ࡱ> 7 (bjbjUU "<7|7|$l8,J|(>>>>>0 J J J J J J J$L (N1J<^<<1J>>BFJ< >l> J< JEl, J>  @ H/G* J\J0JYGNN JAbstract: Shared life in Go generalized nakade captures Harry Fearnley, 38 Henley Street, Oxford, OX4 1ES, UK. email:  HYPERLINK "mailto:harry@ZZZgoban.demon.co.uk" harry@ZZZgoban.demon.co.uk Abstract: The game of go [weiqi (Chinese/C)/ baduk (Korean/K)] has a number of distinct types of shared life known as seki (Japanese/J) [shuang1 huo2 (C)/ bik (K)]. We examine positions where both players may make (big-eye/nakade) captures which do not guarantee independent life. Both players will usually prefer not to make a capture. We provide a systematic treatment, and describe several unpublished examples. Introduction Here we consider only those positions which involve one particular type of capture one-sided, generalized big-eye/nakade (chijungsu(K)/ HYPERLINK "http://senseis.xmp.net/?edit=ChiJoongSu" zhong1 shou3(C)) capture. After such a capture, the player making the capture does not make guaranteed independent life two eyes, or a seki with an internal lump of stones. We consider various combinations of these, and show which are stable configurations of shared life, and which not. We assume a basic Chinese method of counting, and deal almost entirely with terminal positions. Capture In shared life we sometimes find stones which can be captured perhaps, on the next move. In the simplest of such situations each capture is one-sided i.e. if Black can capture some white stones, then those white stones (if left not captured) cannot themselves capture any black stones. When the stones are captured, the capturer (Black, say) cannot be sure of making independent life. Of the one-sided captures, the simplest types can not lead to ko(jie(C)/ pae(K)). These types are the centre/edge nakade, and the simple corner nakade. Captures of complex corner nakade may lead to ko to remain as seki, the defender must have non-removable ko-threats elsewhere. Simple capture One-, and two-, stone capture can usually be ignored we consider the capture of more than two stones. The most well-known such captures are the nakade captures of 3-6 stones in the centre/edge of the board. Some of the 4-6 stone captures give fewer liberties in the corner. Some simple corner nakade have an eye of their own i.e. the 1-1 point is empty. There are six such cases an extra external/shared liberty is required -- captures are equivalent to 2, 3, 4, and 6 liberties. Complex corner nakade The complex, corner captures all involve the possibility of ko sometimes the attacker has the choice, sometimes the defender has the choice. These positions are bent-four-in-the-corner, Murashimas ko (see corners in Fig 4), and rectangular six. We examine two