@*%(ijy˶例/u-\4oPK!44g2drs/downrev.xmlDMK1E!ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!,%Qdrs/shapexml.xmlUn8/w x-R;^7Ec F`D.z,FeH.I9a_`{'+зHv(^6p~H]Zl5<їBY&!)@[SA<ʍلK*F7BR5)"~s^e"DLTޠ߿P9Pf')̽D0Pc?Ĺu/`3CWH{lQط{=.Pb([sXKV&r8 GK 7,lˎk[&W0[lE"vl^{XLH|8aFV//SKvj1M=0`r N#j;83ɑzq>x'HD|6tNlf!:0PbwZ,$lϸmXqLDfOM*yB; fگyCiM|Û24+<Dzs'!|G ;i8\ ?Xߔ;~5k/k֤x][%BNGɦD*s>E8c}:[|{Y s63Ͱ; ɗ] 9t#E'!O yvq">~_HtC 2Q)I-t8Bs<}MƇ%MPK!uEdrs/downrev.xmlDJAEДFLDtBY%t2m\r/rfu@! \M2Pĥ-^W [<",'3̭?[Up@#ZDz!q{U>8CmcNOF;l9=4}Cp׻b.K|bmٰ%4Xƪx_ԣ50U= F` %d zPK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!,%Q)drs/shapexml.xmlPK-!uEdrs/downrev.xmlPKA`}j  F>___PPT9   How many distinct Latin squares?*! $$     H  z`0e0e?gQ[HrbMn@S 2"C=PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!drs/shapexml.xmlXO7űdfaװm,&v1hf4i*iۧ\!!\z˽|-揽f[BR%5OzO*dɵI %\F*N,OGd̄GX>D4L:rFrM94\=><>dKLd`Ԯ.T~7,z"jF*fPR0)?Iӣ^ᴾ{pEGr],ldUjkrhHe9#9#'lɵ3sHL+{&ss/#c'we}t[D;k%|*~'cz4D;' b /a 19Z $(y2edSݖs8 %2h-{ 9.dM:ޗyީghYd.`'w$.x/ 5Xc+q}zIUܰ =]y/dӐ0 LM@?|b}2 Xy_YS(䡰#[g8cXCIX÷E@%N)wbtC(Yp@>r@\cHcJpv7J%Smמ2 !]+e*DQ"ݠcÑÅeڷC `ux-_[TŻ$1?onE%[ķrCpwj[ϸi,l깓q)LI\BePK!Kdrs/downrev.xmlDKK0!\MqR|P'3 Aks $v.bX$A\9mV۾\=Yc(b~~6Bob-C BP5d1L\Gcඕ7y~'-N vPulWnUonR#HcϬ6EjS^c$d,APK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!)drs/shapexml.xmlPK-!K.drs/downrev.xmlPK2 ` ^V___PPT980 Two Latin squares are distinct if they are different in at least one corresponding entry. We use L(n) to denote the number of distinct Latin squares of order n. L(1) = 1, L(2) = 2, L(3) = 12, L(4) = 576, L(5) = ? (Exercise)N" p0  p08      fff̙___PPT10i.H+D=' = @B +"0   7 / 66L (  LA  L  jB 0e0e? jL 1"óPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!_Ldrs/shapexml.xmlU͎6;{]'+Y0AHl#g [@ߢ3#hC/zF3o>RU;b]/_ 2.E։xUDpXL&wnQi b3 b15W1l5 a=h! %`4P9=Pn6R~[Ui堦|Vz;t@,|}w(d6GPK!Ldrs/downrev.xmlDj@EaxB7R'MAl("h[Rkg晄fބi]˽˙-zӈ-+' k.wS> kl,+yX73̴](EPABI鋊 mcw`ѕR;Did$iPaKݗQ`ڌNK<~ucQv/@xեGW^zrޢDh-APK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!_L)drs/shapexml.xmlPK-!Ldrs/downrev.xmlPKA`}j  F>___PPT9  tStandard Latin squares  | L  zC 0e0e?gQ[HrbMn@S 2 `  A Latin square of order n is called standard (jnW) if the first row and the first column are in increasing order 0, 1, 2, & , n-1.P$ gO8.  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K6 f¯|/eW4V$zybvԧOwBMftV/uڬjF\&KxM0&0#]7"Ӎ">=gxgLR D-ev 9zg^ff^6|'GbƥWBN(Q=`BAE%N7hzĚPZ&ݞ8nؽkE|muwn|DbpOPK!@adrs/downrev.xmlDj@Eax--:g3oTcޡvys9Egq&k ƣqauͥp deRp%ynyJ!STPЦR"~d[3bt/n9I'iPaK(0|m_K&1ϔ3@]gi+QZzpޢDh-AoPK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!d` )drs/shapexml.xmlPK-!