(;GM[1]FF[4]CA[UTF-8]AP[CGoban:2]ST[2]
RU[Japanese]SZ[19]KM[0.00]
PW[White]PB[Black]AW[hr]AB[gq][hq][iq][gr][ir]C[In this file we will discover a formula for how to know if a line of stones on the second line (by white in this case) is alive or dead. The rules apply to lines that are unbroken, or in which the breaks must be approached by the oppnent before they can be pushed into. This discussion also assumes that the surrounding stones (black in this case) cannot be cut, captured or placed in atari. It should be no supprise that a single stone on the second line is not enough to live.]
;B[hs]CR[hs]
;AE[hs]AW[hr][ir]AB[jq][jr]C[two stones is also fairly obvious]
;W[js]CR[js]C[the best white could try for is to escape by crossing under]
;B[is]CR[is]C[but black has an all too familiar tetsuji]
;W[hs]CR[hs]
;B[gs]CR[gs]
;W[is]CR[is]
;B[ks]CR[ks]
;AE[gs][ks]AW[hr][ir][jr]AB[kq][kr]C[So can 3 stones escape?]
;W[ks]CR[ks]
;B[ls]CR[ls]C[although the tetsuji is broken, black can still box white in]
;W[js]CR[js]
;B[hs]CR[hs]C[now black has another tetsuji.]
;W[lr]CR[lr]
;B[is]CR[is]
;AE[hs][is][ls]AW[hr][ir][jr][kr]AB[lq][lr]C[4 stones is the first case that begins to look hopeful]
(;W[gs]CR[gs]
;B[fr]CR[fr]C[simple responses by black however easily keep white surrounded.]
;W[fs]CR[fs]
;B[es]CR[es]
;W[hs]CR[hs]
;B[er]CR[er]
;W[ls]CR[ls]
;B[mr]CR[mr]
;W[ms]CR[ms]
;B[ns]CR[ns]
;W[ks]CR[ks]
;B[nr]CR[nr]
;W[pq]CR[pq]C[Now hite has no eye space and will die.]
;B[js]CR[js]
;W[is]CR[is]
;B[js]CR[js]
;AE[pq][er][fr][nr][es][js][ns]AW[hr][ir][jr][kr][lr]AB[mq]TR[hs][js][ls]C[with five stones one can see that if white has stones at the triangles white will be alive.]
;W[hs]CR[hs]
;B[gs]CR[gs]
;W[ls]CR[ls]
(;B[js]CR[js]C[But black knows what white needs and plays to prevent white from getting it.]
;W[ms]CR[ms]
;B[nr]CR[nr]
;W[ns]CR[ns]
(;B[pr]CR[pr]
;W[ps]CR[ps]
(;B[os]CR[os]C[this doesn't stop white.]
;W[or]CR[or]
;B[oq]CR[oq]
;W[os]CR[os]
;B[qr]CR[qr]
;W[qs]CR[qs]
;B[rr]CR[rr]
;W[rs]CR[rs]
;B[sr]CR[sr]
;W[ss]CR[ss]
;B[is]CR[is]
;W[ks]CR[ks]
;B[js]CR[js]
;W[is]CR[is]
;B[js]CR[js]
;AE[oq][pr][qr][rr][sr][gs][js]AW[hr][ir][jr][kr][lr][mr]AB[nq]TR[hs][ms]SQ[js][ks]C[So what if we have an extra space does that give us the move we need? we need the triangles and either one of the squares.]
;W[hs]CR[hs]
;B[ms]TR[js][ls]CR[ms]C[note that black gets more agressive here. Now white needs to get boht triangles to live.]
;W[ls]CR[ls]
;B[js]CR[js]
;W[ns]CR[ns]
;B[or]CR[or]
;W[gs]CR[gs]
;B[fr]CR[fr]
;W[fs]CR[fs]
(;B[er]CR[er]
;W[es]CR[es]
;B[dr]CR[dr]C[white obviously won't escape that way.]
;W[os]CR[os]
;B[ps]CR[ps]
;W[pr]CR[pr]
;B[ms]CR[ms]
;W[ns]CR[ns]
;B[os]CR[os]C[Does Black need to fight a ko?]
(;AE[dr][er][or][pr][es][fs][gs][hs][js][ls][ns][os][ps]AW[gr]AB[fq]C[so now we have seven stones! Over 1/3 of the board wide! This group can live, but it may also die. Sente (the initiative) makes the difference. Even with 7 stones, life is not guaranteed!]
(;W[ms]CR[ms]C[if white plays first, white lives]
;B[gs]CR[gs]
;W[hs]CR[hs]
;B[ks]CR[ks]
;W[js]CR[js]C[it takes 7 stones to live in GOTE. if there are less than 7 stones white will die unless white can connect along the 1st line.])
(;W[pd]CR[pd]C[But what white really would like is not to have to live in GOTE, but rather be alive and take sente.... 7 stones is not enough however.]
;B[ms]CR[ms]
;W[ls]CR[ls]
;B[gs]CR[gs]
;W[hs]CR[hs]C[can you guess where black plays now?]
;B[js]CR[js]
;W[fs]CR[fs]
;B[er]CR[er]
;W[ns]CR[ns]
;B[or]CR[or]
;W[es]CR[es]
;B[ds]CR[ds]
;W[dr]CR[dr]
;B[gs]CR[gs]
;W[fs]CR[fs]
;B[es]CR[es]
;W[os]CR[os]
;B[ps]CR[ps]
;W[pr]CR[pr]
;B[ms]CR[ms]
;W[ns]CR[ns]
;B[os]CR[os]C[Again black does not have to fight ko. white can never win the white group is dead if white does not play first. Black can wait until the game is over, and all ko threats have been resolved, then take the group. To avoid the tedium of playing out ko's that only one side can win, groups like this may simply be removed at the end of the game without fighting the ko.]