further positions which contain an eye -- step four, and butterfly seven they may challenge Japanese rulesets, but cannot be used in simple seki. The building blocks We have now collected several different captures which give varying numbers of internal liberties: 0 (no eye, no capture!); 1 (one point eye, or 1-, or 2-, stone capture); 2 (3-stone capture, or 2x2-block in corner); 3 (four 5-stone corner captures, corner 4-nakade with eye); 4 (normal 4-stone nakade); 4/1&k- (bent 4 in the corner); 6 (Flower six in the corner, and flower-six nakade with eye); 6/1&k- (rectangular six in the corner); 7 (normal 5-nakade); 9/3k (Murashimas Ko Fig 1); 11 (normal flower 6 nakade). We use combinations of these building blocks to construct sekis. One against one If Black has a single eye containing such a nakade shape, opposed to a single white group, and there are no external liberties, then it is fairly easy to see that the condition for the position to be a stable seki is that Black and White surround nakade of the same size i.e. having the same number of internal liberties (not always equivalent to captures of the same size/shape lump!). Complex nakade (corner only) may give ko as an alternative to seki. One against many We now consider one black group versus several white groups. We have to consider how many liberties each group has. If a configuration is really one of shared life, then we know that a minimum requirement is that, for each group, the total number of (both internal/eye and external) liberties must be equal to the total number for all groups with which it shares liberties. If a group touches another, but does not share liberties with it then it is not part of the same seki configuration. This means that all such inter-related groups must have the same total number of liberties. Consider the case where we have one black group in a fight with several (n) white groups with (w1, w2, , wn) respectively internal liberties, and sharing (l1, l2, , ln) liberties with the black group, which has b internal liberties. We do not include the case when b = 0 (i.e. Black has zero internal liberties) considered separately. If all (internal and external) liberties were the same then the condition for seki would be that, for each white group: wi + 2* li = b + sum_of_all(lj) + n { wi > b > 0, n > 1 } . . . . . Equation 1.m All groups and liberties the same In the simple case where all the wis are the same (w1 = w2 = = wn = w), and all the shared liberties are equal to one (l1 = l2 = = ln = 1), we get w+2 = b + 2*n. This gives the conditions (n = 2, w = b+2), (n = 3, w = b+4), (n = 4, w = b+6), The above condition deals with case where either White, or Black, starts an attack by playing on shared