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A short talk with examples by qq(CDavid Byrne BSc (Hons) PgDipSc CMath CEng CSci MRAeS MIMA MAPM V2$D 4,% DefinitionA magic square has all cols, rows (and diagonals) summing to the same number constructed of one occurrence of all the numbers 1..n2 for a square of side n.$A 3 by 3 exampleBOn a 3x3 what does each row add up to? n(n+1)/2=(9.10)/2=45/3=15. A 3 by 3 exampleContinued& A 3 by 3 exampleContinued&  A 3 by 3 examplePBy some re-arranging one with the diagonals summing to 15 can also be achieved!  Moving on to 4x4zn(n+1)/2=(16.17)/2 =8.17. Now as 4 rows gives (8.17)/4=2.17=34. Firstly write the numbers 1..8 on the leading diagonals.  Moving on to 4x4Now reverse the rest of the numbers from the bottom up. Note again the diagonals don t add up to 34!   Moving on to 4x4 For completeness one that does.   Moving on to 8x84The diagonal process still works. Row sum of 260. There are obviously 8x8s with the diagonals also summing to 260  this process will not generate them! Z Now for some graphs  Now for some graphs  Now for some graphs Now for some graphs JGCould traffic lights help here?"How is this connected to dominoes?##( The envelope problem The envelope problem The envelope problem The envelope problem The envelope problem The envelope problem The envelope problem The envelope problem The envelope problem The envelope problem The envelope problem The envelope problem The envelope problem  The envelope problem !The envelope problem "The envelope problem # The envelope problem $!The envelope problem %"Back to dominoes( &#Back to dominoes( '$ The double 3 ( (%Back to dominoes( )&Back to dominoes( *'Back to dominoes( +( The double 4 ( ,)The double 5 and double 6( -*Domino conclusion( .+ Domino moral ( /,\Magic squares and dominos  where is the link?//( 63Time to meet the horse( 0- Horsey moves ( 1. Horsey moves ( 2/ Horsey moves ( 30 Horsey moves ( 41The Knights Tour Problem( 52The Knights Tour Problem( Sebastian meet Nigel74Sebastian meet Nigel85Sebastian meet Nigel96Back to the maths:7 Start small;8 Start small<9 Start small=: Start small>;Try a little bigger?<Try a little bigger@=Try a little biggerA? The 8 by 8B> The 8 by 8C@ The solutionDB The solutionFCThe science bitEA The solutionGDThe knights tour summaryHE ConclusionIFThe End  0` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> $(    6r  `}  T Click to edit Master title style! !  0u  `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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ZB m< s *1 ?**ZB n< s *1 ?hhZB o< s *1 ?ZB p< s *1 ?ZB q< s *1 ?""`B r< 0o ?``H < 0޽h ? 33___PPT10i.$1+D=' J= @B + 0 %@(  @x @ c $ `}   RB  @ s *Dm** RB !@ s *D * LB "@ c $D*  #@ 0d   Xy=mx+c 2 $@ 0(  `  1x 2 %@ 0tmtT 1y 2H @ 0޽h ? 33___PPT10i.$1+D=' J= @B +  0 (  H (  Hx H c $ `}    H 0T  tAdhelm Ethelward ? 2$ 0 H 0 d  HEdward the Elder Ethelward Elfleda Ethelswithe EthelgivaI 2IP    H 0$  6Egbert 2  H 0mn T S Ethelwulf 2  LB  H c $D]  @  H 0D  GEthelbald Ethelbert Ethelred I Alfred the Great AethelswithH 2H$ 3 LB  H c $D}}LB  H c $D"  }LB H c $D}LB H c $D}LB H c $D}LB H c $D}P P LB H c $D}LB H c $DppLB H c $DpLB H c $DpLB H c $Dp% LB H c $DLB H c $D  LB H c $D  LB H c $D  LB H c $D  LB H c $D  LB H c $D  LB H c $DP P H H 0޽h ? 33___PPT10i.$1+D=' J= @B + 0   0%L (  Lx L c $ `}    L 0 zSY `0So from a it is possible to drive to g, h and j.1 21 L 0]H 3a b 2  L 00  >c d i j 2  L 0<#   : 2   L 0&20   3f e 2X L 0 X L 0% 0 X L 0CLB L c $DRB L s *DLB L c $D2 2RB L s *DC CRB L s *D2 % RB L s *DLB L c $D RB L s *DZZ RB L@ s *D L 0,0 } 3g h 2 L 01 3k l 2RB L s *D RB L s *DRB L s *D% RB L s *D% RB  L s *D `RB !L@ s *D% 0 % RB "L s *D@J@RB #L s *D@@  @RB $L s *D}@  } %L 0x8f GM hNext slide is an alternate way of looking at this& 5 25H L 0޽h ? 