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Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   u/"-.)*+%'153 +:/rc&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Supposea>0andab>0.Provethatb>0.Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Proofbycontradiction:c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Supposethatb0.Theneitherb=0orb<0.c&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..ifb=0,thenwewouldhaveab=0,whichisfalseM..M..c&d9 Z6Times New Roman Regularc        @  !  c&d9 Z6Times New Roman Regularc        @#0..ifb<0,thenwewouldhaveab<0,whichisalsofalsec&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Ineithercase,wegetacontradiction,thusb>0Lc&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..NotetheuseofcasesinthatproofWPCXX81%XX81%3'XXXXLLxx9j"2*R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y e$ / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc SomeProofs.+%19׿3aQ `fR//RWPCXX81%XX81%3'XXXXLLxx9j"u/"u/"W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   u/"-.*+%'153 +:/ c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Provethatyoucannottilean8'8checkerboardwith1'3trominosc&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Proof:Supposeyoucould.Then64wouldbeamultipleof3,whichisfalse.(Proofbycontradiction)Lc&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Nowprovethatifyouremoveasinglecornerfromthe8'8checkerboard,thenyoucannottilewhatremainswith1'3trominosWPCXX81%XX81%3'XXXXLLxx9j"2R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y e$ / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc ATilingProof.+%19׿3aQ `fR//RWPCXX81%XX81%3'XXXXLLxx9j"2)R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y e$ / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc ConverseversusContrapositive.+%19׿3aQ `fR//RWPCXX81%XX81%3'XXXXLLxx9j"u./"u./"W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   u./"-.)*+%'153 +.:/&c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Wehavethefollowingtheorem:c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Ifaregionofthecheckerboardcanbetiledwith1'3trominos,thenthata3-coloringofthatregionmusthavethesamenumberofeachcolorM..Mc&d9 Z6Times New Roman Regularcxxx        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Buttheconverseisnotnecessarilytrue:c&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..Namely,thatifaregionhasthesamenumberofeachcolor,thenitcannecessarilybetiledc&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Ingeneral,conversesmightbetrue,butyoucantcountonitM..Mc&d9 Z6Times New Roman Regularcxxx        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Thecontrapositiveisalwaystrue,though.c&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..Ifa3-coloringdoesnothaveanequalnumberofeachcolor,thentheregioncannotbetiledWPCXX81%XX81%3'XXXXLLxx9j W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   -.)*+%'153 `:&`c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..So,howdoweprovetheregionwithabluecornerremovedcannotbetiled?Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Supposeitcould!c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Wheredoyouthinkwegofromhere?Lc&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Thenwecouldrotatethattilingclockwise90degreestoobtainatilingwiththewhitecornerremoved...Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Clearlyacontradiction.WPCXX81%XX81%3'XXXXLLxx9j"2)R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y e$ / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc Dealingwiththebluecorner.+%19׿3aQ `fR//RWPC XX81%XX81%3'XXXXLLxx9Hj"Vum/"um/"W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   um/"-.)*+%'153 `+m:/c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Howmanydifferentwaysaretheretodominizea2'13rectanglewith1'2dominos?Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Herearesomesampledominizations:WPCXX81%XX81%3'XXXXLLxx9Hj"2)R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y `$ / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc Dominizinga2'nrectangle.+%19׿3aQ `R//RWPCPXX81%XX81%օ3'XXXXLLxx90"?[eY1eY1W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   eY1-.)*+%'13 Յe0c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Thisisnotaproof-techniqueLc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Sometimes,whenyouhavetosolveahardproblem,itmakessensetotrytosolveasimplerproblemc&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Howwouldyouchangetheproblemtomakeitsimpler?c&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..Tryadifferentsizegrid...M..M..c&d9 Z6Times New Roman Regularc        @  !  c&d9 Z6Times New Roman Regularc        @#0..Trymanydifferentsizedgrids,andseeifyoucanfindapattern!WPCXX81%XX81%օ3'XXXXLLxx90"2)R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y e6 / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc AProblem-SolvingStrategy.+%19׿3aQ `fR// R WPC XX81%XX81%3'XXXXLLxx90" hL1 hL1W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$    hL1.*+'13 h0c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Itlookslikeeachnumberinthesequenceisthesumofthetwopreviousnumbers:f(n)=f(n!1)+f(n!2)Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Howcouldweprovethatthispatternwillcontinue?WPCXX81%XX81%3'XXXXLLxx90"2)R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y e6 / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc ProvingaConjecture.+%19׿3aQ `fR// R 6/%e%4Z-l @y Y WPC-%XX81%u0-3'XXXXLLxx ? ///333???OOO___fffooof1?O/_??ߟ߿c3f3&f3?LU`l w#'*-/257:=T bjs({3=GQ[s3&?1L; UFaQl[xfp{#',04̳9׿Q.V5b;lAvGLT ` h o rux|(-8DOZfq|3fH uUb#p/:EP\fuɒџزf3uPYcl v  *D_yf(u/5;AG L R X ^!l4zH[ofu    1%MAj^z=.O;`LrVcp̙ܥ$DYނB(#G-)L20P76U<=ZAC_FJdLPiQWmV]r[cw`j|epjwo}tufp{lws}yź3">)L3$U8*`@/lG5wO:V@^EoZn̲9pu  I"0-+1*||U"D  1*ͅ||U"D rrrrrrD D D 6r6r6rrrr6666r6r6rD # r# r# rrrr# # # # r# r# rD rrrrrrrrrD r# r# r# $ # $ # $ # $ r$ r$ rrrrrrrr# r# r# D # # # 6# 6# 6# 6r6r6rrrr# # # D $ $ $ $ $ $ D $ $ $ $ $ $ D rrrr$ r$ r$ $ $ $ D …D Å# # # # # # # # # D ą666666666D Ņ555656565666555D ƅr5r5r5$ 5$ 5$ 5$ $ $ rrrr5r5r5D Džrrrr6r6r6666D ȅ666666D Ʌ666666D ʅ666666666D ˅# # # # # # # # # D ̅15*ㅢ||U"D ΅ttt$ $ $ $ $ $ ttttttD υ666666D ЅrrrrrrrrrD хrrrrrrrrrD ҅rrrrrrrrrrrrrrrD Ӆ# # # # # # rrrrrr# # # D ԅ# # # # # # # # # rrr# r# r# r# # # # # # D Յ5# 5# 5# # # # rrr5r5r5r5# 5# 5# D օ6666# 6# 6# # # # D ׅttt$ $ $ $ # $ # $ # t# t# t# tttD ؅rrrrrrD مD څD ۅ555555555555555555D ܅# 5# 5# 5555# # # # 5# 5# 5D ݅555555555D ޅttt$ $ $ $ 6$ 6$ 6t6t6t6tttD ߅666666666666D D D rrrrrr+15 u# "1u# "1W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$ o2  I  .3  Hint:Whatifwecolorthecheckerboardasshowntotheright?  u# "1.1 +# !=c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Hint:Whatifwecolorthecheckerboardasshowntotheright?.1 ꅄ!!W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$ _2  I  .3  Thereare21red,21whiteand22bluesquares.  ꅄ!.1 : !-c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Thereare21red,21whiteand22bluesquares.WPC4XX81%1%3'XXXXLLxx ? ///333???OOO___fffooof1?O/_??ߟ߿c3f