@adrs/downrev.xmlPKA`}j  F>___PPT9  $Ryser s Conjecture   `  zx 0e0e?gQ[HrbMn@S 2 `j F>___PPT9  &For each Latin square of order n, there are at least n-1 distinct entries which are in  transversal positions (no two are in a row or column). If n is odd, then we can find n distinct entries as a transversal. 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This is a great problem to work.^D    fff̙80___PPT10."     h (  hL  h  j`?0e0e? jL 1"ÿPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!lY drs/shapexml.xmlV͊H/;4}]&xLlF$ f0#CIjي[ݽ%g qoy@"U-Cf!"CJU]?_gZ&R(ۢ2DZ\<" 5*;￝qp.0v\!q!2\+s+^YR{Gybdvԧ'wGBUjtV/uҬjF\&K\^*Y[lױ]7"Ӎ">=崋gθ$j;uϿv{njp2.rBFc ZAmq"G&W/߽5ě L1=>(p\Fp[N7>|PK!ZMdrs/downrev.xmlDAK@a؍DbZm&n_Al1rb띁I\mj/{PQY#'0~p[:Rq1GHkRwQR NON3.=4Ӫb|+|X7oc./(QRv_ԓ50ux___PPT9  sPartial Latin squares   h  z?0e0e?gQ[HrbMn@S 2@  QA partial Latin square of order n based on S is an nxn array such that each element of S occurs in each row and respectively each column at most once. 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g?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!@drs/downrev.xmlPK  ( ?B Lp s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!@drs/downrev.xmlDJ1!lUMK+i?0MfwɚDrfΜs|p)DYxT Xg1!l= EϮX-Tl±DMJ])e 9#[ÔPK]+b"Z vڐڝk?VV_-+no DC}|~9kM` g?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!@drs/downrev.xmlPK))( ?B Mp s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!@drs/downrev.xmlDJ1!lUMK+i?0MfwɚDrfΜs|p)DYxT Xg1!l= EϮX-Tl±DMJ])e 9#[ÔPK]+b"Z vڐڝk?VV_-+no DC}|~9kM` g?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!@drs/downrev.xmlPK )?B Np s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!@drs/downrev.xmlDJ1!lUMK+i?0MfwɚDrfΜs|p)DYxT Xg1!l= EϮX-Tl±DMJ])e 9#[ÔPK]+b"Z vڐڝk?VV_-+no DC}|~9kM` g?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!@drs/downrev.xmlPK ( )(  fff̙___PPT10i.P +D=' = @B +"   !  @t (  t t  js0e0e? jL 1"ËPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK! I$drs/shapexml.xmlTn@#4-6R):#zw]iȍ'C-;!pgv~9(*ky/2-*Lk)BSF%^fٹ' cUn|j/Se5Dd DֽAګ2r&usOT~{QHaF?8N `fu袀p= @c!1 9u+F`tBViqq{i93|c-t4,b露 x N7! %CiՄDf{r۹ęI]N!.;A$⣶ڦ+p`3 yl !^ݠF>5 z!)ĩɣPt!OhV_\5_(|)@y0+0KlycrvD(6fG}| |W뿥e^'͚-5@^aDbƧ=uSW4&He>n:Θ߭q egH.i-j3ilʔkԪ4u+~2tS%uEesomtT+`;6t{Ui6r%X]q:q;jHK#bDWlOWŝtoHǔlQ$EmEͳ2;y=I{䶙r ܆J푎d <<ےljgy!x~C"PK!jdrs/downrev.xmlDMK@Eax+ľ"Cjk̼ 3c;t˽Mz۪=8AdXJgfOB$1:a#//FwױR "!':.:5[ ױn缥tHpa=jK:~^X{)ff5ӆ_s~ *r˰<aj8n}cV"{L%/PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-! I$)drs/shapexml.xmlPK-!j{drs/downrev.xmlPKA`}   \Which one is possible?  