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Player 2 places his pieces by reflecting the position of Player 1 across both the horizontal and vertical axes.`(L} Even-by-Even $ Even-by-Even Case Symmetry$ Even-by-Odd  $ Player 1 s optimal winning strategy is to place his first tile in center of the board. Then he continues to play according to symmetry. By placing his piece in the center of the board Player 1 has reduced the game to a  win state .(Even-by-Odd Case Symmetry$Linear Cram: 1 x n case*$While developing strategies for Linear Cram we discovered that placing the first piece, reduced the game into two smaller games Subsequently, each piece that was placed reduced the game into smaller and smaller sub games( Linear Cram $ Strategy for 1 x n case: Each time a domino is placed on a new board, the game is broken into two separate games, 1 x m1 and 1 x m2, where m1 + m2 = n-2.p()w   Linear Cram $ ?Example: 1 x 11 case becomes two distinct cases 1 x 4 and 1 x 5@)@ Linear Cram $ pA 1 x n case that has been broken up into k cases, will be broken into k + 1 cases by placing the next domino. q(q Linear Cram Strategy$Each case has a particular value known as  Grundy numbers Grundy numbers can tell us which player will win a case when the players play perfectly. Theorem: Player 2 wins if and only if the Grundy number = 0.(!Linear Cram Strategy$*Since linear cram is made up of several cases played simultaneously, the Grundy numbers must be  combined. Exclusive Or Function Example: 3 5 = 6 B()"Linear Cram Strategy$Theorem: Player 2 wins a  multi-pile case of k piles if and only if G(1) G(2) & G(k) = 0, where G(i) is the Grundy number of the ith case. Winning strategy: If there are k cases, move in such a way that G(1) G(2) & G(k) G(k+1) = 0. (K 3M $$((,,004488n#Strategy Example: 1 x 7 case$FG(2) = 1, G(3) =1. G(2) G(3) = 0.($( % Calculating Grundy Numbers$@By definition: G(0) = 0 and G(1) = 0 G(2) = 1 G(3) = 1 G(4) = 28()(A  $Grundy Numbers$Definition: G(0) = G(1) = 0 and G(n) = mex({G(i) G(n  2  i) | i = 0, & , n  2}).  mex is the minimum excluded value of the set, (i.e., the smallest nonnegative integer not in the set.) @()2,('g&!Calculating Grundy Numbers$We wrote a program to calculate the first 300 Grundy numbers (G(0) to G(299)) using the previously stated algorithm. Our data indicated that the Grundy numbers of the 1 x n case attained a periodicity of 34.(,"THEOREM$ In linear cram, the second player wins only on a board that has a number of blocks equal to 0, 1, 15, 35 or (modulo 34) to 5, 9, 21, 25, and 29. "(-#PROOF$a(To show): The Grundy numbers are periodic of period 34. [ie. G(n) = G(n + 34) for all n 68]|;('(;       ,<.$ QInductive step- Assume true for no. - [ie. G(n) = G(n + 34) for all 68 n no]p$(.)! >(/%0&1'2( Future Research$jOdd-by-Odd Case Other Cases - wrap around case - other polyomino cases - variations of the domino cases((O)k9 )B) REFERENCES Acknowledgements$YDan Rudolph, PhD Marshall Cohen, PhD Demond Moy Joelle Davis Carter Turtle, turtle!!!!!! (Y()Z>%$/T'(3  ` f3f3` ff̙` ___` ff` 3ff3f>?" dd@,?lPd@   Z l<@ d`"  n?" dd@   @@``PR    @ ` ` p>> 91 (  ^T y  y"Z  S B7 C DEHFPv @?q 6 6 n(dx  G b q %(@B y"2  ZG1?"  T/ gֳgֳ ?P  T Click to edit Master title style! !"  TD0 gֳgֳ ?``  =*"  T0 gֳgֳ ?`   ?*"  T1 gֳgֳ ?`   ?*"$  0d1    RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S"`  Hd޽h? ? f3f3 Soaring  ` (  T t  t"Z  S BsC5 DEHFPv @?3 r4 rv$[I_l3HxF 3 %(@ "2  3 CRENGCJRQ1? `TO`TR`TO`TRR`TRRt"  T$ gֳgֳ ?/O  T Click to edit Master title style! !"  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"  $ s * J^ ,$D 0 % s * ,$D 0 & s * ~ ,$D 0H  0޽h ? ̙33"___PPT10+QD' = @B D' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*%Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*&D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*&+;  0\(  x  c $$i*P     c $i* <$D 0  H  0޽h ? ̙33___PPT10.