33___PPT10i.$1+D=' J= @B +  0 #@3P (  Px P c $$< `}    P 0@@p]  1d 2 P 0D   1f 2 P 0PH  1a 2 P 0,L   1e 2 P 0O" 3c  2 P 0S   1g 2  P 0HW Z  1i 2 P 0Z 1b 2 P 0PC   1h 2 !P 0a   QSo all of a sudden the traffic light problem looks a bit easier to think about. R 2R "P 0e% w,  1j 2 #P 0i  1l 2 $P 0m- 1k 2LB %P@ c $D- LB &P@ c $D LB 'P c $D ` LB (P@ c $DF LB )P@ c $D  LB *P c $D LB +P@ c $DLB ,P@ c $DS  LB -P@ c $DS `  LB .P c $D  LB /P c $D LB 0P c $D - LB 1P@ c $D F LB 2P c $D  LB 3P c $D H P 0޽h ? 33___PPT10i.$1+D=' J= @B + 0  @f(  @x @ c $x `}    @ BAB< >0 H @ 0޽h ? 33___PPT10.$1+nDh' = @B D'  =0BcancelBubble@BA?%<(<*@%<(<*@_%(D' =%(D' =%(D9' =4@BBBB%(D' =2u;BtogglePause*<3<*@D'  = B`B?<*@%(/%,( < +/%,( < +)?+ 0 P.GT<(  Tx T c $~ `}    T 0~G bDefinition time: A domino ring is defined as a complete ring constructed of all dominoes of a set.c 2c8   T  T 0<  10 2 T 0T  11 2F    T  j  !T 0܌  12 2 "T 0  11 2F   #T  h  $T 0|  10 2 %T 0  12 2L   &T# t  'T 0  11 2 (T 04  11 2L   )T# %   *T 0  12 2 +T 0ԧ  12 2L   ,T#   -T 0\  10 2 .T 0t  10 2XR /T 0 G F   0T P = 1T 0  10 2 2T 0  10 2F   3T lY 4T 0  11 2 5T 0   11 2F   6T l^ Y 7T 0  12 2 8T 0`  12 2F   9T l3 Y :T 0   12 2 ;T 0  11 2L   T 0x  10 2L   ?T# . cPF  @T 0  10 2 AT 0  11 2 BT 0lC"? @ #  52 2 CT 0@C"?   52 2 DT 0C"?` C  52 2 ET 0`` CG 10 2 FT 0f tWM 11 2 GT 09 z ]  12 2H T 0޽h ? 33___PPT10i.$1+D=' J= @B +q 0 `2X(  Xx X c $< `}    X 0~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR X 0 x 0X 0C"? s LB 1X c $D s LB 2X c $D s H X 0޽h ? 33___PPT10i.$1+D=' J= @B + 0 `l(  `x ` c $lh `}    ` 0(~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR ` 0 x ` 0C"? s LB ` c $D s LB ` c $D s LB ` c $D# ) H ` 0޽h ? 33___PPT10i.$1+D=' J= @B + 0 0( d(  dx d c $  `}    d 0 ~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR d 0 x d 0C"? s LB d c $D s LB d c $D s LB d c $D# ) LB  d c $D# # H d 0޽h ? 33___PPT10i.$1+D=' J= @B +m 0 | h(  hx h c $t `}    h 0~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR h 0 x h 0C"? s LB h c $D s LB h c $D s LB h c $D#  LB  h c $D# # LB  h c $D #  H h 0޽h ? 33___PPT10i.$1+D=' J= @B + 0  lh(  lx l c $ `}    l 0~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR l 0 x l 0C"? s LB l c $D s LB l c $D s LB l c $D#  LB  l c $D# # LB  l c $D #  LB  l c $D H l 0޽h ? 33___PPT10i.$1+D=' J= @B + 0 ,$ p(  px p c $) `}    p 0+~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR p 0 x p 0C"? s LB p c $D s LB p c $D s LB p c $D#  LB  p c $D# # LB  p c $D #  LB  p c $D LB  p@ c $D# H p 0޽h ? 33___PPT10i.$1+D=' J= @B +i 0 x t(  tx t c $6 `}    t 07~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR t 0 x t 0C"? s LB t c $D s LB t c $D s LB t c $D#  LB  t c $D# # LB  t c $D #  LB  t c $D LB  t@ c $D# LB  t c $Dm# H t 0޽h ? 33___PPT10i.$1+D=' J= @B + 0 xd(  xx x c $XB `}    x 0C~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR x 0 x x 0C"? s LB x c $D s LB x c $D s LB x c $D#  LB  x c $D# # LB  x c $D #  LB  x c $D LB  x@ c $D# LB  x c $Dm# LB x c $DmH x 0޽h ? 33___PPT10i.$1+D=' J= @B + 0 ( |(  |x | c $(O `}    | 0O~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR | 0 x | 0C"? s LB | c $D s LB | c $D s LB | c $D#  LB  | c $D# # LB  | c $D #  LB  | c $D LB  |@ c $D# LB  | c $Dm# LB | c $DmLB |@ c $D#  H | 0޽h ? 33___PPT10i.