) GSP! ?N ۃJ ۀlۀ"lۀ $۲۲lۀۀ4ۀۀl۲ ۀ A  B !-:@ UKThis sketch traces the paths of four ants as they chase each other. Ant A takes one step towards ant B, ant B towards ant C, ant C towards ant D, and ant D towards and A. This process is repeated with each ant reassesing the position of the ant that they are chasing after each step.2@ F@@:LppL s i`tyV F-:@C Cty"H E-:@AC8~= D-:@CC7n<s C-:@CCYV^[ B-:@CBRW A-:@CBq~o\ A-:@ &Bill Finzer, Key Curriculum Press 1993rr$s}=c4  y-:@CCCB?6gs  x-:@CCCC?XU'W@  w-:@CBCC?Q'  v-:@CBCB?sy'H step-:@ACC C?M, t,y1>  Htep-:@ B8C t y%8CH C1ep-:@ CsCj>:W\ B1ep-:@ CrC ?;X]rk M A1ep-:@ C4B @<y 24ep-:@C#C0oA@?5%F?5%FE)4AL 23ep-:@CsCjA@?5%F?5%FFMeh 22ep-:@CrC A@?5%F?5%FGNf 21ep-:@C4B A@?5%F?5%FH M  f1ep-:@C#C0oC4B ?HE8C  e1ep-:@CsCjC#C0o?EF'\  d1ep-:@CrC CsCj?FGW 'C  c1ep-:@C4B CrC ?GH  b1ep-:@CwCC#C0o?E52=:O  a1ep-:@C CyCsCj?F6|V]@  z1ep-:@CB)CrC ?G7T]  y1ep-:@C(̲BC4B ?H8 H1ep-:@ C@GC~MI1368 G1ep-:@ CELCL%NJUZ F1ep-:@ CCOK[`w1 E1ep-:@ C@BePL|  28ep-:@C@GC~A@?5%F?5%FU')?A 27ep-:@CELCL%A@?5%F?5%FVKc9|x 26ep-:@CCA@?5%F?5%FWQi 25ep-:@C@BeA@?5%F?5%FXb.C  n2ep-:@C@GC~C@Be?XU68  m2ep-:@CELCL%C@GC~?UV'Z   k2ep-:@CCCELCL%?VWZ'   j2ep-:@C@BeCC?WX "$  k1ep-:@C#C0oC@GC~?UE028C  j1ep-:@CsCjCELCL%?VFT\  h1ep-:@CrC CC?WGW`  g1ep-:@C4B C@Be?XH M1ep-:@ C +mCsr`]Y-'2, L1ep-:@ C[C^ZSXrenc K1ep-:@ CIuC"x_[`e J1ep-:@ CK;C B2ep-:@ CCMzD3 A2ep-:@ CsZB  48ep-:@C$CJpA@?5%F?5%F % P 47ep-:@C~ʾCi6A@?5%F?5%F4L 46ep-:@CCMA@?5%F?5%Fp 45ep-:@CsZBA@?5%F?5%FC'  d3ep-:@C$CJpCsZB?@  c3ep-:@C~ʾCi6C$CJp?'C  b3ep-:@CCMC~ʾCi6?y'%=@  a3ep-:@CsZBCCM?36(  z3ep-:@C%CS̡C$CJp?# u2  y3ep-:@CKCC~ʾCi6?=J  x3ep-:@CnCCCCM?q  w3ep-:@CjEBgCsZB? H2ep-:@ C-!qCA9 G2ep-:@ CuCyN6;+C1 F2ep-:@ C#GCVTx E2ep-:@ C{ CN5D 52ep-:@C-!qCAA@?5%F?5%F 51ep-:@CuCyNA@?5%F?5%F,D-H 50ep-:@C#GCVA@?5%F?5%Fx/Bg 49ep-:@C{ CN5A@?5%F?5%F'/Tz  k4ep-:@C-!qCAC{ CN5?/t  j4ep-:@CuCyNC-!qCA?