t  z@Y0e0e?gQ[HrbMn@S 2 `  @Theorem If a partial Latin square is of size at most n-1, then it can be completed. (Evan s conjecture) Theorem If a partial Latin square is of size n, then it can be completed except some exceptions (such as two examples of last page).2b  fff̙___PPT10i.P@K+D=' = @B +"     Pxw (  x x  jY0e0e? jL 1"}wPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!ndrs/shapexml.xmlTn0 ;i~$S N3mˉY$9M {wۓ [4apH"?~$uy)%[ "=p&TB-"~?m]p< V"[K3te&KͰvRN m%xTmpBy𘨔^s.P|:63KR:],+8SPb_?uyq}Ltr ','5aF!Jjo b񅗂#MApv2X{aaYl;aXbMƞyXgOZEprt}lJ=$Uc-*ŨOأS]coF"#6~%`=q9 jM~+[)4QBO vF.r!pr[zY&9$ap!_K 8z6/A-p7wN=FS=d^d.Cqa^&iWE9^xrHcKz݂W!u k?7^wCљPK!z$Ddrs/downrev.xmlDKK1!\LGǦ|lji+wɝIt I:op9|oL:A;+ dȖNj[ x?<] Y0RltQ5K bl C`l* }ͥ>MçYv jVT~OFZy{o\}f!./HCߌFW)` z?; L%PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!n)drs/shapexml.xmlPK-!z$Dfdrs/downrev.xmlPKlA`}^  :2___PPT9  > t x  zY0e0e?gQ[HrbMn@S 2 `  `Now, we consider a special class of partial Latin squares which are very useful in applications.aa(a  fff̙___PPT10i.PPz+D=' = @B +"?  F>`|&(  |Z | ^ s??Text Box 4I  A critical set C in a Latin square L is a partial Latin square obtained from L with the following two properties 1. L is the only Latin square of order n which C can be completedand 2. no proper subset of C has property (1).  b/-Fp  A  /   | lY??Rectangle 8%% R    | np??Rectangle 10 R    | `Y??Text Box 11f   lA critical set    | rA??latin05Picture 12latin05 w  fff̙___PPT10i.G:+D=' = @B +"  p(    ^`Y??Text Box 4<w>  nO 00 NbWQv N PMOn NiQ Pgb_bN PLuƖ(Critical Set)0\N bYN N/fLuƖ6q YN(3 P)_NSN|~~kXb/UNvbNec02R gJcb          pA??latin06Picture 5latin06/  fff̙___PPT10i.p)J;+D=' = @B +",   KC+(    pA??latin07Picture 4latin07* 6   pA??latin08Picture 5latin08 V6   xA??latin07-2Picture 6latin07-2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!"drs/downrev.xmlDOk@t㟊FWK)F?$}d&݂03aVA,+ Ĺ .`deRC6 Sm[># )*(Cp>/ɠZGm (B7'L8.hWR~Ͼɻlo_>w^v "PV0Y,c\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!"drs/downrev.xmlPK* 7 ,$D  ^ (]??Text Box 7V v  tCritical sets, n=5 2   fff̙yq___PPT10Q.+7{D%' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(DM' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B-randombar(horizontal)*<3<*+"  "(    ^z??Text Box 4d Fv A critical set C of a Latin square L provides minimal infos from which L can be reconstructed. Deciding whether a partial Latin square is a critical set is NP-complete. (From completion point of view.) Denote the minimum size of a critical set of order n by M(n). Then M(n) is at least [n2/4]. [D. Curran & G.H.J.van Rees, Cong. Numer. 1979] (Conjecture!) x` x xk x x x' Aw   lY??Rectangle 6 R    fff̙___PPT10i.@;+D=' = @B +"}  |d(     ~0e0e?Rectangle 2A`}   " 0   x0e0e?Rectangle 3 `  dApplications in Cryptography<  fff̙___PPT10i.