$1+D=' J= @B + 0 V(  x  c $[ `}     0p]~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR  0 x  0C"? s LB  c $D s LB  c $D s LB  c $D#  LB  c $D# # LB  c $D #  LB  c $D LB  @ c $D# LB  c $Dm# LB  c $DmLB @ c $D#    08` =C  6Voila! 2H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 V(  x  c $$g `}     0h~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR  0 x  0C"? s LB  c $D s LB  c $D s LB  c $D#  LB  c $D# # LB  c $D #  LB  c $D LB  @ c $D# LB  c $Dm# LB  c $DmLB @ c $D#    0Hp =C  6Voila! 2H  0޽h ? 33___PPT10i.$1+D=' J= @B +1 0 H@ (  x  c $w `}     0@y~G xHIs it possible to draw an envelope without lifting the pen from a paper?I 2IXR  0 x  0C"? s LB  c $D s LB  c $D s LB  c $D#  LB  c $D# # LB  c $D #  LB  c $D LB  @ c $D# LB  c $Dm# LB  c $DmLB @ c $D#    0؀ = But what if we wanted to get back to where we started also? Rather than Trial and Error can we use something that we have learnt so far? 2H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0  0 (  x  c $ `}   8  0@~G Think about the connectivity. For the top of the envelope the purple route is the only route that can be taken so the problem now comes down to the top of the envelope. 2LB  c $D#  LB  c $D# # LB  c $D #  LB  c $D LB  @ c $D# RB  s *Df3>m# RB  s *Df3>mLB @ c $D#  H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 @ (  x  c $A `}  A 8  0ЙA~G Think about the connectivity. For the top of the envelope the purple route is the only route that can be taken so the problem now comes down to the top of the envelope. 2LB  c $D#  RB  s *D># # LB  c $D #  LB  c $D LB @ c $D# RB  s *Df3>m# RB  s *Df3>mLB  @ c $D#    0إA2t  sTake the top left corner as an example. There are three legs remaining. Lets assume that we take the blue route in.t 2tH  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 P a(  x  c $ `}     0 ~G VNow to come out of the purple we have a choice of 3 ways. Assume take the green route.W 2WLB  c $D#  RB  s *D># # LB  c $D #  LB  c $D RB @ s *DԔ# RB  s *Df3>m# RB  s *Df3>mLB  @ c $D#    042t  sTake the top left corner as an example. There are three legs remaining. Lets assume that we take the green leg now.t 2tH  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 ` p(  x  c $l `}   RB  s *Dfo#  RB  s *D># # LB  c $D #  LB  c $D RB @ s *DԔ# RB  s *Df3>m# RB  s *Df3>mLB  @ c $D#    0'A2t  jTake the top left corner as an example. There are three legs remaining. Now all we can do is to go orange.k 2kH  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 p x(  x  c $ `}   RB  s *Dfo#  RB  s *D># # LB  c $D #  LB  c $D RB @ s *DԔ# RB  s *Df3>m# RB  s *Df3>mLB  @ c $D#    02t  rIt is quite clear at this point that we cannot get back to where we started crossing each line once and only once.s 2sH  0޽h ? 33___PPT10i.$1+D=' J= @B +  0   U (  x  c $D `}   RB  s *Dfo#  RB  s *D># # LB  c $D #  LB  c $D RB @ s *DԔ# RB  s *Df3>m# RB  s *Df3>mLB  @ c $D#    02t  RThink a bit about the number of lines in and out of each point. The problem comes down to the fact that some of the nodes have an odd number of edges going in to them. If it is an odd number it has to be an end or a start. As 2 nodes have an odd number of edges a cycle just isn t going to happen.* 2*  0/ 12 2  0 t 14 2  0s Z  13 2  0T  13 2  0 t 14 2H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 @K(  x  c $h `}     0~Ge BA set that goes to 2F    zj0 g  0  13 2  0  13 2x / 0C"?G LB 0 c $DG LB 1 c $DG  2 0"dG  50 2 3 0  51 2 4 0P   52 2 5 0@  53 2 7 0 7E UIn graph speak this is a K4 2H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 (  x  c $, `}     0~Ge BA set that goes to 2F    zj0 g  0   13 2  02A  13 2  0C"?P 3  13 2x  0C"?G LB  c $DG LB  c $DG   0"dG  50 2  0  51 2  0   52 2  0  53 2x  0C"?    