(;GM[1]FF[4]CA[UTF-8]AP[CGoban:3]ST[2] RU[Japanese]SZ[19]KM[0.00] PW[White]PB[Black]LB[aa:2][ab:2H][ac:3][ad:3H][af:3C][ag:4C][ai:ML][aj:MLH][ak:MLO][al:MLI][an:LTO][ao:LTI]C[This is a collection of capturing races intended to explore the effective liberty status of a variety of different corner groups. Not all correct moves are shown in all problems. There are often multiple ways to achieve the result (each with slightly different endgame and ko-threat characteristics). Sometimes more than one way is shown, but often only one is shown. This collection is obviously not comprehensive. Slight variations on the situation, including possible escape threats, descents, clamps, first-line hanes not explored in this set, etc. may change the result depending on the surroundings and the defects of the enclosing groups. Also, no guarantees that there aren't tactical mistakes here! :) originally by lightvector, some comments added by Daniel Hu] (;B[aa]LB[ca:A1]C[Two stones. Notes: Nothing unusual. Usually: 3 liberties.] ;W[ca]C[A1] ;AE[aa][ca]AW[db][eb][fb][bc][cc][bd][ce][bf]AB[bb][cb][gb][dc][ec][fc][gc][jc][cd][dd]C[Status?] ;B[]C[Black can kill. White can kill.] (;B[ab]C[DH: Playing solidly at the boundary is the most profitable way to kill. Since it makes a one eye kills no eye, this is better by 4 points than D19/E19/F19 (which are all the same here).]) (;W[ab]C[DH: Only move to kill unconditionally because the other liberties are B19 and C19. B19 is not connected enough so playing C19 is ko to capture it. C19 is very bad when met with B19 after which E19, a one approach move (in B's favour) ko is the only chance for white to survive. This is because W needs an approach move at A17 to approach from the left whereas connecting C19 back with D19 fills in a boundary liberty, which costs a liberty.])) (;B[ab]LB[ca:A1][da:A2][cb:B1]C[Two stones with a hane. Notes: The hane almost adds a liberty, "almost" being that it's in the case where it matters, it gives favorable 2-stage ko rather than a full kill. Usually: 3 liberties + favorable stage or direct ko resistance ] (;W[ca]C[A1] ;AE[ca][ab]AW[db][eb][fb][bc][cc][bd][ce][de][cg][ci]AB[bb][cb][gb][ac][dc][ec][fc][gc][jc][dd][fe]C[Status? ] ;B[]C[Black can kill. White can make an unfavorable two-stage ko or direct ko, black's choice.] (;B[ab]C[DH: of course. Solid connection is generally good for liberties and making an eye gains points. In this case connecting at 2-1 is particularly important because if white plays there forcing black to capture, the capture move is at 1-1, which loses the group one liberty (at the 1-1 point, since the 1-1 point's only neighbours are liberties already counted for the group, with no new liberties). However, if B intended to attack W on the left and W getting A16 in sente was important for eyespace, then a move at F19 may be better. (It helps if B is komaster)]) (;W[ba]C[DH: Tesuji. White must short the liberties of the B18 group since the liberty count is 3 for 3. So A16 is too slow. This is the key shape point since white can't easily approach from the D19 group since playing mutual space/liberties loses you liberties. Hence W needs to approach from the left with A16 and A18. Even after that, W may need to fill in A17 if B still has one or more mutual liberties remaining. This move prevents black playing B19 themselves to set up such a situation. For example if white played A18, then B19 would unconditionally win the race. In addition it threatens to play C19 to unconditionally win, or to play a ko for A18 to continue shorting B's liberties. ] (;B[ca]C[DH: After this exchange, the B18 group has only 2 liberties, so W can approach from the outside.] ;W[ad]C[DH: tricky position] (;B[aa] ;W[ab]C[Two stage ko, white takes first. DH: If W is komaster, W may want to play F19 to win the ko in order to settle that {1|-1} endgame to zero but watch out for under the stones! The two stages are that there are two different kos to take here (A17-A18) and (A19-B19). After white takes