`)<+D=' = @B +"  t(  <  ^@v??Text Box 4jI PAn (s,t)-secret sharing scheme is a system where s pieces of information called shares or shadows of a secret key K are distributed so that each participant has a share such that: 1. the key K can be reconstructed from knowledge of any t or more shares; and 2. the key K can not be reconstructed from knowledge of fewer than t shares. Q QQ   fff̙___PPT10i.PА+D=' = @B +"q  xpX(  -   ~`p0e0e?Rectangle 2A`}   [ A sharing scheme(    x0e0e?Rectangle 3 `  Key: A Latin square L From L we design a critical set C for the time being. (This critical set can be changed.) The shares are partial Latin squares which are contained in C. >6u  fff̙___PPT10i. +D=' = @B +"  (  R  ^??Text Box 4F  Rce!j_ (Sharing Scheme) 00 00(WяN g_Y͑'Yvzl[ pNxOzlV{N zl gfu 8^gc(u1uY PN Tav`l NMbWLOYё^ |v\8hH_& 0@bN ^zN P|q}O_\vNxe!ql_U/f g[v_'`0dr cgccXcF  <     fff̙___PPT10i.g:+D=' = @B +"  l(  4  ^Y??Text Box 4*  @LuƖvxd g_Y ! \eƖvN NY0 ! XR4x㖦^ ! !& !   fff̙___PPT10i.T+D=' = @B +"s  z r Z (  T    j@0e0e? jL 1"ôPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!WMdrs/shapexml.xmlU8 ?dLStL[ =-'ndI+ə$_L:(K}IGնb|I䶨27g/Lȝ +7Nf.ݸ J[ Jkey1P{~WCe2=IfEU$r C__,esh}|,dQxҾI@c )1 u+wsUJb~Dr0zA֛3|c-t?4,b兩?~gB"G!632|q9@şZSiՄ㩉4s[;3C\l?wDGmMW;f"sj CLN`=g\6 w, 8T@"zSG/EdCЎkPZ_  v=2_g͚-5(`Gv͆1qħ]uSWT14HeާH7gWgx \`"kIt-j3ilʒkDU΋؊ EګmsomrQc)qT6{SiR Ȉe]/#n,U.c|*E.*s8Σ^E*YL1Ƞ#Ǒ,y~-Y~S}oi,X(uM_@Dw_z  ׃ C$ %d rPK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!WM)drs/shapexml.xmlPK-!Gdrs/downrev.xmlPKA`}j  F>___PPT9  Latin square changes my life*$p$p     zи0e0e?gQ[HrbMn@S 2'  It is a story now! I went to Auburn University (1977) with a teaching assistantship to learn Topology which I felt very interesting when I was an undergraduate student (as a x[_ of +^). Auburn was a power house of General Topology in 70 s. (Not Football)]p0Ig g Ep0T    H   fff̙___PPT10i.a\+D=' = @B +"     (  D    j ]0e0e? jL 1"ôPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK! Mdrs/shapexml.xmlU͎6;{]'+Y0AHlɒ#g [@ߢ3#hC/z7PIv:T]D•]J"-8SuQ}յFa>UD? b WkKʅa=AGm5f0 +2T!TTb|N.ՠ7lXb,dQxҞ%X]$@R8ec)7 <5(}ƣxj4y͙ZHeIڸqav Ӝ4 b*y@ǟZTM.cj&fjDT?ض;$` $Αvtp\lȵq',>zw.ۀbEDF穭l;؉/P/"N? M½fG-"YTRjyvgװ7a+*g!rhzvM\BŲdv2GhG=1KӓԾ/zmMX׼͎g8o^k6L {!>GBnۦnܧZfTi{!#\X}{$ym,k)[I2TՁTy識]d^ F28W\"Fx պژݛL]KAo@Mu}3+]U@Bu6C6Co^re3uꘇF,u~9z-3,l}[NrUP:4. 