0# 7E UIn graph speak this is a K4 2LB  c $D    0d(@ #  50 2  0|, 51 2  00g n N  52 2  03g N  53 2  0 8C"?mT 13 2  0;C"?   13 2  0 C"?F -  13 2RB  s *Dfo    0B P E Y)Lets start with playing the (2,1) domino.* 2*H  0޽h ? 33___PPT10i.$1+D=' J= @B +) 0 @8(  x  c $dJ `}     0K~Ge BA set that goes to 2F    zj0 g  0O  13 2  0S  13 2  0WC"?P 3  13 2x  0C"?G LB  c $DG LB  c $DG   0\"dG  50 2  0,^  51 2  0b   52 2  0f  53 2x  0C"?    0j 7E UIn graph speak this is a K4 2LB  c $D    0DR@ #  50 2  0r 51 2  00vg n N  52 2  0yg N  53 2  0}C"?mT 13 2  0C"?   13 2  04C"?F -  13 2RB  s *Dfo    0 P   = Now the (1,0) 2RB @ s *Dfo H  0޽h ? 33___PPT10i.$1+D=' J= @B +/ 0 F>(  x  c $ `}     0L~Ge BA set that goes to 2F    zj0 g  0,  13 2  0ؚ  13 2  0dC"?P 3  13 2x  0C"?G LB  c $DG LB  c $DG   0"dG  50 2  0  51 2  0Ъ   52 2  0Į  53 2x  0C"?    0 7E UIn graph speak this is a K4 2RB  s *Dfo    0,@ #  50 2  0H 51 2  0@g n N  52 2  0Xg N  53 2  0C"?mT 13 2  0C"?   13 2  0C"?F -  13 2RB  s *Dfo    0 P   = Now the (0,3) 2RB @ s *Dfo H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 0(  x  c $4 `}     0~Ge BA set that goes to 2F    zj0 g  0>A  13 2  0  13 2  0C"?P 3  13 2x  0C"?G LB  c $DG LB  c $DG   0t"dG  50 2  0  51 2  0   52 2  0  53 2x  0C"?    0 7E UIn graph speak this is a K4 2RB  s *Dfo    00@ #  50 2  0$ 51 2  0Xg n N  52 2  0pg N  53 2  0 C"?mT 13 2  0XC"?   13 2  0C"?F -  13 2RB  s *Dfo    0 P   = Now the (3,2) 2RB @ s *Dfo RB @ s *Dfo  H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 (  x  c $T `}     0 !~Ge BA set that goes to 2F    zj0 g  0%  13 2  0 )  13 2  01AC"?P 3  13 2x  0C"?G LB  c $DG LB  c $DG   0/"dG  50 2  04  51 2  07   52 2  0;  53 2x  0C"?    0: 7E UIn graph speak this is a K4 2RB  s *Dfo    0@D@ #  50 2  0\H 51 2  0TLg n N  52 2  0lPg N  53 2  0SC"?mT 13 2  0WC"?   13 2  0ZC"?F -  13 2RB  s *Dfo    02 P E [+Now the (2,0). Game over and 1 domino left., 2,RB @ s *Dfo RB @ s *Dfo  RB  s *Dfo H  0޽h ? 33___PPT10i.$1+D=' J= @B +  0   )Q (  x  c $f `}     0h~Ge BA set that goes to 2F    zj0 g  01A  14 2  0p  14 2  0lt 7K UIn graph speak this is a K5 2R  0dxm   Based on what has been said so far it is clear that as each node has an even number of edges this should be possible. Rather than drawer numerous powerpoint slides I ll leave this for you to try out yourself. 2 5LB  c $DmJ LB  c $DmJ6 LB  c $Dt6 LB ! c $Dt`LB "@ c $D` LB # c $Dt LB $ c $DmtJLB % c $DmJ`LB &@ c $D`6 LB ' c $D 6 H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 N(  x  c $` `}     07wm 6Extending this theory it can be seen that the double 5 will not work (you ll have 6 dominoes not used) and the double 6 will work.  2H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0   (  x  c $` `}     07wt ZBe it by chance or otherwise I don t know, but to get the most out of your dominoes always have a set where the maximum number is an even number. This may be the reason that sets go up to a double-6 or double-12. You can buy sets to other sizes (8 and 9s exist) but if possible avoid the odd doubles. . 2.H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 0N(  x  c $ `}     0P7w6 6If your stuck on a desert island and making a game and it takes 4 days to make a domino out of a coconut don t use every husk  only use enough husks to make an even game. & but& & more importantly& & remember to drink the coconut milk and eat the coconuts!  