+.V8D' = @B DJ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*;D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*;D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*;D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*;D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+ +   }@(  x  c $k*P     c $$l* <$D 0   " n #  " n,$D 0#"  <.?" ( n O6 = 110 @`  <d1?" (  O5 = 101 @`  <1?"   O3 = 011 @``B  0o ?"  ZB   s *Ԕ ?" ( ( `B   0o ?" nn`B   0o ?" " `B   0o ?  `B   0o ?" ( " n`B  0o ?" " ( `B  0o ?( n`B  0o ? ( H  0޽h ? ̙33___PPT10.+<D' = @B DC' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*lD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*lD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+ +   P\(  x  c $1P     c $1 <$D 0  H  0޽h ? ̙33  ___PPT10l .+.V8D ' = @B D ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+ +7,  `4(  x  c $1P     c $1 <$D 0  B   D #  D,$D 0#" r  <d1? D T  @`  <1?w D T  @`  <$1?R wD T  @`  <1?. R D T  @`   <1?  . D T  @`   <D1? D T  @`   <1? D T  @``B   0o ?  `B   0o ?DD`B  0o ? DZB  s *1 ? DZB  s *1 ?  DZB  s *1 ?. . DZB  s *1 ?R R DZB  s *1 ?w wDZB  s *1 ? D`B  0o ? D  61 R ;,$D 0 8P1  6d1 uJ,$D 0 8P2  61 J,$D 0 8P1H  0޽h ? ̙33;3___PPT10.+ajD ' = @B D' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ ++0+ +W  22(  x  c $a*P  1   c $~F  <$D 0 1 x  c $F p  1 @  D # Sy,$D 0#" r  <dF?3 D P  @`  <F? 3D P  @``B  0o ?  `B   0o ?DD`B   0o ? DZB   s *1 ?3 3D`B   0o ? Dn  6  #   ,$D 0#"   <$F? 6 P  @`  <F? 6 P  @`  <F? 6 P  @``B  0o ?  `B  0o ?66`B  0o ? 6ZB  s *1 ? 6ZB  s *1 ? 6`B  0o ? 6  D #  L ,$D 0#" r  <DF? D P  @`  <F?@ D P  @`  <F? @ D P  @`  <dF? D P  @``B  0o ?  `B  0o ?DD`B  0o ? DZB  s *1 ? DZB   s *1 ?@ @ DZB ! s *1 ? D`B " 0o ? D  D ##  a,$D 0#" r $ <ĂF? D P  @` % <$F?@ D P  @` & <F? @ D P  @` ' <F? D P  @``B ( 0o ?  `B ) 0o ?DD`B * 0o ? DZB + s *1 ? DZB , s *1 ?@ @ DZB - s *1 ? D`B . 0o ? D / s *fp,$D 0 0 s *  ,$D 0 1 s * %C ,$D 0 2 s ** uu,$D  0H  0޽h ? ̙33;;___PPT10;.+YDE;' = @B D;' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*&D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*&D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&/%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*&/D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*&/D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*/8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*/8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8A%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8AD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8ADn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*/D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*/Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*1D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*1Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*#D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*#Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*2D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*2+8+0+ +  p0(  x  c $1P  1 x  c $1  1 H  0޽h ? ̙33  0(  x  c $FP  1 x  c $dF   H  0޽h ? ̙33   (    c $FP<$D 0  1   c $DF  <$D 0 1 H  0޽h ? ̙33___PPT10j.+ D' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+p+0+ ++0+ +  (  R  0ĈF ,$D 0 F(By induction): Basis step- The statement is true for 68 n 265.: 7 >   <   c $$F  <$D 0 )   c $FP<$D 0  ) `  c $A F?? F`  c $A J?? 0 J`  c $A K??  KH  0޽h ? ̙33___PPT10w.+#{D ' = @B D ' = @BA?