liberties. Black never wants to start an attack by capturing (playing on an internal liberty). When White starts the attack by playing an internal liberty, Blacks defence is to play on as many of these internal liberties as possible. If the black group, and this white group, now have nakade of the same size, then Black has created a temporary stalemate between the two groups now Black can eventually kill all the white groups. If Black cannot do this then the black group dies. With this extra condition added to those above, the only remaining possibilities are: n = 2, w = b+2, giving: (w=3, b=1), unstable seki Fig 1 - White plays at any time to get a terminal seki, and gets 14 points; (w=4 , b=2), possible unstable seki Fig 2 if either (or both) of the black lumps is the pyramid-4 shape, then White captures eight black stones, and obtains another seki, gaining 11 points -- see also [Feldmann2005a]; (w=6, b=4), simple seki Fig 3; (w=9, b=7), Fig 4: White choice seki, or ko to kill Black; (w=11, b=9), not a seki Black wins - white liberty shortage(damezumari(J)/ gong-bae-me-u-gi(K)). n = 3, w = b+4, gives the single case (w=11, b=7) Fig 5 see also [Feldmann2005a].   Fig 1: Equal 1-3 Fig 2: Equal 4-6   Fig 3: Possible unstable seki Fig 4: Equal 7-9  Fig 5: Largest centre/edge combination Fig 6a is unusual -- if White plays first then all white stones die, whereas if Black plays first then one white group lives Black decides which one lives. Four other configurations exhibit similar behaviour. The most extreme example is shown in Fig 6b. We tabulate outcomes of all positions involving centre/edge nakade only.   Fig 6a: One-or-two die Fig 6b: Four-or-five die Different sized captures; different shared liberties One against two In Equation 1.m, with two white groups, and l1 = 1, l2 = 2, we get: b+5 = w1+2 = w2+4 ( w1 = b+3, and w2 = b+1, and similarly, if (l1 = 2, 3, ; l2 = l1+1), Other combinations of values of l1, and l2, are not possible -- this would give wi <= b, which violates a condition. Whites only possible attack is first to capture; Blacks defence is to take those internal liberties, and to threaten a temporary seki. This gives us three families of seki: 1) b = 1, w1 = 4, w2 = 2, l1 = l, 2, 3, , l2 = l1+l seki Fig 7 -- see also [Feldmann2005a] 2) b = 3, w1 = 6, w2 = 4, l1 = l, 2, 3, , l2 = l1+l simple seki see Fig 8 3) b = 6, w1 = 9, w2 = 7, l1 = l, 2, 3, , l2 = l1+l Fig 9 - White is safe, but can kill Black with ko.   