2H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 @r(  x  c $ `}     07w \In chess there is a piece called the knight& . / 2/L  C $A bluem    0PAd tD& or to others he is just a horse.# 2#  B; 0http://www.classichorse.com/clipart/cartoon.html11)  0"@ g  : 2 H  0޽h ? 33___PPT10i.$1+D=' J= @B +q 0  (  x  c $\ `}   L  C $A bluem    B; 0http://www.classichorse.com/clipart/cartoon.html11)  0x"@ g  : 2   0H"m  dMy name is Sebastian  I ll do what I can to help?3 23r  JB CDElFAA[[1Kxx`#M).8G\xh}K.h_YP/@8hH0&[ 0 M # ( ( 9 r` z ?    @ 1   8 E PZ v }UI d(D7<HdXX?@@                              (  H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 P(  x  c $ `}   4  07wm The knight moves in a regular pattern  2 steps to the front, 1 step to the side. S 2SL  C $A blue)  LB  c $D}G }LB  c $DpG pLB  c $D6 G 6 LB  c $D) G ) LB  c $D}GG LB  c $D}:: LB  c $D}-- LB  c $D} LB  c $D} LB  c $D G H  0޽h ? 33___PPT10i.$1+D=' J= @B +; 0 RJ`(  x  c $ `}   4  07wm The knight moves in a regular pattern  2 steps to the front, 1 step to the side. S 2SL  C $A blue  LB  c $D}G }LB  c $DpG pLB  c $D6 G 6 LB  c $D) G ) LB  c $D}GG LB  c $D}:: LB  c $D}-- LB  c $D} LB  c $D} LB  c $D G RB  s *D) f H  0޽h ? 33___PPT10i.$1+D=' J= @B +; 0 RJp(  x  c $ `}   4  0A7wm The knight moves in a regular pattern  2 steps to the front, 1 step to the side. S 2SL  C $A blue LB  c $D}G }LB  c $DpG pLB  c $D6 G 6 LB  c $D) G ) LB  c $D}GG LB  c $D}:: LB  c $D}-- LB  c $D} LB  c $D} LB  c $D G RB  s *D) f H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 <(  x  c $ ? `}  ? 4  0?7wm The knight moves in a regular pattern  2 steps to the front, 1 step to the side. S 2SL  C $A bluep LB  c $D}G }LB  c $DpG pLB  c $D6 G 6 LB  c $D) G ) LB  c $D}GG LB  c $D}:: LB  c $D}-- LB  c $D} LB  c $D} LB  c $D G RB  s *D)S f RB  s *D>S S H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 ,$(  x  c $ ? `}  ?   0?7wm Is it possible for a knight to visit every square on a chess board once, and only once, returning to where the knight started from?.  2L  C $A blue}   0?`>  3? 26H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0  Z(  x  c $!? `}  ?   0P#?7wm Is it possible for a knight to visit every square on a chess board once, and only once, returning to where the knight started from?.  2L  C $A blue}   0H)?`>  3? 26  0'?} tk  I m not the most heroic horse for the challenge  I ll go fetch Nigel.G 2Gr  JBCQDElFAA[[IK?_4%Y"CJXqz9X1x-*%T  ,d|%%YHQg >SXnzKy oQ%D7$dP:&mrN'?ZJQCF2!U"ElN1(~1f1?@@                              P  H  0޽h ? 33___PPT10i.$1+D=' J= @B + 0 PO(  r  S 5? `}  ? L  C $A blueZ   07?G5 3? 26N  C &A SensePf X  06 W H  0޽h ? 33___PPT10i.$`"+D=' J= @B + 0  e(  x  c $?? `}  ? L  C $A blueZ   0@?G5 3? 26N  C &A SensePf   0$D?  ~NHi Nigel  can you help with a problem?( 2("  BxCDE0FpAALL 7@NS`|o?s&e  09>H~WG/pO'87HOQSW[`_ofoo';P'_doxX';q#X7%HluP"b d@58@                         y8 X  06 W H  0޽h ? 33___PPT10i.$`"+D=' J= @B +  0  C(  x  c $,L? `}  ? L  C $A blueZ   0M?G5 3? 26N  C &A SensePf   0LP?  ~NHi Nigel  can you help with a problem?( 2(  0DU?C~  vSure Sebastian  I ll do what I can. I rely on Horse Sense.< 2<(  BxCDE0FpLL 7@NS`|o?s&e  09>H~WG/pO'87HOQSW[`_ofoo';P'_doxX';q#X7%HluP"b d@58@                         y8 "  BCDE0FpAALLx40@;590v+4 Q]P/73E` fvj2HH/{ w18@VTpx8X<`dP/$ 88OdpB.x0``58@                         ph H  0޽h ? 33___PPT10i.$`"+D=' J= @B +o 0 ~ (  x  c $,\? `}  ? N  C &A Senset   0^?P  xNow we have a horse on board with horse sense we ll be fine.= 2=H  0޽h ? 33___PPT10i.$`"+D=' J= @B + 0 !$-(  x  c $|e? `}  ? N  C &A Sense( D  0xg?7; As with most things in life it is best to start small  take the humble 4 by 4 chessboard.