ࡱ> ij  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root Entryb.@pj@SlideShowP jP jMain gPerfectOffice_MAIN&g_ Original Slide Show  Oh+'Oh+'0 H T ` Rob Hochberg 3c:\Corel\Suite8\Programs\Masters\Color\DEFAULT.MST      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghWPC ~   |"XXXX'3XXXX3' > | - ~YR FPWPC ##2  I  .3  ProofbyContradiction..2  A  .3  Proofbycontradiction....2  1  .3  Assume:......2  a  .3  p F alse....2  1  .3  Conclude......2  a  .3  p..2  A  .3  Weshowthatifpisnottrue,thenweobtainacontradiction.Thisimpliesthatpmustbetrue..2  A  .3  Example:....2  1  .3  Provethatan8'8checkerboardwithtwooppositecornersremovedcannotbetiledwith1'2dominos'@ '2  I  .3  TilingaPrunedCheckerboardwithDominos1' 12  I  .3  ProofbyCases..2  A  .3  Provethatifanm'ngridcontainsanevennumberofsquares,thenitcanbecoveredbydominos....2  1  .3  Anm'ngridcontainsmnsquares.Ifthisquantityiseven,theneithermmustbeevenornmustbeeven.......2  a  .3  Howwouldyouprovethat?....2  1  .3  Ifmiseven.........2  a  .3  Laythedominoshorizontally....2  1  .3  Ifmisnoteven,then.........2  a  .3  nmustbeeven,andwecansetthedominosvertically-#!-2  I  .3  SomeProofs..2  A  .3  Supposea>0andab>0.Provethatb>0...2  A  .3  Proofbycontradiction:....2  1  .3  Supposethatb0.Theneitherb=0orb<0.......2  a  .3  ifb=0,thenwewouldhaveab=0,whichisfalse......2  a  .3  ifb<0,thenwewouldhaveab<0,whichisalsofalse....2  1  .3  Ineithercase,wegetacontradiction,thusb>0..2  A  .3  Notetheuseofcasesinthatproof6,!  62  I  .3  ATilingProof..2  A  .3  Provethatyoucannottilean8'8checkerboardwith1'3trominos....2  1  .3  Proof:Supposeyoucould.Then64wouldbeamultipleof3,whichisfalse.(Proofbycontradiction)..2  A  .3  Nowprovethatifyouremoveasinglecornerfromthe8'8checkerboard,thenyoucannottilewhatremainswith1'3trominos"!  "2  I  .3  -#!   -2  I  .3  ConverseversusContrapositive..2  A  .3  Wehavethefollowingtheorem:....2  1  .3  Ifaregionofthecheckerboardcanbetiledwith1'3trominos,thenthata3-coloringofthatregionmusthavethesamenumberofeachcolor....2  1  .3  Buttheconverseisnotnecessarilytrue:......2  a  .3  Namely,thatifaregionhasthesamenumberofeachcolor,thenitcannecessarilybetiled....2  1  .3  Ingeneral,conversesmightbetrue,butyoucantcountonit....2  1  .3  Thecontrapositiveisalwaystrue,though.......2  a  .3  Ifa3-coloringdoesnothaveanequalnumberofeachcolor,thentheregioncannotbetiled6,     62  I  .3  Dealingwiththebluecorner..2  A  .3  So,howdoweprovetheregionwithabluecornerremovedcannotbetiled?..2  A  .3  Supposeitcould!....2  1  .3  Wheredoyouthinkwegofromhere?..2  A  .3  Thenwecouldrotatethattilingclockwise90degreestoobtainatilingwiththewhitecornerremoved.....2  A  .3  Clearlyacontradiction.6,!  