$%;  k3ep-:@C#GCVCuCyN?'k  j3ep-:@C{ CN5C#GCV?0.6  h3ep-:@C$CJpC-!qCA?x  g3ep-:@C~ʾCi6CuCyN?5C   f3ep-:@CCMC#GCV?y   e3ep-:@CsZBC{ CN5??. M2ep-:@ C6C:   L2ep-:@ CmC_-2a K2ep-:@ CC^:/L J2ep-:@ CVC 0 56ep-:@C6C:A@?5%F?5%Fi 55ep-:@CmC_A@?5%F?5%F#;0 54ep-:@CC^:A@?5%F?5%F 53ep-:@CVC A@?5%F?5%FC'  u4ep-:@C6C:CVC ?w   t4ep-:@CmC_C6C:?]2  s4ep-:@CC^:CmC_?'R  r4ep-:@CVC CC^:?fD  q4ep-:@C-!qCAC6C:?  p4ep-:@CuCyNCmC_?,;  n4ep-:@C#GCVCC^:?  m4ep-:@C{ CN5CVC ?0 R2ep-:@ C@C4wM0 Q2ep-:@ Cf#C~Se#(0 P2ep-:@ CACd ` N2ep-:@ C> Cv/ Hz 60ep-:@C@C4wMA@?5%F?5%F / 59ep-:@Cf#C~SeA@?5%F?5%F1=@ 58ep-:@CACdA@?5%F?5%F 57ep-:@C> CvA@?5%F?5%F'  c4ep-:@C@C4wMC> Cv?jL  b4ep-:@Cf#C~SeC@C4wM?;(  a4ep-:@CACdCf#C~Se?'p  z4ep-:@C> CvCACd?  y4ep-:@C6C:C@C4wM?   x4ep-:@CmC_Cf#C~Se?"2  w4ep-:@CC^:CACd?   v4ep-:@CVC C> Cv?g|  V2ep-:@ CL~C0g U2ep-:@ Ca6CsX  T2ep-:@ CCi}=  S2ep-:@ CXLC#$ 64ep-:@CL~C0A@?5%F?5%F 63ep-:@Ca6CsA@?5%F?5%F& 62ep-:@CCi}=A@?5%F?5%F 61ep-:@CXLC#A@?5%F?5%FV'  j5ep-:@CL~C0CXLC#?\  k4ep-:@Ca6CsCL~C0?s  j4ep-:@CCi}=Ca6Cs? 'cyP  h4ep-:@CXLC#CCi}=?p  g4ep-:@C@C4wMCL~C0?//  f4ep-:@Cf#C~SeCa6Cs?(/  e4ep-:@CACdCCi}=?/ /8  d4ep-:@C> CvCXLC#? Z2ep-:@ CW~C-;  Y2ep-:@ C]nCh/   X2ep-:@ CCk  W2ep-:@ CwC/} 68ep-:@CW~C-;A@?5%F?5%F 67ep-:@C]nCh/A@?5%F?5%F3< 66ep-:@CCkA@?5%F?5%F  65ep-:@CwC/}A@?5%F?5%F'  t5ep-:@CW~C-;CwC/}?  s5ep-:@C]nCh/CW~C-;?   r5ep-:@CCkC]nCh/?'t  q5ep-:@CwC/}CCk?  p5ep-:@CL~C0CW~C-;?  n5ep-:@Ca6CsC]nCh/?   m5ep-:@CCi}=CCk? '  k5ep-:@CXLC#CwC/}? D3ep-:@ CcC.1 C3ep-:@ C\t3C\> B3ep-:@ CCjd  A3ep-:@ CdC;} 72ep-:@CcC.1A@?5%F?5%F 71ep-:@C\t3C\>A@?5%F?5%FB 70ep-:@CCjdA@?5%F?5%F 69ep-:@CdC;}A@?5%F?5%F '  b5ep-:@CcC.1CdC;}?O  a5ep-:@C\t3C\>CcC.1?2  z5ep-:@CCjdC\t3C\>?'Y  y5ep-:@CdC;}CCjd?   x5ep-:@CW~C-;CcC.1?&c  w5ep-:@C]nCh/C\t3C\>?T2  v5ep-:@CCkCCjd?   u5ep-:@CwC/}CdC;}?' H3ep-:@ CoVvC1   G3ep-:@ C^\CPe  F3ep-:@ CwCCft  C E3ep-:@ C٪CG? B 76ep-:@CoVvC1A@?5%F?5%F4( 75ep-:@C^\CPeA@?5%F?5%F4@ 74ep-:@CwCCftA@?5%F?5%Fk4 73ep-:@C٪CG?A@?5%F?5%Fe'Ct  k5ep-:@CoVvC1C٪CG??  j5ep-:@C^\CPeCoVvC1?  h5ep-:@CwCCftC^\CPe?'2L  g5ep-:@C٪CG?CwCCft?C  f5ep-:@CcC.1CoVvC1?U  e5ep-:@C\t3C\>C^\CPe?  d5ep-:@CCjdCwCCft? $  c5ep-:@CdC;}C٪CG?? M3ep-:@ Cx2C840 L3ep-:@ Cd-:CEm K3ep-:@ CnC^sx? J3ep-:@ CCQ   80ep-:@Cx2C8A@?5%F?5%F% 79ep-:@Cd-:CEmA@?5%F?5%F&Y 78ep-:@CnC^sxA@?5%F?5%F' 77ep-:@CCQA@?5%F?5%F('Ni  s6ep-:@Cx2C8CCQ?