Fig 7: Unequal 1-2-4 Fig 8: Unequal 3-4-6  Fig 9: Unequal 6-7-9 In the three families above, the only ones which are terminal seki occur when l1 = l. When l1 > l we have (l1 - l) one-sided, non-removable, ko threats for Black. Further analyses and conclusions We continue the above analysis of unequal groups and liberties for one black group simultaneously against many white ones. We further extend all the previous analyses from one black group to two, or more, black groups e.g. consider Fig 10, below. We consider more aspects of the involvement of ko, including complex corner nakade involved in fights with bigger nakade (with, or without, ko). More work remains to be done -- both including more complicated topologies, and with more groups.  Fig 10: 2 black versus 3 white : 1-3   ;<|};=n o !"618bgKO#$'(./`adekl(JK]^bcklFپپپپپپپپپپپپپپپپپCJH* 6CJ] 5CJ\ jCJU>*CJCJ5CJ50JCJjCJU jCJUCJ5CJ K;<no|n o ~ n o !"6dev$a$$a$$a$$a$((((&tL,MFGHZ[PQ| $ & Fa$ & F$a$FOPXY[Pz{| d!e!!!!!!3"`"a"h"i"~""""""""""""""""""""""$#&#$$ $$$$CJOJPJQJ^J jCJOJQJ^JCJH*OJQJ^JCJOJQJ^J5CJOJPJQJ\^J j/CJU j'CJUCJ jgU jU jU 5CJ\ j U jU; Y!Z!!!!!""#"3"w""#Y$$%%'%(%%%%%%%P&$a$ $[$\$a$$a$$%$'$*$+$d$f$l$n$t$v$$$$$$$$$$$$$$$%%%%$%%%6%%%%%%%%& &&&Q&r&g(h(i(((((((((((CJ$B*CJ0mH phsH  B*phjCJUmHnHsH u jkRUCJH* j{JU 5CJ\ j@U j9UCJCJH*OJQJ^JCJOJQJ^J9P&Q&r&g(h(j(k((((((((((((((((((((( $a$$a$$a$((((((((((((((((((((((  1h/ =!"#$%DyK harry@ZZZgoban.demon.co.ukyK Dmailto:harry@ZZZgoban.demon.co.uk1Dd  C |Ad..\..\Folders_Folder\seki-paper\images\x2-1-3.pngb=vDy|13vnvDy|1PNG  IHDR<ϫ̱,tEXtCreation TimeMon 20 Jun 2005 17:24:34 -0000qtIMEVn pHYs  ~gAMA aWIDATx\r1 \PP0P|EZ~._pJ'`B .r7x-˖n*ke˒,:&7OV|~Q.B߾y's'p;G64ii^G . :8@`HDH0w0xb.kwLM`I$?@ocnwR ۤ΃B#EeJLI9X@:]R &gZ$He =gYdM.I$ jY&J;>wvOB[ x']T!dnyAό5y=zC. sI0Vs @!z$#evId+b ** T,XBhC55(t :Dz#HRG<3b?V!xP3gMQ'Q$Oy$uW2hCwSH$Yjޓs% |*CRSr&il-2Og9\_wPǖ-C'r-?E+<*zɂL(d>UkT j !9cQQ,'ztX1d$YlV^ RYٞC["H*.yUPl N2W^eN ~FgeZH.飲&j/rʭ72ٳ2Cjkso0zW[_P]7$%sƮ76/=}&48 WL;SǮV_ 95cךwCICgUJjqɜkFE,z5F"4vU!p.s$A,ŵ(FZU9N-_+dQtU&VՌV5Z Y]U3Q*Zw+rN;Q+̼W`ӯ:Vrؤzܑ_Gފl.h/6wmrЦ\]s{y+[5~ȩpU;8BU|'2V\V]5omVknmݻlЦ]s ںk&r6]^576;:У#=z:w{}&>}x2Եs;~lIENDB`Ddx  C Aj..\..\Folders_Folder\seki-paper\images\equal-4-6.pngbb,KF߻bdvnZb,KF߻PNG  IHDRFe,tEXtCreation TimeTue 28 Jun 2005 15:10:06 -0000,StIME b} pHYs  ~gAMA aIDATx\A6 &7(Eok,9BΓ%Z,EER , LS%, !MQMw_^v/!1!.iZ^ڋa,HD/fi8>t /樈5^RS4$".Rm{%?Y:]V(-8 BGfsx3>7<ډ-]^qa]R> dunɢ7m$5М2MS*+xnSmjb3Z~6Qc8(zF%P QȘhT\Qh %1޻e D8 o~+kdےq)/aQщwpQ,{ MdpdQG:xv%E]GU%zuvF_p7m3n Fv(rqE> ڶbW7/f_QYDV̴%Zژu/KdL5'62Y-m f#,%n/ =b e;]UuzPaDFF"դK VCPyqe5Šj6FtVba NA5&gwAqIs#n qLCѶMY߭ˋv?!F&n ~i'ހOPcZ=PpnXnM 1ȚNt997C]Tw&؞S+2qF`šHϖe]Ҕ+E5(|WFbuWNȊ}:xG+N)VME̢Lw8n̙Oah\tr8zEd QO{}_E ~YV؆Y}5I(6ZafAk`s0&-QnFM)q%)0(jAw}UbιHBTOufue˜8>vct ΨgFXYJ|^gqvy*fZ{k>i\>k k XsW NkBYZkv'51\Sfg;muvh䫧[n+Vl>ayKDܶAv_;Vksw{{q|Ehi2A`mr8kp,L͹3_:f;ȹ-g\ DJ3y gv ,tEXtCreation TimeMon 20 Jun 2005 20:45:36 -0000KitIME6iϲ pHYs  ~gAMA aIDATx\10 udab=8N CTKvQd[,;(Rdٲ_dgw?