%,( < +O%,( < +DN' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*;D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*;D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;a%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*;aD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*;aD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+p+0+ ++0+ +  (    c $I ` @<$D 0    0I,$D 0 N.For (no + 1): -Show that: {G(i) G([no + 1]  2  i) | i = 0, . . ., no  1} = {G(i) G([no + 1]  34  2  i) | i = 0, . . ., no  34  1}                      *     `  c $A L??0 L`  c $A M??G MH  0޽h ? ̙33OG___PPT10'.+r/D' = @B DJ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$R%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$RD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$RD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+p+0+ ++0+ +i &(  z  0$Ipw Since the periodicity starts at 68, (no + 1)  34  2  i must be > 68 jK 2&" " * "  "    0I :,$D 0 `We know that i must be 0. So 0 i < no  1031 2"  "  " " * "  ,  0DI@0 -,$D 0 no  35  i > 68^ 2" * "  "  2  0I` @p ,$D 0 i < no  103j 2"  " " * "  `  c $A N??0 NH  0޽h ? ̙33{s___PPT10S+(lvD' = @B DJ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+ ++0+ ++0+ + <4(  H  0I  In order to get the mex{G(i) G(n  i  2) | i = 0, . . ., n  2} values of i such that 0 i are needed because once i exceeds the Grundy calculations will agree with a prior calculation because i = n  i  2 when i = .  2"  " " " " " " "  ,N  c $A 9??p (x 9d  c $A '?? P x 'd  c $A 5?? p x ` x 5d`  c $A P?? w  P`  c $A Q??0  QH  0޽h ? ̙33L B(     0$I,$D 0 x(With this being the case, 4) 2"  "    0Ip  ,$D 0 no > 204 ^ 2" * "  "  l  0I = ,$D 0 &no  2 < 2no  206  2" * "  " " * " "  Zl    ,$D 0  s *A (??tx (d  6dI  < no  103.^ 2 " " * "  H  0޽h ? ̙336.___PPT10+wD' = @B D=' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+p+0+ ++0+ +# q(  *  0D1 ,$ 0 By induction the statement holds true as long as no > 204. By computation the statement is true for 68 d" n d" 265. v 2(22" * "  A"     0dI ` ,$D 0 XQuod 2   0ďI 0  ,$D  0 iErat . 2g "   0I = ,$D  0 zDemonstrandum 6 2 g G  H  0޽h ? ̙33___PPT10+lTD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Du' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*++0+ ++0+ ++0+ +   D$(  Dr D S $ P   r D S     H D 0޽h ? ̙33n  (  l  C IP  I   0I@0  Knotted Doughnuts and Other Mathematical Entertainments. Martin Gardner. W. H. Freeman & Co., 1986.  The G-values of Various Games. R. K. Guy and C. A. B. Smith in Proceedings of the Cambridge Philosophical Society, Vol. 52, Part 3, 1956, pages 514-526. 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Player 2 places his pieces by reflecting the position of Player 1 across both the horizontal and vertical axes.`(L} Even-by-Even $ Even-by-Even Case Symmetry$ Even-by-Odd  $ Player 1 s optimal winning strategy is to place his first tile in center of the board. Then he continues to play according to symmetry. By placing his piece in the center of the board Player 1 has reduced the game to a  win state .(Even-by-Odd Case Symmetry$Linear Cram: 1 x n case*$While developing strategies for Linear Cram we discovered that placing the first piece, reduced the game into two smaller games Subsequently, each piece that was placed reduced the game into smaller and smaller sub games( Linear Cram $ Strategy for 1 x n case: Each time a domino is placed on a new board, the game is broken into two separate games, 1 x m1 and 1 x m2, where m1 + m2 = n-2.p()w   Linear Cram $ ?Example: 1 x 11 case becomes two distinct cases 1 x 4 and 1 x 5@)@ Linear Cram $ pA 1 x n case that has been broken up into k cases, will be broken into k + 1 cases by placing the next domino. q(q Linear Cram Strategy$Each case has a particular value known as  Grundy numbers Grundy numbers can tell us which player will win a case when the players play perfectly. Theorem: Player 2 wins if and only if the Grundy number = 0.