(%'  r6ep-:@Cd-:CEmCx2C8?%&  q6ep-:@CnC^sxCd-:CEm?&'  p6ep-:@CCQCnC^sx?'(4   n6ep-:@CoVvC1Cx2C8?%3  m6ep-:@C^\CPeCd-:CEm?&  k6ep-:@CwCCftCnC^sx?'  j6ep-:@C٪CG?CCQ?(3 R3ep-:@ C~SCCT-) Q3ep-:@ Cn[C?.*L P3ep-:@ Ci2CSQ/+$ N3ep-:@ CzCW)0, & 84ep-:@C~SCCTA@?5%F?5%F5 83ep-:@Cn[C?A@?5%F?5%F65X 82ep-:@Ci2CSQA@?5%F?5%F7 81ep-:@CzCW)A@?5%F?5%F8'X/  a6ep-:@C~SCCTCzCW)?85 '  z6ep-:@Cn[C?C~SCCT?56   y6ep-:@Ci2CSQCn[C??67  x6ep-:@CzCW)Ci2CSQ?78.  w6ep-:@Cx2C8C~SCCT?5%A  v6ep-:@Cd-:CEmCn[C??6&&  u6ep-:@CnC^sxCi2CSQ?7'g  t6ep-:@CCQCzCW)?8( V3ep-:@ C|"COy=9 U3ep-:@ Cy7CB>:? T3ep-:@ ClfCG?; S3ep-:@ CnCT@</ 88ep-:@C|"COyA@?5%F?5%FE 87ep-:@Cy7CBA@?5%F?5%FFxA 86ep-:@ClfCGA@?5%F?5%FG 85ep-:@CnCTA@?5%F?5%FH5  j6ep-:@C|"COyCnCT?HE'a   h6ep-:@Cy7CBC|"COy?EFo'   g6ep-:@ClfCGCy7CB?FGi,  f6ep-:@CnCTClfCG?GH  e6ep-:@C~SCCTC|"COy?E5C(  d6ep-:@Cn[C?Cy7CB?F6  c6ep-:@Ci2CSQClfCG?G74  b6ep-:@CzCW)CnCT?H8' Z3ep-:@ CpVCS:MI Y3ep-:@ C{CNcNJ X3ep-:@ Cw8CCoOK W3ep-:@ ClCH/PL 92ep-:@CpVCS:A@?5%F?5%FU 91ep-:@C{CNcA@?5%F?5%FV  90ep-:@Cw8CCoA@?5%F?5%FW 89ep-:@ClCH/A@?5%F?5%FXV6<  r7ep-:@CpVCS:ClCH/?XUf6D  q7ep-:@C{CNcCpVCS:?UV'C  p7ep-:@Cw8CCoC{CNc?VWT'  n7ep-:@ClCH/Cw8CCo?WX  m7ep-:@C|"COyCpVCS:?UE  k7ep-:@Cy7CBC{CNc?VFU  j7ep-:@ClfCGCw8CCo?WG  k6ep-:@CnCTClCH/?XH D4ep-:@ ClCH]Y?5%F C4ep-:@ Cq+CS ^Z6$ B4ep-:@ C|CN_[ A4ep-:@ Cwm&CC`\fB 96ep-:@ClCHA@?5%F?5%Fe 95ep-:@Cq+CS A@?5%F?5%Ff$ 94ep-:@C|CNA@?5%F?5%Fg 93ep-:@Cwm&CCA@?5%F?5%Fh'  z7ep-:@ClCHCwm&CC?heN  y7ep-:@Cq+CS ClCH?ef|  x7ep-:@C|CNCq+CS ?fg'  w7ep-:@Cwm&CCC|CN?ghŨe  v7ep-:@CpVCS:ClCH?eU  u7ep-:@C{CNcCq+CS ?fV?5%F  t7ep-:@Cw8CCoC|CN?gW  s7ep-:@ClCH/Cwm&CC?hXP H4ep-:@ CwoCCxmib G4ep-:@ ClCHnj$ F4ep-:@ Cq(BCSoki E4ep-:@ C| .CNplv 100p-:@CwoCCxA@?5%F?5%Fu 990p-:@ClCHA@?5%F?5%FvD 980p-:@Cq(BCSA@?5%F?5%FwC3 970p-:@C| .CNA@?5%F?5%Fx'B  h70p-:@CwoCCxC| .CN?xu'  g70p-:@ClCHCwoCCx?uv]8  f70p-:@Cq(BCSClCH?vwl"  e70p-:@C| .CNCq(BCS?wxC  d70p-:@ClCHCwoCCx?uePD  