62  I  .3  Dominizinga2'nrectangle..2  A  .3  Howmanydifferentwaysaretheretodominizea2'13rectanglewith1'2dominos?..2  A  .3  Herearesomesampledominizations:1' 12  I  .3  AProblem-SolvingStrategy..2  A  .3  Thisisnotaproof-technique..2  A  .3  Sometimes,whenyouhavetosolveahardproblem,itmakessensetotrytosolveasimplerproblem....2  1  .3  Howwouldyouchangetheproblemtomakeitsimpler?......2  a  .3  Tryadifferentsizegrid.........2  a  .3  Trymanydifferentsizedgrids,andseeifyoucanfindapattern!1'! 12  I  .3  ProvingaConjecture..2  A  .3  Itlookslikeeachnumberinthesequenceisthesumofthetwopreviousnumbers:f(n)=f(n!1)+f(n!2)..2  A  .3  Howcouldweprovethatthispatternwillcontinue? ^(q <G W2$8L`"=؝QBackground BlankBackground 3Background 4 TitleBulleted ListTextOrg ChartData ChartCombinationWPCT}C:P\\TIN-3'XXXXKKKK5/ @1 'Q; ? ///333???OOO___fffooof1?O/_??ߟ߿c3f3&f3?LU`l w#'*-/257:=T bjs({3=GQ[s3&?1L; UFaQl[xfp{#',04̳9׿Q.V5b;lAvGLT ` h o rux|(-8DOZfq|3fH uUb#p/:EP\fuɒџزf3uPYcl v  *D_yf(u/5;AG L R X ^!l4zH[ofu    1%MAj^z=.O;`LrVcp̙ܥ$DYނB(#G-)L20P76U<=ZAC_FJdLPiQWmV]r[cw`j|epjwo}tufp{lws}yź3">)L3$U8*`@/lG5wO:V@^EoZn̲9WPCXX81%l 3'XXXXLLxx5139_JWPCXX81%A-3'XXXXLLxx5139_JWPC8XX81%X0%$3'XXXXLLxx5139 #X0%153**`!?'u[/;~s%+%+s+153**`"`l\M&2&2BBWPC XX81%)/#x3'XXXXLLxx ? ///333???OOO___fffooof1?O/_??ߟ߿c3f3&f3?LU`l w#'*-/257:=T bjs({3=GQ[s3&?1L; UFaQl[xfp{#',04̳9׿Q.V5b;lAvGLT ` h o rux|(-8DOZfq|3fH uUb#p/:EP\fuɒџزf3uPYcl v  *D_yf(u/5;AG L R X ^!l4zH[ofu    1%MAj^z=.O;`LrVcp̙ܥ$DYނB(#G-)L20P76U<=ZAC_FJdLPiQWmV]r[cw`j|epjwo}tufp{lws}yź3">)L3$U8*`@/lG5wO:V@^EoZn̲9_J"Ws.0 .0 W ` ` &d9 Z6Times New Roman Regular -O;@YW ` ` &d9 Z6Times New Roman Regular -O;@YW ` ` &d9 Z6Times New Roman Regular -O;@YW ` ` &d9 Z6Times New Roman Regular -O;@YW ` ` &d9 Z6Times New Roman Regular -O;@YW ` ` &d9 Z6Times New Roman Regular -O;@Y)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$ =2  I  .3  TitleofShow  W.0 .%O;3Y v/c&d9 Z6Times New Roman Regularc` ` `      O;   @Y  Q Qc&d9 Z6Times New Roman Regularc` ` `  TitleofShow.+%19׿3aQ `w.0.0??WPC.$XX81%n/"3'XXXXLLxx ? ///333???OOO___fffooof1?O/_??ߟ߿c3f3&f3?LU`l w#'*-/257:=T bjs({3=GQ[s3&?1L; UFaQl[xfp{#',04̳9׿Q.V5b;lAvGLT ` h o rux|(-8DOZfq|3fH uUb#p/:EP\fuɒџزf3uPYcl v  *D_yf(u/5;AG L R X ^!l4zH[ofu    1%MAj^z=.O;`LrVcp̙ܥ$DYނB(#G-)L20P76U<=ZAC_FJdLPiQWmV]r[cw`j|epjwo}tufp{lws}yź3">)L3$U8*`@/lG5wO:V@^EoZn̲9_"u/"u/"W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  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" " ! 1 `3' " #P U!P ! 1]@3 & X&@ X&a & 1 33& [$ & & [$ 1 `3' & X&a $a [$ 1]@3 ^) )@ )r ^)" 1 33& ' ^) ^)" '" 1 `3' ^)" )r (r '" 1]@3 , -@ - ,1 33&  + , , +1 `3' ,- [+  ++%13 ! !, .13 # >#O, \AZ"Arial RegularO        @  1stQtr.13 % %, .13 S#&O, \AZ"Arial RegularO        @  2ndQtr.13 _( _(, .13 &)O, \AZ"Arial RegularO        @  3rdQtr.13 + +, .13 !