7jx)BQ>~蚏!,+Z ږK@38Fjsj<<ꧧ$PK!2drs/downrev.xmlDAK@aM[*9Z4;MBawmS}z|&޶>4N KL#T$Z'pńr㎲:V*A$PZfKa:휷S6 n[=ʲ;mPSǏ5EM7ۯ9}luQPD"4qW^ YQ[2MgPK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-! M)drs/shapexml.xmlPK-!2drs/downrev.xmlPKA`}j  F>___PPT9  vFortunate or unfortunate  v   z\0e0e?gQ[HrbMn@S 2 `  They did not have enough students to open the class of Topology that year, they need at least three. I was informed to take another class besides Algebra (I) and Analysis (I) which I have chosen already. I did have many courses to choose. So, it comes  Combinatorics . N  O     fff̙___PPT10i.T+D=' = @B +"      (  J    j?0e0e? jL 1"÷PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!Pdrs/shapexml.xmlU͎FG;vgAcV;HDB#4fUnv]fg [,R"Ue(!j]Wwڪ .їZrDX^*"w&_]7(:I1ߘ+G҇a=h! `4P9=#Wn6R~[Ui堦.7lX|,dQxҞ%.P ` )q m6  Z)}ǣxj4y͙ZHeIڸav ӌ4 bZ=-Jb'15e3~}5Mxɞj\[Ͻe &:;s}.\lȵ4!jBX$||ƈ``'6bk CQԋ8R6 "D8T "vMScЪ0 #hcgo`nV{9A^~蚏3ӭN_% UzjS翥 ?;9;[ݽ!hm7%PK!\6Cdrs/downrev.xmlDAK@a؍DcJ!*ަiΆݵM.xMfԞ}h \2P,3/[P!0‘̦'ʍ;HuUH ϵUÖ,9o)km<vze7R+顡>_^o~{˷%0ybU E=1q[SR[2M?PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!P)drs/shapexml.xmlPK-!\6Cdrs/downrev.xmlPKA`}j  F>___PPT9  yThe first subject I learned  ,   z@q0e0e?gQ[HrbMn@S 2 `  ^The first subject I learned from Prof. C. C. Lindner was  Latin square . He claims that Latin square is the most important combinatorial configuration in combinatorics. It took more than one month to complete his lectures on Latin square at that year.('   fff̙___PPT10i.Nr+D=' = @B +"   (     ~@[0e0e?Rectangle 2A`}   <From C. C. Lindner    x[0e0e?Rectangle 3 `  fDo whatever you enjoy the most not others. No one can stop your research unless you don t want to do it anymore. He is 75 now and he is working as hard as any young researchers. 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LindnerPowerPoint ²  ϥΦr  GDDO OLE A{ vD* 8@ _PID_HLINKSAH=http://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714.svgFhttp://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714_solution.svg=http://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714.svg=http://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714.svgFhttp://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714_solution.svg=http://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714.svg=http://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714.svgFhttp://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714_solution.svg=http://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714.svgFhttp://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714_solution.svg=http://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714.svg=http://en.wikipedia.org/wiki/File:Sudoku-by-L2G-20050714.svg_$USERUSER  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnpqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-.012345689:;<=>KRoot EntrydO)PicturesGCurrent User7SummaryInformation(PowerPoint Document(o7DocumentSummaryInformation8/