(;AE[pd][dr][or][pr][ds][es][fs][hs][js][ls][ns][os][ps]AW[cc][qc][ed][jd][od][ge][me][jf][fr]AB[eq]TR[hs][is][js][ks]SQ[ca][da][ea][fa][ga][ha][ia][ja][ka][la][ma][na][oa][pa][qa][cb][db][eb][fb][gb][hb][ib][jb][kb][lb][mb][nb][ob][pb][qb][ec][fc][gc][hc][ic][jc][kc][lc][mc][nc][oc][gd][hd][id][kd][ld][md]CR[fm][gm][hm][im][jm][km][lm][mm][dn][en][fn][gn][hn][in][jn][kn][ln][mn][nn][on][bo][co][do][eo][fo][go][ho][io][jo][ko][lo][mo][no][oo][po][qo][ap][bp][cp][dp][ep][fp][gp][hp][ip][jp][kp][lp][mp][np][op][pp][qp][rp][aq][bq][cq][dq][oq][pq][qq][rq][sq][ar][br][cr][dr][or][pr][qr][rr][sr][as][bs][cs][ds][os][ps][qs][rs][ss]C[It takes 8 stones to be 100% alive! That is a LOT of stones. Flat groups are very inefficient. This is one of the best ways to see why making good shapes is important. 8 stones in a line is a very slow, tedious shape. Which looks better, the 8 stones at the top, or the 8 white stones at the botom? The triangles at the bottom represent  probable white territory of 4 points. The squares at the top represent probable white territory of 47. The circles represent a conservative estimate of the influence of the black stones. Influence isn't territory, but every time white thinks about playing on one of the circles the black stones pose a very real danger. White must keep a distance and have supporting stones before moving into this area. black influences 80 intersections on the board, and because of this the black group would find it very easy to live, and to punish nearby white groups. 

To this point our discussion has been very narrow.  Frequently black doesn't have a complete wall that wraps all the way around the end. obviously if white can escape through the wall everything changes, but now we will look at what happens if one or both ends are not capped.]
;W[pj]CR[pj]
;B[ms]CR[ms]
(;W[ls]CR[ls]
;B[fs]CR[fs]
;W[gs]CR[gs]
;B[is]CR[is]
;W[js]CR[js])
(;W[fs]CR[fs]
;B[ls]CR[ls]
;W[ks]CR[ks]
;B[is]CR[is]
;W[hs]CR[hs]))
(;AE[pd][dr][er][nr][or][pr][ds][es][fs][hs][js][ls][ns][os][ps]C[But if black has only capped one end... ]
;W[pj]CR[pj]
(;B[nr]CR[nr]C[obviously this doesn't work for black it is white's turn with 7 stones and we already have seen that.])
(;B[ns]CR[ns]
;W[ms]CR[ms]C[much the same.])
(;B[ms]CR[ms]
;W[nr]CR[nr]
;B[or]CR[or]
;W[os]CR[os]
;B[pr]CR[pr]
;W[ns]CR[ns]C[still worse]
;AE[pj][nq][fr][mr][nr][or][pr][ms][ns][os]C[now 6 stones and 2 uncapped ends.]
;B[fr]CR[fr]
;W[gs]CR[gs]
;B[mr]CR[mr]
;W[ls]CR[ls]
;B[js]CR[js]
;W[is]CR[is]C[So the rule of thumb, is that it takes 8 to be alive, but you can live with one less for each uncapped end, and sente is also worth a stone. so 5 and sente with no capped ends is the least that can be alive. 

The magic Formula:
if    (Stones + uncapped ends + 1 if you have sente) >= 8    the group is allive.]
;AE[mq][fr][lr][mr][gs][is][js][ls]
;B[dj]CR[dj]C[five with sente....]
;W[lr]CR[lr]
;B[mr]CR[mr]
(;W[ms]CR[ms])
(;W[ls]CR[ls]
;B[fr]CR[fr]
;W[gs]CR[gs]
;B[js]CR[js]
;W[is]CR[is])))))
(;W[ms]CR[ms]C[No! This ko can never be won by white because if white fills this ko white dies anyway.]))
(;B[dr]CR[dr]
;W[ds]CR[ds]
;B[cs]CR[cs]
;W[es]CR[es]
;B[cr]CR[cr]
;W[er]CR[er]
;B[eq]CR[eq]))
(;B[qs]CR[qs]C[even better for black...]
(;W[or]CR[or]
;B[oq]CR[oq]
;W[os]CR[os]
;B[qq]CR[qq])
(;W[os]CR[os]
;B[qr]CR[qr]
;W[or]CR[or]
;B[oq]CR[oq])))
(;B[os]CR[os]C[black cannot play this]
;W[or]CR[or]
;B[ps]CR[ps]
;W[pr]CR[pr]C[white would live this way.]))
(;B[ms]CR[ms]C[this is lax play by black, allowing white to live.]
;W[js]CR[js]))
(;W[hs]CR[hs]C[This will not due for white we do not have enough room to make 2 eyes...]
;B[gs]CR[gs]
;W[ks]CR[ks]
;B[ls]CR[ls]))