~J%[4M˕v\/?ibFt-ac NнEj`ʼn\l7A a ir_r'd.;aYiBY%9{:`kMNcmE'3e#젶ydr"Z>NG3F3>)6݂6C&1|9 Kח7UVYtCjw0HW}PTRF%̵#^$8XČ/2<|: q"Bc <Kk ۍɍàc&W[s]5Z۳bAXfZSnlU62mi)hM[q\ʴȩfoΔIJv L[47m\xn*$`=OsIٌ]!Tb.WIv'x8S-Xڌӌ ا;ڋ_Biaá79֑hAy5/V 3xĹi{%[l\QnAL{Qa-+&WF)ʏE:\\jS-a[QSkld>(4 nNF0qӂQ3,W񞅄'-4y ±8ƳaszLg0nij>qE< /gY,9pƢ%yT 1=Ks4ϒ0{I4l,mF ƛ>s;/Cۣ&cd(> F`Xڼڊ8n6.{_Dt6Ƿ;ԟei+z=ĨȎoE6Y[G.GKvi$+s9mr̪YpȈmdY`TdGA32-W7c.eѓ;W =V7=EB5b"uSܘ;MGY0jLc0W6c.9_- U7\٥+|R\L;z=y@orsIENDB`NDd  C |Ad..\..\Folders_Folder\seki-paper\images\x2-7-9.pngbZ;׮%Uږ.tl6]vn.;׮%Uږ.tlPNG  IHDRG ,tEXtCreation TimeMon 20 Jun 2005 17:27:08 -0000x*tIME#* pHYs  ~gAMA atIDATx\5 N&5M9ț@yҤ@)R-KLv,Y4~|Ghx79/Iz9| <t:e#^'})iZk/ť? qmzJ7"zWniשZ"o =ƧK% gn;Eq8 z^zS=BuoU:/˭|r: om$6 (6mEMPOh.s!5Ct(R8k!1*l%;fA?MD2y7v M<*]=)hf;Y@-#aOcn[K${WCH-/=>}VHoVƞPgZINm=_k,j"hḒBD1Duj3deQճ1uat5:5=E8-3aȂ貁 pݝYS2hށ ؤC!/&j^z;Y\?V e61J7먆_kCژ6 jb.ͻsApZuƴiWsi6SnE8@U}PPDB,2̑*}~|_ef3gСO󗯤02}Wvf 6Owe@.`xWt$Ք$CƴiWsitOc4쫉hIZips0 m$ " $&F5@%("i۩Yh)4IXr`>1MI+CM4C+m|˕0$-S^_:~K*iCo h4k%ap$+A V1zv q~GW.]yi̊*7T^+i{Р'5~jzw-<3Yʸխ31o;&1LJvnBZ.J>;&1LPNG  IHDRQ{5,tEXtCreation TimeMon 20 Jun 2005 20:42:23 -00007F)tIME4*5\ pHYs  ~gAMA aIDATx]=7 LK 4R5 ,r7{d@K$E ET/>Eqa7(CQx=篾uÏp}:DzW4=_ z bSYb{+͙5CJgYOW 6O&(P 68; H?W߿߿i@'9db" Wu~nF0r*Fd49j$:p(ׇq ׅ(ATۣ,g,n*_,GQbJIʶ,瘒+DmY.5 ىrfA[6],?x93*i:>l_P&ulF%%;*'YfR9*ml7n54je-OmKlڟ̨TSf=0Ů UP9|*9u 4*o;sh8((BE]c>< Em7(m,ܹag"(E)=4@Xr*Fcm=Q4lwNocsgTmf0$H̚?kŠʱyee %gepg e@_I޺wz}4 N4D-+Kjԝ-7H<(b:}? -( kQبPw̎\h-+KZ@۶OZ;!OiҾD+fQ`YY½drl=-l=3"3[V0'<ճ(<7詥V#v*%AS d0H$+|F5|7= U%Rq̊Ҩ\O*i|Ʋcn/ʊT#Ked{1דʣ!Ѱ~̑>JۘKb-T̗;ޘ;Fի檕xcِlE亠4|74Mzv)/+v.f7rj>chkCXlHYQokXۦIj6iTsez4;e˻3ʩsfĶUEOfW;e˻3J\ِUdzN5w1˖wg[={N=af{M)V_SIʨaU!JФKed!5%Qb_hV?\(-IHcV5Q V[lt\}jZtnԢǐmZtbYjcHE!=tXMfIENDB`>Dd  C At..\..\Folders_Folder\seki-paper\images\one-or-two-die.pngb:$'}S (vn$'}S PNG  IHDRqpG8,tEXtCreation TimeMon 20 Jun 2005 20:49:56 -0000[ `tIME7:ˏ pHYs  ~gAMA aTIDATx]=0 vޤ܆~g(8p=%` f(zd,˶8b'zY,{;|}"qf<|&3dMo^M移є17ӂMefGNJ?hvop%'$M*(^?Sn}z92[H39~J@OcAd7[W ifAmMvt \|#n'eܐDRcYMMA$J z?Dh|JTȸaYet5ǭ"eP08OjkY@Pg"ށ B// ڔVX3P!rIĜ쬨YvJbU$40rV H3QH*ň*U%N:-!MLz.OaF S0rȐ(PÕ:#ERWhdJs5≚[RW+n%o|3rFķ 4OydlySu+zJ}16[G?]ww=-k?PNx#׵o,׏EVe][ʶh׾۠}+ƽ6 .9c=WOe4F)u;➘-!#q:.1@ DH3 7c 9E`Jvum28Ÿ|%:ZDZGt#aq_5C.5>ibSw)_;F<`'t{)6 sx7#$G0P~ Lٚ"f)zG4kԋxdZLj]grܧ3m"2tSg?әIkٝ3 487ęIq&vEřgbfI6  flƙȄ8:rf*"vowVqELj˯k#(yXB.Hjš<7 n!ѪMKڮqGޜ%6ǟuƼΠSԌ\^۵$͛\Rj\$V]EKoΨodӓ/h'2TΖ.Q}\ RYGWq)੣%tŪc~XآKkS/ucfDu@}QI[GQ:&ZQGkg+{ Q|%1 QQQQQQ3hqfhŏg&QQQW \__(rObT)!6ؔbS~}߆n׿ij@IENDB` Dd=   C Ax..\..\Folders_Folder\seki-paper\images\four-or-fivr-die.pngb JR1XT6# 9_ [/vnW JR1XT6# 9PNG  IHDR#u,tEXtCreation TimeMon 20 Jun 2005 20:47:04 -0000V$tIME89g pHYs  ~gAMA aIDATx]4 h`ٚfgOh` -*,۱H:S$ˊ{wy/_~ϯ?ng06Bs\bip*Xfk%DPR/|?5>`ޛ J1z`"x<xlvm6.L+@4 i7 Me+y7I+%3$-j>fred1EԄ)Roapt=Qh!1;gdyJ F,wj6; ٮV2}z3 5^;gD^@bHQF: ۝SyWA ##RwbeN@ AWd E ϙ"x]!FZXgEM[MpͧAE=uMQLMZ&8'u l˗ IqfmA{:ԮRzR+Fnct𒣻<<7 gP#e!