c70p-:@Cq+CS ClCH?vfZpd  b70p-:@C|CNCq(BCS?wg  a70p-:@Cwm&CCC| .CN?xhBvr M40p-:@ C| !CN}yBrB L40p-:@ CwpCCc~zL K40p-:@ ClCH{ J40p-:@ Cq(CS|Br 104p-:@C| !CNA@?5%F?5%F2< 103p-:@CwpCCcA@?5%F?5%FCZ 102p-:@ClCHA@?5%F?5%Fº~ 101p-:@Cq(CSA@?5%F?5%FmD  q81p-:@C| !CNCq(CS?'0D  p81p-:@CwpCCcC| !CN?'C"  n81p-:@ClCHCwpCCc?^/D  m81p-:@Cq(CSClCH?  k81p-:@CwoCCxC| !CN?uBX~  j81p-:@ClCHCwpCCc?v  k71p-:@Cq(BCSClCH?wBDm  j71p-:@C| .CNCq(CS?x R41p-:@ Cq(CSD/ Q41p-:@ C| CNS P41p-:@ CwpCCd1 N41p-:@ ClCH} 108p-:@Cq(CSA@?5%F?5%F.l 107p-:@C| CNA@?5%F?5%FzC 106p-:@CwpCCdA@?5%F?5%FC~ 105p-:@ClCHA@?5%F?5%F^D  y85p-:@Cq(CSClCH?m  x85p-:@C| CNCq(CS?'  w85p-:@CwpCCdC| CN?' Cm  v85p-:@ClCHCwpCCd?  u85p-:@C| !CNCq(CS?7HE  t85p-:@CwpCCcC| CN?3dD  s85p-:@ClCHCwpCCd?C  r85p-:@Cq(CSClCH? V45p-:@ ClCHtC U45p-:@ Cq(CS T45p-:@ C| CN S45p-:@ CwpCCdB t 112p-:@ClCHA@?5%F?5%F= 111p-:@Cq(CSA@?5%F?5%FV 110p-:@C| CNA@?5%F?5%F0C 109p-:@CwpCCdA@?5%F?5%F'Cm  g89p-:@ClCHCwpCCd?^C  f89p-:@Cq(CSClCH?mCb  e89p-:@C| CNCq(CS?'CT!  d89p-:@CwpCCdC| CN?24  c89p-:@Cq(CSClCH?Ⱦ  b89p-:@C| CNCq(CS?1  a89p-:@CwpCCdC| CN?Cq  z89p-:@ClCHCwpCCd?6 Z49p-:@ CwpCCdxC Y49p-:@ ClCHB  X49p-:@ Cq(CS3$ W49p-:@ C| CNCG 116p-:@CwpCCdA@?5%F?5%F0 115p-:@ClCHA@?5%F?5%F.C 114p-:@Cq(CSA@?5%F?5%FB 113p-:@C| CNA@?5%F?5%F'BM  p93p-:@CwpCCdC| CN?'XC  n93p-:@ClCHCwpCCd?^  m93p-:@Cq(CSClCH?m  k93p-:@C| CNCq(CS?Be  j93p-:@ClCHCwpCCd? lC  k83p-:@Cq(CSClCH?C  j83p-:@C| CNCq(CS?C  h83p-:@CwpCCdC| CN?Ct D53p-:@ C| CN0h C53p-:@ CwpCCdC~ B53p-:@ ClCHBK A53p-:@ Cq(CS , 120p-:@C| CNA@?5%F?5%F 119p-:@CwpCCdA@?5%F?5%Ff 118p-:@ClCHA@?5%F?5%F6 K51p-:@ C| CN/hC J51p-:@ CwpCCdL 128p-:@ClCHA@?5%F?5%FpB} 127p-:@Cq(CSA@?5%F?5%F 126p-:@C| CNA@?5%F?5%FCw 125p-:@CwpCCdA@?5%F?5%F'C?  n10p-:@ClCHCwpCCd?^BD  m10p-:@Cq(CSClCH?m%FA  k10p-:@C| CNCq(CS?'  j10p-:@CwpCCdC| CN?4,ì  k90p-:@Cq(CSClCH?  j90p-:@C| CNCq(CS?o  h90p-:@CwpCCdC| CN?zÒ  g90p-:@ClCHCwpCCd?- R50p-:@ CwpCCdB C@ Q50p-:@ ClCH P50p-:@ Cq(CSC N50p-:@ C| CNAV  s10p-:@ClCHCwpCCd?1  r10p-:@Cq(CSClCH?AV  q10p-:@C| CNCq(CS?C,݌  p10p-:@CwpCCdC| CN?