[ 2[LB  c $D}G }LB  c $DpG pLB  c $D6 G 6 LB  c $D) G ) LB  c $D}GG LB  c $D}:: LB  c $D}-- LB  c $D} LB  c $D} LB  c $D G   <Hq? 2  <t? 3  <x? 3  <Pt?z >  2  <?  3  <x?o V  3  <L? f  2  <?  4  < ?l  4  <l? y  4  <̔? ls  4  <,?b I  3  <?b I  3  <?l 0  3  <L?o n 2 V  3  <?V z > =  2~ " 0 ?P t7   # 0?   {KHere the numbers show the number of ways I can go in and out of each squareL 2L $ jBC DEFAAaaO|cKC#{p]E0{kSMxb(K] s=Cu6xkh{`PG90 S *(KNp5l Ea  >xEyv[<HXhx?@!P[Kpsk:SCD@                                = H H  0޽h ? 33___PPT10i.$`"+D=' J= @B + 0 !!(  x  c $@? `}  ? N  C &A Sense(   0  2  <?  3  <|?o V  3  <? f  2  <?  4  <P?l  4  <? y  4  <? ls  4  <p?b I  3  <?b I  3  <0?l 0  3  <?o n 2 V  3  <?V z > =  2~  0P?P t7     0$? s  tDIt might help to bring Sebastian back. Allow me to go to the stable.E 2E ! pBC DEFaaO|cKC#{p]E0{kSMxb(K] s=Cu6xkh{`PG90 S *(KNp5l Ea  >xEyv[<HXhx?@!P[Kpsk:SCD@                                  H  0޽h ? 33___PPT10i.$`"+D=' J= @B + 0 %'((  x  c $h? `}  ? N  C &A Sense( LB  c $D}G }LB  c $DpG pLB  c $D6 G 6 LB  c $D) G ) LB   c $D}GG LB   c $D}:: LB   c $D}-- LB   c $D} LB   c $D} LB  c $D G   <C 2  <C 3  < C 3  <Cz >  2  <C  3  <Co V  3  <C f  2  <XC  4  <Cl  4  <"C y  4  <x%C ls  4  <(Cb I  3  <8,Cb I  3  </Cl 0  3  <2Co n 2 V  3  <X6CV z > =  2~  09CP t7     0;Cd  pHere Seb has gone a, b then c. 2 ! pBC DEFaaO|cKC#{p]E0{kSMxb(K] s=Cu6xkh{`PG90 S *(KNp5l Ea  >xEyv[<HXhx?@!P[Kpsk:SCD@                                `&  L " C $A blueW  L # C $A blue L $ C $A blue Z  % 0AC  1a 2 & 0EC f  1b 2 ' 0,IC Z=s  1c 2  ( 0MC w XNow from c Seb can go a variety of ways but at some point he has to head to the corners.Y 2Y KH  0޽h ? 33___PPT10i.$`"+D=' J= @B + 0  &)(  x  c $TC `}  C LB  c $D}G }LB  c $DpG pLB  c $D6 G 6 LB  c $D) G ) LB  c $D}GG LB   c $D}:: LB   c $D}-- LB   c $D} LB   c $D} LB   c $D G   <[C 2  <_C 3  <(cC 3  <^Cz >  2  <$jC  3  <mCo V  3  <0qC f  2  <tC  4  <wCl  4  <P{C y  4  <~C ls  4  <Cb I  3  <pCb I  3  <ЈCl 0  3  <0Co n 2 V  3  <CV z > =  2~  0CP t7  L ! C $A blueW  L " C $A blue L # C $A blue Z  $ 0C  1a 2 % 0pC f  1b 2 & 0МC Z=s  1c 2 ' 0\C} J  tDNow there is only 1 way to get to  d , via  a and  c . As  a and  c are already needed for  b it is impossible to do the Knights tour on a 4 by 4 chessboard.  2L ( C $A blueM w  ) 0XCM 0  1d 2H  0޽h ? 33___PPT10i.$`"+D=' J= @B + 0 0)2 =(   x   c $  3   <C  3   <|Co V  4   <C f  3   < =  4~   0PCP t7   '  0$CM4 NNow going to look at a 5 by 5. 2LB (  c $D} YLB )  c $DYt Y *  <Ca %  2 +  <C   3 ,  <Co V  4 -  <8C  w  3 .  <D   2 /  <D  2 0  <X D  3 1  <, D l 4 2  <D _  3H   0޽h ? 33___PPT10i.$`"+D=' J= @B +e 0 |t@,, (  x  c $ D `}  D LB  c $D}G }LB  c $DpG pLB  c $D6 G 6 LB  c $D) G ) LB  c $D}GGYLB  c $D}::YLB  @ c $D}--YLB   c $D}YLB   c $D} YLB   c $D G    <D 2  <\C 3  <$#D 4  <'Dz >  3  <*D  3  <|&Do V  4  <x0D f  3  < 5D  4  < 8Dl  6  <;D y  6  <x?D ls  8  <HBDb I  4  <FDb I  6  <|IDl 0  4  <LDo n 2 V  6  < =  4~  0SDP t7  L  C $A bluep%   0pUDM4 NNow going to look at a 5 by 5. 2LB   c $D} YLB ! c $DYt Y " <[Da %  2 # <_D   3 $ <cDo V  4 % <fD  w  3 & <jD   2 ' <,fD  2 ( <qD  3 ) <0uD l 4 * <xD _  3 + 0zD w  IDo you think I can do it? 2 , BCDEFAAjj.b 6z@YkDxxppnyh`L1  P(0 2GPujpxkg`YNHCD@-(i `8Gpw-\x8TOh 8xH`TDh0z@p/W%5 hhhIL@                                    H  0޽h ? 33___PPT10i.$`"+D=' J= @B + 0 P,/(  x  c $D `}  D LB  c $D}G }LB  c $DpG pLB  c $D6 G 6 LB  c $D) G ) LB  c $D}GGYLB  c $D}::YLB  @ c $D}--YLB   c $D}YLB   c $D} YLB   c $D G    <D 2  <pD 3  <D 4  <Dz >  3  <8D  3  <Do V  4  <D f  3  <D  4  <PDl  6  <D y  6  <D ls  8  <pDb I  4  <дDb I  6  <0Dl 0  4  <Do n 2 V  6  <DV z > =  4~  0PDP t7    0$DM4 NNow going to look at a 5 by 5. 