̙>xQX0xe$UwB0^Ԩj~ Er+1Ģ nj:˿9&Q9fpH64i`ы:259v^CEAR*?`)9y;4Ĺyg. kdբNO_QD/ A<饂qAJqjF XKl Tدj̚yi؅:`4{S*OȪ[ ce6ao2^>k.UU*Đ72^FA72>5r4|+Ys$ nR$ H&2Wt^:뙫HL/4IȬ&Mnx ^ޝp$ݪwe>\F;.sk0 ŷ;jy\v>ܠ"!/ocdFh Wl|xeFh &;LmJ&#s jq(ɔ!iqؙ $]o\fz#pEa.3 xYMeǡzmwԟyC: OyA= Ӕ_AN)ܠHi&L9r^~x7_|aT/Wx?Q0?\fBIt$̏Ia5|~V"5c-f>beƚKMc-fv.Z$]0hg3>85gn t4j$]@U$ԟLM&2Idf5I*n2!.Y^I3p3ow\t.~owLa%ݴ)132tӦĴ=e8I 6%d)I`V4 wA,KM؞=zͣk^C.枢kdkQ3ضHr{S"zܼ^]Dw =n^yDyhڤ,S:y jѕ*a5nN6Hz͊D4^Hvyx,=[<%-D.b;уL|z <YQ蜇zyxͧl@DWVD׽גv*^ރ;uމ.zӻ"ڽDMnh߉.zs"ڽ]8 ϒk.6A%-<(sxn uXgz5 f> Ƽ<QDb5p6o%<7j揻;ja[Dx4Aj{ j5#ljq^5ލ j+yX{UR8:R5e'NE{7h'?5ϩ)^;<9530 ݫ8XP ;:EG9T{5xmpm̫ w}{:"ڪWְ%-ۉAJvHtcOxdrtѪԢ{Uyt@s~/:a.CfY8wIENDB` Dd    C A r..\..\Folders_Folder\seki-paper\images\unequal-1-2-4.pngbKZ]ހ@G體9vnKZ]ހ@G體PNG  IHDR==O,tEXtCreation TimeTue 28 Jun 2005 15:04:21 -0000tIME&Lf pHYs  ~gAMA a!IDATx]r:;0i(h2#PѤ!x<iR0 B̤H=տc^$?˹s7W_QAxe/.n7)FQDbe4{te}tg^'0 @D.#(:I]d}cg`-\R(" mϭ'ff1S%BX  r¢"q=ܛP(41QH W8 6E8mhos=/^c+s@&ޜp/"B<\`q{{pY-Q}<G%ĝh7YfgD|!Ph\6{E%gS'Oz":St%k_p =[撋FæSìԾҊes%T1KbqsU&<5M-3s/{;OcN \8\U8[9.y 'ؙl+Jγ(䄣)4g{ْFfiMÚ$^gZc ObҐ {!$i010'=̫uC5ȸb{ID1$D`5;-V0sL/9,i+ǥ.AL-I[np++Ew֐XN=Ё_YqymLQL VAϴ$\%y]Q#hʬiX N-lzikZ`/[4 B4%V^Mٴ|EOTM+fh"1 ִ$'la H]mZqXVLYEƝ Nid1EњV5Ҧu"Du\pSHvКEmVڱ}xwvosťoBVF)*%g=ȦFrZ18sςA9~ Zg|YҎx~t|rz.UY^_̞\Cy1݋YJg|Ey9sֺBqJDre*ZsY=+h.YYmcf#ؖ/A0I;*Ϲdrp\2q<1\窓6se8WVc.ǹje\|8WӦrHMR"t"մ Bc71 OJՋF Br׊ѝfEj$|@%\+%H>bkڰ "rr8H@i9̾Vb60s T%6 ]Q 4aXfAZ7Xf {Eد)Y%F`+y&du@~/OjGЍ4AUr 5`鄠 !qRS!"XEf%;xag!h@D6.;x:!ll̹Ӂk]- E[.^|f ^|w1cѹs+u#7Sv&닅dv+1i=@  Tĥr x I(@\Y|u\6L&W@½-  BB|"W% #!aeK! O\ yVwԗuS O\ D#0 sk= _G r$䦯)^%MA֡%$pE[g%EK5AE >%*|JbP`^QU8\JH2b\p(rBzER1K%$"u!$kS"Fjq3"Nq\hΣH%qu(ɲ鸂,pt\AMKMő+}%fqK s2_+}mpV.kAY…0 FnڇQZ%YUF(wKg^y}( еmP…0 FzI _B(B _6t0wԭ$ Кp'^(鱬N0D5EBkGEL 5CfRg:Ay $tTt̥)Ma>d;BJ;m̕چz?QSB G#GF`Jdžz?\DE= ﮈ aWs3 qyt5,hIJo[ӽ0 %9lJHq*OU"H173&zcDo>>~߷_~|_Vؓ0 />y?2 IENDB`Dd   C A l..\..\Folders_Folder\seki-paper\images\xne2-6-7-9.png b_1"W}Jvn_1"W}PNG  IHDRHXZ]s,tEXtCreation TimeMon 20 Jun 2005 17:22:21 -0000I~.tIME)T pHYs  ~gAMA aIDATx\16 jԹLd&Enk,973NiI$3)\Z.OAR AݛWï_}!* En^}".T\)&߂BZLmF]66ERqSP0"Tf6GBXܔ#ypӻ&hMg#-C,N<|L 1ɯME&Ôx'6 +q )nmr8\fC ],a"#OT t3)?G)jg6.=k Dژ E!ݑmPBBqs42z ϟ@ܭP&Im Nj&ϩ7VdKW ёY vY$6ff#BB_CWVw/OثH A I}+@Lt剆1?ՅyD$20CpZ;wܸP /PZ "!bIc:5׶4p54f8Q0:@ I3$Ns4n[ƝX{؛ ;$]*RC,:j}a4_ž? \g6"2 VnǬ"CEH4Qi;g$-+4ՈPĨ^t> $ق ;9q\mwz0BHhK@mO/n>a"*ۋlA҇ah!m[loZm8ZSpφj Ov qf hwg~l:o;ڜQ3fiӫkh=–nYň- Ϸϙvm(WX;⑯Օf)}E+2-{32y/f6GNi++uPD<*˷*#cz-hWꞩ8WjS0ૹ0sjFiь0>Ϧݣ^8g1Àhlia"zhe#?xSqcmsnў =8"z@"Z=cAzFm8"zhUMoV{sfo`*9GTy&h ϦG--Sڑ@`xڑ@*!jTy dd 4HF 0h@o??}z&ah֋_*G9aHIENDB` Dd   C ~Af..\..\Folders_Folder\seki-paper\images\x2bvs3w.pngb ӍHnk7SRvnӍHnk7SPNG  IHDRgAb,tEXtCreation TimeMon 20 Jun 2005 17:22:57 -0000 ktIME)ET pHYs  ~gAMA a:IDATx]4 :}R1C z^0ad.uYVlKD؉"ےYVd>⫇G ~p~:uPA>}s]^N]^gO*ыȗB39[ Y^ܗ‰ݖ֠O|2^]d|8\|e7>!9tsOpov.zg_BCW.