first, this is essentially an even situation. Neither side is willing to invest a move on the mutual liberties (D19 or F19) since the ko is more valuable. But this ko takes a long time to finish so moves here are less valuable than it looks. If B spends 2 more moves locally (A17, F19), B can win the semeai, but still lose the ko.]) (;B[fa]C[DH: This move seems silly at first since white takes first. Normally that would mean this move is dominated by A19. However, it is a common ko technique to delay playing the ko in order to exploit the ko rule. This means that after white A18, black takes A19 first and white can't take back. So white needs to find a ko threat first. In addition, black aims to connect at A18 to win the ko rather than retreating with D18.] ;W[ab] ;B[aa]C[Black can also make direct ko, taking first.])) (;B[da]C[DH: mentioned by Bill Spight. This risks the A18 direct ko, which is probably a slightly gain for W since it threatens to take B cleanly off the board. Black could have played for a non-direct ko where black had the advantage.])) (;W[ca]C[Probably a mistake.] ;B[ba] ;W[ea] ;B[da]C[This is an unfavorable two-step ko for white, which is worse than an unfavorable two-stage ko. DH: Actually quite a bit worse. Modern terminology is to call this an approach ko, i.e. a two approach move ko, since "step" is ambiguously conflated with "stage" in different texts. This means that white needs to play two moves locally (with black passing) for this to become a direct ko (where both sides can settle the position with one move. In this case that means both sides are in atari after the opponent takes the ko). The two moves are C19 and A18. Black also must win the ko in order to survive since the A19 eye isn't helpful when white is directly approaching from the upper side. They say that a three approach move ko is essentially unconditionally settled. In order to win such a ko when it is not in your favour, you need to be able to play a ko sequence of 4 moves in a row while the opponent plays elsewhere on the board. This requires your big ko threats to be plentiful, generally needing a weak opponent dragon or two or multiple large ko threats that the opponent doesn't have time to add a move to defend. After all, the opponent can spend 3 moves solidly connect ko threats in their position while you play the approach moves. Your ko threats need to be as big as you are aiming to profit by. However, your opponent only needs to play ko threats bigger than around 1/5 the size of the ko. (3 approach moves + 2 moves to end the ko, one for each side)])) (;W[da]C[A2] ;AE[da][ab]AW[db][eb][fb][bc][cc][bd][ce][de][cg][ci]AB[bb][cb][gb][ac][dc][fc][gc][jc][dd][fe]C[Status? ] ;B[]C[Black can make a favorable two-stage ko or direct ko. White can kill.] (;B[ec]C[DH: This shape analysed above.] ;W[ba] ;B[ca] ;W[ad] (;B[aa] ;W[ab]C[Two-stage ko, white takes first.]) (;B[fa] ;W[ab] ;B[aa]C[Direct ko, black takes first.])) (;W[da]C[DH: A nice move that just wins unconditionally. A19 and C19 are miai. So B can't make an eye without losing a liberty. Playing at A16 instead would fall to A19 for a one approach move ko in white's favour. White also has one locally ko threat at D19 (for F19). Playing C19 would fall to D19 then E19 and B19 for a one approach move ko in white's favour. Why is this the vital shape? Again, solid shape is key since if B plays here, then W can't approach except with E19 which loses some liberties (more than zero and less than two but not necessarily always one). After this move, although white seems to have gone from 4 liberties to 4 liberties, the space at E19, F19, G19 means black needs an approach move to short white's liberties in that direction. Combining this with the space for mutual liberties at C19, this move is worth a nonzero amount of liberties, since B finds it difficult to approach from either side. After D19, the position is a definite 4 liberties to 3, a win for white. Previously, it was like a soft golden chicken