*L-O, \AZ"Arial RegularO        @  4thQtr.13  i .13 }% 4 O, \AZ"Arial RegularO        @  0.13  i .13 F U O, \AZ"Arial RegularO        @  20.13  i .13 g vO, \AZ"Arial RegularO        @  40.13  +i+.13 O, \AZ"Arial RegularO        @  60.13  MiM.13 O, \AZ"Arial RegularO        @  80.13  nin.13 Ð^O, \AZ"Arial RegularO        @  100 0 ̐ ,n.+%19׿33& Đ 6 n!n!6.+%13 Ő*#/%>O, \AZ"Arial RegularO        @North.+%133& Ɛt'6t'n(n(6.+%13 ǐ)/,>O, \AZ"Arial RegularO        @South.+%1ɒ33& Ȑ  !!.+%13 ɐ*#%O, \AZ"Arial RegularO        @East.+%1 33& ʐt't'((.+%13 ː),O, \AZ"Arial RegularO        @WestSATev3##4s#6'' '#''('Ņ'''''''΅'υ'Ѕ'х'҅'Ӆ'ԅ'&q.JRUjGl goS y+|),ݫ5''9!ZvWPCXX81%XX81%3'XXXXLLxx90"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/".*+'13 +:/\c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Proofbycontradictionc&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Assume:c&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..p F alsec&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Concludec&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..pLc&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Weshowthatifpisnottrue,thenweobtainacontradiction.ThisimpliesthatpmustbetrueLc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Example:c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Provethatan8'8checkerboardwithtwooppositecornersremovedcannotbetiledwith1'2dominosWPCXX81%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 ProofbyContradiction.+%19׿3aQ `fR// R WPCXX81%XX81%13'XXXXLLxx90"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/#*+%`L'13Y eg/ #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc TilingaPrunedCheckerboardwithDominos.+%19׿3aQ `fR//4R4WPCXX81%XX81%Dž3'XXXXLLxx90"az$dz$dW @@&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$   z$d.*+'13 ƅ0^#c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Provethatifanm'ngridcontainsanevennumberofsquares,thenitcanbecoveredbydominosc&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Anm'ngridcontainsmnsquares.Ifthisquantityiseven,theneithermmustbeevenornmustbeeven.c&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..Howwouldyouprovethat?c&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Ifmiseven...c&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..Laythedominoshorizontallyc&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Ifmisnoteven,then...c&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..nmustbeeven,andwecansetthedominosverticallyWPCXX81%XX81%Dž3'XXXXLLxx90"2(R/#R/#W &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LYW &d9 Z6Times New Roman Regular +`LY)?  9׿aQ 4 @ @ @ @ @ @ f" dddddd$   R/#.%`L13Y 6 / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc ProofbyCases.+%19׿3aQ `R// R WPCXX81%XX81%&3'XXXXLLxx9"%2)R/# R/# %R/#.%`L13Y #6 / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc SomeProofs.+%19׿3aQ `$R// R WPChXX81%XX81%&3'XXXXLLxx9""u/" u/"  "u/".'13 !