2LB   c $D} YLB ! c $DYt Y " <Da %  2 # <D   3 $ <xDo V  4 % <D  w  3 & <tD   2 ' < D  2 ( <D  3 ) <D l 4 * <D _  3N - C &A SenseVn  . 0HDFK eAgain look at the corners only 1 way in and one way out and you need to use the middle squares so no.f 2f / BC DEFAAsshK-h YG8rrQP M0*xc@> eZP`ox hDAU$8P' g1'8h>H H KcLPXryl c 0x =          o        ;L q+ cF)8dUnT"OP@                                      ` xH  0޽h ? 33___PPT10i.$`"+D=' J= @B +: 0 ::VW 9(   x   c $tD `}  D ~   08DP t7  b8 )   # #"2& GFGGFGFG  D   <|D?N G2 @`   <?N G3 @`   <h?h G4 @`   <?h G4 @`   <? G4 @`   <&? G4 @`   </? G3 @`   <7?) G2 @`   < @?N G3 @`   <H?N G4 @`   <Q?h G6 @`   <Z?h G6 @`   <\? G6 @`   <k? G6 @`   <d? G4 @`   <}?) G3 @`   <?N< G4 @`   <p?<N G6 @`   <?h< G8 @`   <?<h G8 @`   <?< G8 @`   <?< G8 @`   <L?< G6 @`   <?)< G4 @`   <?N< G4 @`   <?N< G6 @`   <<?h< G8 @` !  <?h< G8 @` "  <?< G8 @` #  <?< G8 @` $  <D?< G6 @` %  <$ ?)< G4 @` &  <$?N G4 @` '  <t?N G6 @` (  <"?h G8 @` )  <?h G8 @` *  <d4? G8 @` +  <d6? G8 @` ,  <E? G6 @` -  < ??) G4 @` .  <TW?Nj G4 @` /  <TY?jN G6 @` 0  <|h?hj G8 @` 1  <q?jh G8 @` 2  <y?j G8 @` 3  <?j G8 @` 4  <?j G6 @` 5  <?)j G4 @` 6  <0?N$j G3 @` 7  <x?$Nj G4 @` 8  <x?h$j G6 @` 9  <a?$hj G6 @` :  <?$j G6 @` ;  <X?$j G6 @` <  <l?$j G4 @` =  <?)$j G3 @` >  <|?N $ G2 @` ?  <? N$ G3 @` @  <t?h $ G4 @` A  <? h$ G4 @` B  <? $ G4 @` C  < ? $ G4 @` D  <D? $ G3 @` E  <?) $ G2 @``B F  0o ?)  ZB G  s *1 ?)$$ZB H  s *1 ?)jjZB I  s *1 ?)ZB J  s *1 ?)ZB K  s *1 ?)<<ZB L  s *1 ?)ZB M  s *1 ?)`B N  0o ?)`B O  0o ?) )ZB P  s *1 ? ZB Q  s *1 ? ZB R  s *1 ? ZB S  s *1 ? ZB T  s *1 ?h hZB U  s *1 ? ZB V  s *1 ?N N`B W  0o ? H   0޽h ? 33___PPT10i.$`"+D=' J= @B +@ 0 ??`_z?(  x  c $0  `}   ~  0jP t7  \8 )   #"2&GFGGFGFG   m < ?N G2 @` l <;?N G3 @` k <B?h G4 @` j <D?h G4 @` i <@T? G4 @` h <|M? G4 @` g <e? G3 @` f <g?) G2 @` e <v?N G3 @` d <?N G4 @` c <x?h G6 @` b <T?h G6 @` a <И? G6 @` ` <L? G6 @` _ <ȩ? G4 @` ^ <D?) G3 @` ] <?N< G4 @` \ <<?<N G6 @` [ <?h< G8 @` Z <4?<h G8 @` Y <?< G8 @` X <,?< G8 @` W <?< G6 @` V <$?)< G4 @` U <?N< G4 @` T <?N< G6 @` S <?h< G8 @` R <?h< G8 @` Q < ?< G8 @` P <)?< G8 @` O <1?< G6 @` N <:?)< G4 @` M <B?N G4 @` L <K?N G6 @` K <|S?h G8 @` J <[?h G8 @` I <td? G8 @` H <l? G8 @` G <lu? G6 @` F <}?) G4 @` E <?Nj G4 @` D <T?jN G6 @` C <?hj G8 @` B <ؠ?jh G8 @` A <آ?j G8 @` @ <D?j G8 @` ? <?j G6 @` > <?)j G4 @` = <?N$j G3 @` < <4?$Nj G4 @` ; <p?h$j G6 @` : <?$hj G6 @` 9 <?$j G6 @` 8 <$?$j G6 @` 7 < ?$j G4 @` 6 <` ?)$j G3 @` 5 <t ?N $ G2 @` 4 <0 ? N$ G3 @` 3 <# ?h $ G4 @` 2 <(, ? h$ G4 @` 1 <4 ? $ G4 @` 0 < = ? $ G4 @` / <E ? $ G3 @` . <N ?) $ G2 @``B n 0o ?)  ZB o s *1 ?)$$ZB p s *1 ?)jjZB q s *1 ?)ZB r s *1 ?)ZB s s *1 ?)<<ZB t s *1 ?)ZB u s *1 ?)`B v 0o ?)`B w 0o ?) )ZB x s *1 ? ZB y s *1 ? ZB z s *1 ? ZB { s *1 ? ZB | s *1 ?h hZB } s *1 ? ZB ~ s *1 ?N N`B  0o ?   0T &9 9 Sum to 26 2   0, <zF? N$ H26 @` ?, <H?h H44 @` ( <FH? H57 @` ( <OH? G2 @` ( <WH? H47 @` ( <`H?) G6 @` ( <hH?N< H42 @` ( <qH?<N H17 @` ( <yH?h< H32 @` ( <H?<h H53 @` ( <|H?< G8 @` ( <H?< H45 @` ( <tH?< G4 @` ( <H?)< H59 @` ( <lH?N< H29 @` ( <H?N< H54 @` ( <dH?h< H41 @`  ( <H?h< H20 @` !( <\H?< G1 @` "( <H?< H60 @` #( <TH?< G7 @` $( <H?)< H48 @` %( <LH?N H16 @` &( <H?N H39 @` '( <I?h H28 @` (( < I?h H33 @` )( <I? H52 @` *( <0I? G9 @` +( <#I? H22 @` ,( <(,I?) H61 @` -( <4I?Nj H27 @` .( < =I?jN H36 @` /( <EI?hj H13 @` 0( <NI?jh H40 @` 1( <VI?j H21 @` 2( <_I?j H64 @` 3( <gI?j H49 @` 4( <pI?)j H10 @` 5( <xI?N$j H38 @` 6( <I?