%Cc&/ky;zPv)ddxe]]SWWM*xڲVf}̀St elk '"Ju#$ft_&].X-o “lJqÂ"xˣd82LAW^ $S]bH0Q$yM/ ߬xY!W՟` yI]Q$vpquD !娻!ĵqa##EJ+lD/$NpK*sI]㑚vƨD㩜AK`S+ OJJC%:ǡ[+iAŽ+@SN`@veu}zLvj8|)(+$߾,HHV| o |ɿO 1=YW7}x‘/q6ipaYW G)(R@^|*pll9S -HR 7_ nx׃< o5zFAx4:f! n^CtTB"Dl% *g1^#AEPȏ *ʂ4 L`l'hgֈ}C*]=r7A;#mXPivvF McpC?I+4ܡt5if6< 4 ,K*Iӕv>e条"A! M2rv ˤgĩ"d, G\S! {O(+T M2ix؃wp_?|^<_oNjܓsud79Sls&,rΙ8dPs ZKaV5xɩ=hy [˵G=5?`r||TSg\7跻D2Q\:uuSrjp\Ge:uurj\:uV}^cf_qgV'r3ZH+g9L<mchV()挫}Dp'#@g.?Eԩg ˩ \Sgr9PNm*(1'=ާn]hu*O=^SY,6}\pD[ƝۣNMY$bQ:'{9fSv$9˩ |4S6G}嘂p6ofQ~ mf:A+rj$Sc%Z"guj2ӳ]S]թ,ȇO9U:"g9"OΨ,rqhT7h7ϥ1ErjC4E{f.ELP NIENDB` i8@8 NormalCJ_HaJmH sH tH 2@2 Heading 1$@&5\00 Heading 2$@&CJ6@6 Heading 3$@& 5CJ\<< Heading 4$$@&a$ 5CJ\0@0 Heading 5$@&CJ0@0 Heading 6$@&CJ8@@@ Heading 7$$@&a$ B*CJ0ph<@< Heading 8$$@&a$ 5CJ\< @< Heading 9 $$@&a$ 5CJ\<A@< Default Paragraph Font.U@. Hyperlink >*B*phJ^@J Normal (Web)dd[$\$OJPJQJ^J>V@> FollowedHyperlink >*B* ph,@", Header  !, @2, Footer  !,>@B, Title$a$5CJ @C@R@ Body Text Indent h^hCJ$$ <;<no|no~n o ! " 6 dev(&tL,MFGHZ[PQ|  YZ"#3wY !!'!(!!!!!!!P"Q"r"g$h$j$k$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$00000000i000000i0i00i 0i x0i 0 0 (0i 0 0 00_0_0_0_0_0_0_0_ 0_0_0_0_0_0_ 0_000000000000j00j00j0j0j00j0j0j0j0j0j00000j00000@0j0000x0j00 0 0 0@0@0@0@0@0@0@0@0@0@0@0@0 0J00000000000000000000000rZ000 F$( P&(((|;<$XX@  @ 0(  B S  ?V>(  Z  S  N  3  N  3  u!p5u8u _Hlt107049803 _Hlt107049804$@@$hkA L   -/jlIMjl#%$$$$fiuz M O FH(,IKMN-.NOwx#%\ ] 9";"##$$$$$$333333333333333333333333333333333Harry FearnleySC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of abstract_nakade.asdHarry FearnleySC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of abstract_nakade.asdHarry Fearnley&C:\WINDOWS\Desktop\abstract_nakade.docHarry FearnleySC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of abstract_nakade.asdHarry Fearnley&C:\WINDOWS\Desktop\abstract_nakade.docHarry FearnleySC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of abstract_nakade.asdHarry FearnleySC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of abstract_nakade.asdHarry Fearnley&C:\WINDOWS\Desktop\abstract_nakade.docHarry FearnleySC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of abstract_nakade.asd Jennet BattenKC:\WINDOWS\Desktop\Harryf\gobangoseki\nakade\wm_as_sent_abstract_nakade.docIV#pH(= V#pk i@V#p="p\ (t`),y-v.~I5X):<&sI/LQJ(wX$9xZZM "I&]87@d0F j 6p-.Uh^`)^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.h ^`OJQJo(^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.^`o()^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.h ^`OJQJo(^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.