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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!$hdrs/downrev.xmlLAO@$͘xbLJݡ-vg߻@8y/{ӊ#9XV8J@V7\)z>Yckrq7cs:>Cu]&/k2G#:!FWIᦕOIJ Dž;z+FOsy[ts?R D><'t-/O lW_} wDh rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!$hdrs/downrev.xmlPKV W =  g4  E   ( <`5 "PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!]@drs/downrev.xmlDAO1&fHAWDW 1&Enp7^*^βiFerdeRp"jG.PF$! nɠځ8u; ]#c^f\8-8sKo7 z+w3vZ(u=O O^3M+H&:]Ηd,A.PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!]@drs/downrev.xmlPK W V =  g5  E   ( <u5 "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.xmlDN0EH5HUVx[6icuAw\zr)DYA9+@^[7 1!k=#EX./Xi-ԉ 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L is the only latin square of order n which has symbol k in (i,j)-cell for each and 2. no proper subset of c has property (1).X!0  3fb#  j  ]  / ~  BA a?Object 5&/ a  lF??Rectangle 8%% Z  ~  BA b?Object 7  b  n? ??Rectangle 10 Z  ~  BA! c?Object 9p Z  c  `$??Text Box 11f   A critical set*0    rA"??latin05Picture 12latin05 w  fff̙___PPT10i.G:+D=' = @B +"   P(    ^`4 ??Text Box 4<w>  O 00 NbWQv N PMOn NiQ Pgb_bN PLuƖ(Critical Set)0\N bYN N/fLu6q YN(3 P)_NSN|~~kXb/UNvbNec0NQ0 g g3f&gb         pA#??latin06Picture 5latin06/  fff̙___PPT10i.p)J;+D=' = @B +"   e ] `E (     x ( 0e0e?Rectangle 2"ÚPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!F/drs/shapexml.xmlTn1}G,4Dݠ6R)nȻMxmc{4_{> ərrq*7Fg3KS5zۓK|hG7Wv-c6t0F¿7VjjZ:"Q|ЊFL]nnt2~~vεh^b$Aı0 ri!S4 kF5bOjJ Q&(>d<9MI~'t&oL8 ﹠@w v!8E#xRȌ.F`sI\vnTFV3^pyץi ے)|X gGÝ7ʨ v 纑M0CQnD#O+" >jC+y$Zc8dE~'Cҗbo6[BxEbTXD9Y}VhbR$e\%--l1l+עnŕMY_5l錩#0WRD;tjs(9x.#%J Ð -YXr.HoV݋*.O)٪I:2oAn]]·;h;N B(mW\-J`qYS)^ HҿRÌ}OA4up+EpӊPK!)ƃdrs/downrev.xmlDN0EH5Hã uBjEMC6mcνgM/|gY$A\Yq`yYcorq~6L#PFD ! jɠ؁8fuC<]#c^$I* vZ豥2 _E=7]azۢRD1?|_ Now^DhMA.~PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!F/)drs/shapexml.xmlPK-!)ƃdrs/downrev.xmlPKA`}   _Critical Set for Sudoku  2   x$0e0e?Rectangle 5 `  fIf we expect the solution of a Sudoku puzzle is unique, then the partial latin square shown must have contain a  critical set in the sense of satisfying the requirements of a Sudoku game. Sometimes, we did find more than one solution for some game.B pAA  fff̙80___PPT10.w`"H  OGp/(  s  ^$??Text Box 4d aFact 1 A critical set C of a latin square L provides minimal infos from which L can be reconstructed. Fact 2 Deciding whether a partial latin square is a critical set is NP-complete. (From completion point of view.) Fact 3 Denote the minimum size of a critical set of order n by M(n). [D. Curran & G.H.J.van Rees, Cong. 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The seven  lines are the 3 outer edges of the triangle, the 3 lines passing through the middle, and the circle.    5  *??6Slide Number Placeholder 3_   p*( A   33`(  @(     0e0e?4Slide Image Placeholder 1      @+0e0e5%?(Notes Placeholder 2 0   kThe sets are the lines. Every two sets have exactly one point in common. Every point is in exactly 3 sets. l  5   -??6Slide Number Placeholder 3_   p*( A   33f) &`(     0e0e?4Slide Image Placeholder 1      +0e0e5%?(Notes Placeholder 2 0   SComplete graph with each edge directed. Another example of a combinatorial design. T TT  5  ,??6Slide Number Placeholder 3_   p*( A   33z* :2(     0e0e?4Slide Image Placeholder 1      -0e0e5%?(Notes Placeholder 2 0   What we study is the structure, i.e., how to make a schedule for the tournament, not the outcome of a particular game or tournament.   5  `.??6Slide Number Placeholder 3_   p*( A   33+ c[C(     0e0e?4Slide Image Placeholder 1      *0e0e5%?(Notes Placeholder 2 0   Whist is a type of doubles tournament. Whist is a card game similar to bridge. Design theorists are interested in existence of whist tournament with the given structural constraints. Whist tournaments exist whenever the number of players is a multiple of 4, or 1 more. Each game requires 4 players so if all players play in every round then the number of players is 4K. If one player sits out each round then the number of players is 4K+1. The first round is used to generate the other rounds by adding 1 with the convention that inf+1=inf. Point out particular players. Notice that each player plays in each round. Player 0 partners each player once and opposes each player twice.   5   ؏ ??6Slide Number Placeholder 3_   p*( A   33, ^V>(     0e0e?4Slide Image Placeholder 1      ' 0e0e5%?(Notes Placeholder 2 0   Theoretical techniques from combinatorics, algebra, number theory, geometry. Experimental techniques: write a computer program to check all possibilities (brute force).   5  8 ??6Slide Number Placeholder 3_   p*( A   33. Ps(     0e0e?4Slide Image Placeholder 1   (   @&0e0e5%?(Notes Placeholder 2 0   Sudoku puzzles are very popular. Here is a simplified example. Each of the numbers 1,2,3,4 must occur in each row, column and 2 by 2 subsquare. 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