(!Linear Cram Strategy$*Since linear cram is made up of several cases played simultaneously, the Grundy numbers must be  combined. Exclusive Or Function Example: 3 5 = 6 B()"Linear Cram Strategy$Theorem: Player 2 wins a  multi-pile case of k piles if and only if G(1) G(2) & G(k) = 0, where G(i) is the Grundy number of the ith case. Winning strategy: If there are k cases, move in such a way that G(1) G(2) & G(k) G(k+1) = 0. (K 3M $$((,,004488n#Strategy Example: 1 x 7 case$FG(2) = 1, G(3) =1. G(2) G(3) = 0.($( % Calculating Grundy Numbers$@By definition: G(0) = 0 and G(1) = 0 G(2) = 1 G(3) = 1 G(4) = 28()(A  $Grundy Numbers$Definition: G(0) = G(1) = 0 and G(n) = mex({G(i) G(n  2  i) | i = 0, & , n  2}).  mex is the minimum excluded value of the set, (i.e., the smallest nonnegative integer not in the set.) @()2,('g&!Calculating Grundy Numbers$We wrote a program to calculate the first 300 Grundy numbers (G(0) to G(299)) using the previously stated algorithm. Our data indicated that the Grundy numbers of the 1 x n case attained a periodicity of 34.(,"THEOREM$ In linear cram, the second player wins only on a board that has a number of blocks equal to 0, 1, 15, 35 or (modulo 34) to 5, 9, 21, 25, and 29. "(-#PROOF$a(To show): The Grundy numbers are periodic of period 34. [ie. G(n) = G(n + 34) for all n 68]|;('(;       ,<.$ QInductive step- Assume true for no. - [ie. G(n) = G(n + 34) for all 68 n no]p$(.)! >(/%0&1'2( Future Research$jOdd-by-Odd Case Other Cases - wrap around case - other polyomino cases - variations of the domino cases((O)k9 )B) REFERENCES Acknowledgements$YDan Rudolph, PhD Marshall Cohen, PhD Demond Moy Joelle Davis Carter Turtle, turtle!!!!!! (Y()Z>%$/T'(3  (    c $t&1  <$D 0  R  0&1 ,$D 0 F(By induction): Basis step- The statement is true for 68 n 265.: 7 >   <   c $&1P<$D 0  ( `  c $A J?? 0 J`  c $A K??  K`  c $A |?? |H  0޽h ? ̙33___PPT10w.+#{D ' = @B D ' = @BA?%,( < +O%,( < +DN' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*;D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*;D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;a%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*;aD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*;aD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+p+0+ ++0+ +jxUkA$6дa/K6-ƣSiIr g\=Tkͣ]t 7"JKe{oޏ8wg a1x~PHȞڟX;)øy{٦ nDoL j^B޽/^{ɹz*6P3 D4%&۱Gc3ҧr$k|USOtd1;{&nA^ {)< 8;bν??+n}бVSsv>CfYҪqxukTg4xe>ǡJʁCX5XsjUfzRv5U,g4f)cVgDmJlSyv)1S}i?Dچ'J^۝b|esRF ˽nObؽCnN|t" >qWԌL%*x !Η{N.%˜#B_,_IwttHJ▊O!X%| .Ⲍ ij]ظ;([!:0GTpOFw*|l8ҭls$/2.GmgfrK-C3#KCEquation 3.0CorelEquation! 2.0 EquationTILING GAMES Presented by:Overview of PresentationWHAT IS A TILING GAME? HISTORY: OBJECTIVEEven by Even Case Even-by-EvenEven-by-Even Case Symmetry Even-by-Odd Even-by-Odd Case SymmetryLinear Cram: 1 x n case Linear Cram Linear Cram Linear CramLinear Cram StrategyLinear Cram StrategyLinear Cram StrategyStrategy Example: 1 x 7 caseCalculating Grundy NumbersGrundy NumbersCalculating Grundy NumbersTHEOREMPROOFNo Slide TitleNo Slide TitleNo Slide TitleNo Slide TitleNo Slide TitleFuture Research REFERENCESAcknowledgements  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles 8@ _PID_HLINKSA8 http://images.google.com/imgres?imgurl=www.polar.umd.edu/data_archive/UMD_logo_3.JPG&imgrefurl=http://www.polar.umd.edu/data_archive/gifindex.htm&h=444&w=453&prev=/images%3Fq%3Duniversity%2Bof%2Bmaryland%26start%3D60%26svnum%3D10%26hl%3Den%26lr%3D%26ie%3DUTF-8http://images.google.com/imgres?imgurl=www.circs.neu.edu/images/spiral.gif&imgrefurl=http://www.circs.neu.edu/alain.htm&h=150&w=150&prev=/images%3Fq%3Dspiral%26svnum%3D10%26hl%3Den%26lr%3D%26ie%3DUTF-8%26oe%3DUTF-8%26sa%3DN