+:/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'XXXXLLxx90"s3R/# R/#  R/#%`L13Y A / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Q c&d9 Z6Times New Roman RegularcATilingProof.+%19׿3aQ `R// R WPCQ XX81%XX81%3'XXXXLLxx90"Q]u/" u/"  u/".*+'13 +:/E c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@  /  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@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Nowprovethatifyouremoveasinglecornerfromthe8'8checkerboard,thenyoucannottilewhatremainswith1'3trominosWPC)XX81%XX81%03'XXXXLLxx9"/2)R/# R/# /R/#.%`L13Y -6 / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc ConverseversusContrapositive.+%19׿3aQ `.R// R WPCXX81%XX81%03'XXXXLLxx9",u@/" u@/"  ,u@/".'13 ++@:/&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,thentheregioncannotbetiledWPC'XX81%XX81%&3'XXXXLLxx9"%2(R/# R/# %R/#.%`L13Y #6 / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc Dealingwiththebluecorner.+%19׿3aQ `$R// R WPCXX81%XX81%&3'XXXXLLxx9 "%%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$   "%.'13 !ac&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.WPC3XX81%XX81%3'XXXXLLxx90"?}R/# R/#  R/#%`L13Y A / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Q c&d9 Z6Times New Roman RegularcDominizinga2'nrectangle.+%19׿3aQ `R// R WPCXX81%XX81%3'XXXXLLxx90"um/" um/"  um/"-.*+%'13 +m:/c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Howmanydifferentwaysaretheretodominizea2'13rectanglewith1'2dominos?Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Herearesomesampledominizations:WPC%XX81%XX81%&3'XXXXLLxx9"%@3R/# R/# %R/#.%`L13Y #6 / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc AProblem-SolvingStrategy.+%19׿3aQ `$R// R WPC%XX81%XX81%&3'XXXXLLxx9""Ou/"u/"W @@&d9 Z6Times New Roman Regular +ނW xx&d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +pP#W &d9 Z6Times New Roman Regular +#W &d9 Z6Times New Roman Regular +#9/!!!!!!  Z9׿aQ 4 @ @ @ @ @ @ L" dddddd$   "u/".'13 !+:/~c&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%&3'XXXXLLxx9"%?@R/# R/# %R/#.%`L13Y #6 / #c&d9 Z6Times New Roman Regularc     `L   @Y  Q Qc&d9 Z6Times New Roman Regularc ProvingaConjecture.+%19׿3aQ `$R// R WPCtXX81%XX81%&3'XXXXLLxx9""u /" u /"  "u /".'153 !+ :/xc&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Itlookslikeeachnumberinthesequenceisthesumofthetwopreviousnumbers:f(n)=f(n!1)+f(n!2)Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Howcouldweprovethatthispatternwillcontinue?        @ނF// 8(#\02Hc f p WPC/XX81%<02413'XXXXLLxx ? ///333???OOO___fffooof1?O/_??ߟ߿c3f3&f3?LU`l w#'*-/257:=T bjs({3=GQ[s3&?1L; UFaQl[xfp{#',04̳9׿Q.V5b;lAvGLT ` h o rux|(-8DOZfq|3fH uUb#p/:EP\fuɒџزf3uPYcl v  *D_yf(u/5;AG L R X ^!l4zH[ofu    1%MAj^z=.O;`LrVcp̙ܥ$DYނB(#G-)L20P76U<=ZAC_FJdLPiQWmV]r[cw`j|epjwo}tufp{lws}yź3">)L3$U8*`@/lG5wO:V@^EoZn̲9 7  d /41,*J0p($ $ ( ,*Jy%($ $ ( +$$1$*ssj[.".4*ssj[B+B+B+"B+"B+"B+4*ssj['''"'"'"'4*ssj[###"#"#"#4*ssj[* * * "* "* "* 4*ssj[www"w"w"w4*ssj["""4*ssj["""$*ssb.".