$Nj H15 @` 7( <|I?h$j H34 @` 8( <I?$hj H25 @` 9( <tI?$j H12 @` :( <I?$j H51 @` ;( <lI?$j H62 @` <( <I?)$j H23 @` =( <dI?N $ H35 @` >( <I? N$ H26 @` ?( <\I?h $ H37 @` @( <I? h$ H14 @` A( <TI? $ H63 @` B( <I? $ H24 @` C( <LI? $ H11 @` D( <I?) $ H50 @``B E( 0o ?)  ZB F( s *1 ?)$$ZB G( s *1 ?)jjZB H( s *1 ?)ZB I( s *1 ?)ZB J( s *1 ?)<<ZB K( s *1 ?)ZB L( s *1 ?)`B M( 0o ?)`B N( 0o ?) )ZB O( s *1 ? ZB P( s *1 ? ZB Q( s *1 ? ZB R( s *1 ? ZB S( s *1 ?h hZB T( s *1 ? ZB U( s *1 ?N N`B V( 0o ?  ^( 0J V&Here every row and column sums to 260.' 2'H ( 0޽h ? 33___PPT10i.$`"+D=' J= @B + 0 4(  4x 4 c $0 I `}  I ~ 4 0FP t7  j 4 0JMK  It is possible to find a solution of the knights tour by using a magic square. If you follow the numbers you ll see it maps out to a pleasing symmetrical cyclical tour. A bigger question is can it be done with a diagonal magic square&  2H 4 0޽h ? 33___PPT10i.$`"+D=' J= @B + 0 87(  8x 8 c $J `}  J ~ 8 0JP t7   8 0JM  GThis was always meant to be just a bit of fun and nothing to deep or meaningful. On magic squares of side 4n where is a whole +ve number use leading diagonals and lay out a magic square. Domino sets yield more interesting games if the maximum number is even. A possible solution to the Knights Tour Problem is a magic square. H 2HH 8 0޽h ? 33___PPT10i.$`"+D=' J= @B +t 0 <(  <x < c $<)J `}  J ~ < 0,JP t7   < 0T-JMD }MHas anyone got any tricks they would like to share? How about magic cubes? N 2NH < 0޽h ? 33___PPT10i.$`"+D=' J= @B +r,'146]HY`hu3 #7>B}GLRW]jd/kHryfȼ#o!"&4;HBI(QEU,`vgqt)(3 FWxnw J1Oh+'0 `h  8 D P\dtNon-diagonal magic square and why domino sets go up to a double 6 or a double 12. A short talk with examples by  cdfreshwater24Microsoft Office PowerPoint@`l.@Й$@d G g  9  y--$xx--'@Arial-.  2 Non."System@;-@Arial-.  2 "-.-@Arial-. .2 %diagonal magic square and .-@Arial-. 02 'why domino sets go up to a .-@Arial-. 92 2!double 6 or a double 12. A short .-@Arial-. (2 =&talk with examples by .-@Arial-. 2 K= David Byrne .-@Arial-. 2 QBSc (S.-@Arial-.  2 Q'Hons.-@Arial-.  2 Q0) .-@Arial-. 2 Q2PgDipSc.-@Arial-. 72 Q@ CMath CEng CSci MRAeS MIMA MAPM.-@Arial-.  2 [LV2.-՜.+,0P    On-screen ShowQinetiQbH JArialDefault DesignqNon-diagonal magic square and why domino sets go up to a double 6 or a double 12. A short talk with examples by DefinitionA 3 by 3 exampleA 3 by 3 exampleA 3 by 3 exampleA 3 by 3 exampleMoving on to 4x4Moving on to 4x4Moving on to 4x4Moving on to 8x8Now for some graphsNow for some graphsNow for some graphsNow for some graphs Could traffic lights help here?#How is this connected to dominoes?The envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemThe envelope problemBack to dominoesBack to dominoes The double 3Back to dominoesBack to dominoesBack to dominoes The double 4The double 5 and double 6Domino conclusion Domino moral/Magic squares and dominos where is the link?Time to meet the horse Horsey moves Horsey moves Horsey moves Horsey movesThe Knights Tour ProblemThe Knights Tour ProblemSebastian meet NigelSebastian meet NigelSebastian meet NigelBack to the maths Start small Start small Start small Start smallTry a little biggerTry a little biggerTry a little bigger The 8 by 8 The 8 by 8 The solution The solutionThe science bit The solutionThe knights tour summary ConclusionThe End  Fonts UsedDesign Template Slide TitlesH$_ cdfreshwatercdfreshwater  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ\]^_`abdefghijlmnopqryRoot EntrydO)Pictures+Current UserkSummaryInformation([PowerPoint Document(ODocumentSummaryInformation8c