^`OJPJQJ^Jo( TT^T`OJQJo(o $ $ ^$ `OJQJo(   ^ `OJQJo( ^`OJQJo(o ^`OJQJo( dd^d`OJQJo( 44^4`OJQJo(o ^`OJQJo(^`OJPJQJ^Jo( TT^T`OJQJo(o $ $ ^$ `OJQJo(   ^ `OJQJo( ^`OJQJo(o ^`OJQJo( dd^d`OJQJo( 44^4`OJQJo(o ^`OJQJo(^`o()^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.^`o()^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.^`o(()^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.^`o()^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.^`OJPJQJ^Jo( ^`OJQJo(o pp^p`OJQJo( @ @ ^@ `OJQJo( ^`OJQJo(o ^`OJQJo( ^`OJQJo( ^`OJQJo(o PP^P`OJQJo(h hh^h`OJQJo(h 88^8`OJQJo(oh ^`OJQJo(h   ^ `OJQJo(h   ^ `OJQJo(oh xx^x`OJQJo(h HH^H`OJQJo(h ^`OJQJo(oh ^`OJQJo(^`o()^`o()pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.h ^`OJQJo(h ^`OJQJo(oh pp^p`OJQJo(h @ @ ^@ `OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh PP^P`OJQJo(h^`.h^`.hpLp^p`L.h@ @ ^@ `.h^`.hL^`L.h^`.h^`.hPLP^P`L.^`o()^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.^`OJPJQJ^Jo( ^`OJQJo(o ^`OJQJo(   ^ `OJQJo( \ \ ^\ `OJQJo(o ,,^,`OJQJo( ^`OJQJo( ^`OJQJo(o ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh pp^p`OJQJo(h @ @ ^@ `OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh PP^P`OJQJo(9xZZF j (=&sI/LQ6pX):v.(wX7@dkI5`),"I&]IH(= i@                                    P.        P.                          %                                    B                                   P.                 @J$@UnknownG:Times New Roman5Symbol3& :Arial;WingdingsI& : ?Arial Unicode MS?5 :Courier New"qhCȘݖ&DJ&@i20d% 2Q/Conditional life in Go  a preliminary analysisPreinshalled Ufer Jennet Batten Oh+'0 , DP l x  0Conditional life in Go a preliminary analysisondPreinshalled Ufern reirei Normal.dotdJennet Battenfe68nMicrosoft Word 9.0 @#%[@e{@S@$J& ՜.+,D՜.+,\ hp|  \@% 0Conditional life in Go a preliminary analysis TitleH 8@ _PID_HLINKSANje(http://senseis.xmp.net/?edit=ChiJoongSu""mailto:harry@ZZZgoban.demon.co.uk|/O2..\..\Folders_Folder\seki-paper\images\x2-1-3.pngl~X5..\..\Folders_Folder\seki-paper\images\equal-4-6.pnglD=..\..\Folders_Folder\seki-paper\images\possible_unstable.pngp/2..\..\Folders_Folder\seki-paper\images\x2-7-9.png'z:..\..\Folders_Folder\seki-paper\images\largest_centre.pngg,d!:..\..\Folders_Folder\seki-paper\images\one-or-two-die.png O!<..\..\Folders_Folder\seki-paper\images\four-or-fivr-die.png/>% 9..\..\Folders_Folder\seki-paper\images\unequal-1-2-4.png!g$% 6..\..\Folders_Folder\seki-paper\images\xne2-3-4-6.png(g% 6..\..\Folders_Folder\seki-paper\images\xne2-6-7-9.png h(3..\..\Folders_Folder\seki-paper\images\x2bvs3w.png  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstvwxyz{|~Root Entry F w#Data [1TableMNWordDocument"<SummaryInformation(uDocumentSummaryInformation8}CompObjjObjectPool w# w#  FMicrosoft Word Document MSWordDocWord.Document.89q