$*ssj[.t4*ssj[L.L.L.LtLtLt4*ssj[ . . . t t t4*ssj[...ttt4*ssj[\.\.\.\t\t\t4*ssj[ . . . t t t4*ssj[...ttt4*ssj[l.l.l.ltltlt$*ssj["."t+1,*JJ!($ $ ( ,*J)H($ $ ( ,*Jɇhq($ $ ( ,*JA'ޣ($ $ ( ,*J͛x#ޣ($ $ ( ,*Jl/,Ĵ($ $ ( ,*Jn($ $ ( ,*Jy%Puv($ $ ( ,*JJ!Puv($ $ ( ,*Jm*Puv($ $ ( ,*Jɇ@($ $ ( ,*JA'vH($ $ ( ,*J͛x#vH($ $ ( ,*Jl/,vH($ $ ( ,*Jpz($ $ ( ,*Jy% ($ $ ( ,* JJ! ($ $ ( ,* Jm* ($ $ ( ,* Jɇ{($ $ ( ,* JA'($ $ ( ,* J͛x#($ $ ( ,*Jl/,($ $ ( ,*J($ $ ( ,*Jy%($ $ ( ,*JJ!($ $ ( ,*Jm*($ $ ( ,*JɇȀ($ $ ( ,*JA'4($ $ ( ,*J͛x#4($ $ ( ,*Jl/,4($ $ ( 1 !q!qW @@&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  Bycoveringwithdominos,wemeanplacingdominosonthecheckerboardsothateverysquareiscoveredwithoutgapsoroverlaps  !q.1 0qh|c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Bycoveringwithdominos,wemeanplacingdominosonthecheckerboardsothateverysquareiscoveredwithoutgapsoroverlaps.1    /'"C9oTc&d9 Z6Times New Roman Regularc        @X"C9J c&d9 Z6Times New Roman Regularc        @X' #K? WK? WW @@&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$ l2  I  .3  Withthesetwosquaresremoved,canwhatremainsbecovered?  #K? W.1 /? _:c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@        @ނ        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Withthesetwosquaresremoved,canwhatremainsbecovered?.1 (<02f <02f 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  A  .3  Ifso,thenlookatwhathappens:Eachdomino,nomatterwhereplaced,coversexactlyoneblueandonewhitesquare.Butthentheremustbeanequalnumberofwhiteandbluesquaresintheportioncoveredbydominos.Thisisfalse.  (<02f .13# .<z1 }c&d9 Z6Times New Roman Regularc        @#a /!!!!!!aM..Mc&d9 Z6Times New Roman Regularcxxx        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Ifso,thenlookatwhathappens:Eachdomino,nomatterwhereplaced,coversexactlyoneblueandonewhitesquare.Butthentheremustbeanequalnumberofwhiteandbluesquaresintheportioncoveredbydominos.Thisisfalse.WPC XX81%#51Dž3'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@^EoZn̲9_  Ņ#51+%1"$$h"$/" $/"+$/"E$/"k$/"$/"$/"$/"$$h"$$h"$$h"$$h"$$h"…a $$h+%'1"Å!>)W ,c&d9 Z6Times New Roman Regularc@@@        @m"ą>)V ,c&d9 Z6Times New Roman Regularc@@@        @nWPC&#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 "11 + !Ac&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@  /  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 : !1c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Thereare21red,21whiteand22bluesquares.WPC.2XX81%0$3'XXXXLLxx ? ///333???OOO___fffooof1?O/_??ߟ߿c3f3&f3?LU`l w#'*-/257:=T bjs({3=GQ[s3&?1L; UFaQl[xfp{#',04̳9׿Q.V5b;lAvGLT ` h o rux|(-8DOZfq|3fH uUb#p/:EP\fuɒџزf3uPYcl v  *D_yf(u/5;AG L R X ^!l4zH[ofu    1%MAj^z=.O;`LrVcp̙ܥ$DYނB(#G-)L20P76U<=ZAC_FJdLPiQWmV]r[cw`j|epjwo}tufp{lws}yź3">)L3$U8*`@/lG5wO:V@^EoZn̲9  *++2]e I"0-+1*g||U"B e f1*~||U"B hrrrrrrB iB jB k6r6r6rrrr6666r6r6rB l# r# r# rrrr# # # # r# r# rB mrrrrrrrrrB nr# r# r# $ # $ # $ # $ r$ r$ rrrrrrrr# r# r# B o# # # 6# 6# 6# 6r6r6rrrr# # # B p$ $ $ $ $ $ B q$ $ $ $ $ $ B rrrrr$ r$ r$ $ $ $ B sB t# # # # # # # # # B u666666666B v555656565666555B wr5r5r5$ 5$ 5$ 5$ $ $ rrrr5r5r5B xrrrr6r6r6666B y666666B z666666B {666666666B |# # # # # # # # # B }15*||U"B ttt$ $ $ $ $ $ ttttttB 666666B rrrrrrrrrB rrrrrrrrrB rrrrrrrrrrrrrrrB # # # # # # rrrrrr# # # B # # # # # # # # # rrr# r# r# r# # # # # # B 5# 5# 5# # # # rrr5r5r5r5# 5# 5# B 6666# 6# 6# # # # B ttt$ $ $ $ # $ # $ # t# t# t# tttB rrrrrrB B B 555555555555555555B # 5# 5# 5555# # # # 5# 5# 5B 555555555B ttt$ $ $ $ 6$ 6$ 6t6t6t6tttB 666666666666B B B rrrrrr+15 $$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  Wecanseethatifweremoveawhitesquare,thenwhatremainscannotbetiledwith1'3trominos2  A  .3  Becausethentherewillbe20white,21redand22bluesquares2  A  .3  Andiftheregioncouldbetiled,therewouldhavetobethesamenumberofeachcolorofsquare2  I  .3  Sameforaredsquare  $1 $ c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Wecanseethatifweremoveawhitesquare,thenwhatremainscannotbetiledwith1'3trominosc&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Becausethentherewillbe20white,21redand22bluesquaresM..Mc&d9 Z6Times New Roman Regularcxxx        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Andiftheregioncouldbetiled,therewouldhavetobethesamenumberofeachcolorofsquareLc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Sameforaredsquare1 00W @@&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  Butwhatifweremoveabluesquare?Thentheresultingregiondoeshaveanequalnumberofeach.2  I  .3  Doesthatmeantheregioncanbetiled?2  I  .3  No!  01 uI0c&d9 Z6Times New Roman Regularc        @a /!!!!!!ac&d9 Z6Times New Roman Regularc@@@  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Butwhatifweremoveabluesquare?Thentheresultingregiondoeshaveanequalnumberofeach.Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..Doesthatmeantheregioncanbetiled?Lc&d9 Z6Times New Roman Regularc@@@        @  /  c&d9 Z6Times New Roman Regularc@@@        @ނ0M..No!WPCXX81%w 03'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@^EoZn̲94H " *0+!!1* D  1*υ 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 Årrrr$ r$ r$ $ $ $ D ąD Ņ# # # # # # # # # D ƅ666666666D Dž555656565666555D ȅr5r5r5$ 5$ 5$ 5$ $ $ rrrr5r5r5D Ʌrrrr6r6r6666D ʅ666666D ˅666666D ̅666666666D ͅ# # # # # # # # # D ΅15* D Ѕttt$ $ $ $ $ $ ttttttD х666666D ҅rrrrrrrrrD ӅrrrrrrrrrD ԅrrrrrrrrrrrrrrrD Յ# # # # # # rrrrrr# # # D օ# # # # # # # # # rrr# r# r# r# # # # # # D ׅ5# 5# 5# # # # rrr5r5r5r5# 5# 5# D ؅6666# 6# 6# # # # D مttt$ $ $ $ # $ # $ # t# t# t# tttD څrrrrrrD ۅD ܅D ݅555555555555555555D ޅ# 5# 5# 5555# # # # 5# 5# 5D ߅555555555D ttt$ $ $ $ 6$ 6$ 6t6t6t6tttD 666666666666D D D rrrrrr+%15,*`1 1 xд/ f.*0*0 .  $" +" WPC XX81%>/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@^EoZn̲9_ "G p  "h x+%1$"JP 0 0 P  "Ӆ | ԅpp   Յ p p  $"օP 0 0 P $"ׅP 0 0 P $"؅P 0 0 P $"مP 0 0 P $"څl P 0 0 P  "ۅ | ܅pp   ݅ p p  $"ޅ P 0 0 P $"߅P 0 0 P $"P 0 0 P  "хh.P( " | pp   … p p   "Å | ąpp   Ņ p p  $"ƅP 0 0 P  "DžL  | ȅpp   Ʌ p p  $"ʅl P 0 0 P $"˅L P 0 0 P $"̅, P 0 0 P  "ͅ | ΅pp   υ p p  $"ЅP 0 0 P  "h x$"JP 0 0 P  " | pp    p p   " | pp    p p   "L  | pp    p p   "  | pp    p p   " | pp    p p   " | pp    p p  +%'1 >/{.c&d9 Z6Times New Roman Regularc@@@        @Canyoudiscoverastrategyforgettingthisanswer?WPC XX81%450G ׅ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@^EoZn̲9_ "΅ +$$%1   +%   +%1 "υ^B +$$%1 p 0 0 p+% p 0  P P +%1 "Ѕ  +$$%1   +%   "P P "…P P +%1 "х~" +$$%1   +%   "Å< P P  ąg g"Ņg g+%1 "҅b( +$$%1  P( P(+%  P( "ʅP P "˅S g g"̅4 g g"ͅ& g g+%13ނ"Ӆ c&d9 Z6Times New Roman Regularc@@@        @ނ#ways:123"ԅzTf.c&d9 Z6Times New Roman Regularc@@@        @ނက58WPC XX81%M13'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@^EoZn̲9_  D(M1+%1 ,, P((/(+%13ނ"'~]/tc&d9 Z6Times New Roman Regularc        @ނn󀀀#ways11223345586137218349551089%3 <(-<(-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  A  .3  Thereare2waystostartdominizinga2'ngrid:2  1  .3  Laythefirstdominovertically2  1  .3  Inwhichcasetherearef(n!1)waystofinishthejob2  1  .3  Laythefirstdominohorizontally2  1  .3  Inwhichcasetheoneaboveorbelowitmustalsoliehorizontally,andthentherearef(n!2)waystofinishthejob2  A  .3  Sincethoseareallthecases,weveproventhatf(n)=f(n!1)+f(n!2)  <(-13# <' c&d9 Z6Times New Roman Regularc        @#a /!!!!!!aM..Mc&d9 Z6Times New Roman Regularcxxx        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Thereare2waystostartdominizinga2'ngrid:c&d9 Z6Times New Roman Regularc        @#M..M..        @  !  c&d9 Z6Times New Roman Regularc        @#0..LaythefirstdominoverticallyM..M..c&d9 Z6Times New Roman Regularc        @  !  c&d9 Z6Times New Roman Regularc        @#0..Inwhichcasetherearef(n!1)waystofinishthejobM..M..c&d9 Z6Times New Roman Regularc        @  !  c&d9 Z6Times New Roman Regularc        @#0..LaythefirstdominohorizontallyM..M..c&d9 Z6Times New Roman Regularc        @  !  c&d9 Z6Times New Roman Regularc        @#0..Inwhichcasetheoneaboveorbelowitmustalsoliehorizontally,andthentherearef(n!2)waystofinishthejobc&d9 Z6Times New Roman Regularcxxx        @#M..M        @    c&d9 Z6Times New Roman Regularcxxx        @#0..Sincethoseareallthecases,weveproventhatf(n)=f(n!1)+f(n!2)SummaryInformation(PerfectOffice_OBJECTS,jjRob189$C:\COREL\SUITE8\PROGRAMS\PRWIN8.EXE