standing one leg situation where W found it difficult to approach from either side. If black plays C19 then after A19, B goes down from 3 liberties to 2 liberties. Watch out for under the stones in that variation. Black's best move locally seems to be (optionally C16 then) F19. ])) (;W[cb]C[B1] ;AE[ab]AW[db][eb][fb][gb][bc][cc][bd][ce][de][cg][ci]AB[bb][cb][hb][ac][dc][ec][fc][gc][hc][jc][dd][fe]C[Status? ] ;B[]C[Black can make a favorable two-stage ko or direct ko. White can kill.] (;B[ga]C[DH: This is exactly the same shape as above. The only addition is G18 for G19. This makes zero difference to liberties unless G19 gets captured, because this only makes a difference to the value of the G19 stone. If black plays say F19, E19 and white captures those three stones with D19 and H19, capturing three stones gains an extra liberty over capturing two stones. However in no good variation would G19 get captured. Therefore the good variations are exactly the same as above.] ;W[ba] ;B[ca] ;W[ad] (;B[aa] ;W[ab]C[Two-stage ko, white takes first.]) (;B[fa] ;W[ab] ;B[aa]C[Direct ko, black takes first.])) (;W[da]C[The same as above.]))) (;B[ac]LB[ca:A1]C[Three stones. Notes: Nothing unusual. Usually: 4 liberties.] ;W[ca]C[A1] ;AE[ca][ac]AW[eb][fb][gb][hb][bc][cc][dc][ce][cg]AB[bb][cb][db][ib][ec][fc][gc][hc][ic][kc]C[Status? ] ;B[]C[Black can kill. White can kill.] (;B[ab]C[DH: solid 2-1 extension at boundary with an extra liberty (A17) at the boundary. All good things (mostly).]) (;W[ab]C[DH: B can't make an eye so this is clearly winning 4 liberties to 3. White can approach liberties from either left or right. Black playing C19 is not a tesuji. Even if black had a small eye, it probably wouldn't make a difference to liberty count. ])) (;B[ad]LB[ca:A1][da:A2][cb:B1]C[Three stones with a hane. Notes: Usually: 4 liberties + direct ko resistance ] (;W[ca]C[A1] ;AE[ca][ad]AW[eb][fb][gb][hb][bc][cc][dc][bd][ce][cf][df][ci]AB[bb][cb][db][ib][ac][ec][fc][gc][hc][ic][kc][dd][de][fe]C[Status? ] ;B[]C[Black can kill. White can make direct ko where black takes first in the corner.] (;B[ab]C[DH: same as above.]) (;W[ea] (;B[da]C[DH: Good, but instinctively, this shorts B's liberties. This can't be called the best move, though in most positions it would be equivalent. Also, in some extreme positions, white might have J19 as a move.] ;W[ba] ;B[ha] ;W[ab] ;B[aa]) (;B[ha]C[ DH: This move seems natural.] ;W[ba]C[Still the only move. However, W now threatens C19 after which A19 is required to play a ko, so black still has to fight the ko by playing D19 or C19.] (;B[da]C[This reverts to the main line.]) (;B[ca]C[DH: This position is essentially the same as a variation above except better for white since although B has an option of taking the ko first, white has a sort of golden chicken shape where black will need to play J19 as an approach move. This means white at the very least has the option of making it a ko where white takes first. I think it is normally somewhat worse for B than the mainline. But if B is ko master, it may be marginally better for B since B gets either the J19 connection or the A18 connection (for one eye kills no eye)] (;W[ad] ;B[ga] ;W[ab] ;B[aa] ;W[ac] ;B[ia]) (;W[ga] ;B[ia] ;W[ad] ;B[aa] ;W[ab])) (;B[ga]C[DH: If Black is truly ko master, they can get away with this move] ;W[da] ;B[aa]C[A direct ko where W takes first. However, when B wins, black can take W off the board. But that still isn't the end because B needs to very much be ko monster to hold onto the A18 until the small endgame to prevent the E18 cut and under the stones!]))) (;W[da]C[Just for reference: sometimes a mistake?] (;B[ea]C[DH: Painful but I think normally best.] (;W[fa] ;B[ba]C[Because black has this additional seki option now.]) (;W[ba]C[ DH: B wins in double ko.])) (;B[ca]C[ This move is playable.] (;W[ab] ;B[aa] ;W[ad] ;B[ea]C[Two stage ko in black's favour.]) (;W[aa] ;B[fa] ;W[ga] ;B[ea] ;W[da] ;B[ea] ;W[ia] ;B[ab]) (;W[fa] ;B[aa] ;W[ad] ;B[ha]C[double ko])))) (;W[da]C[A2] ;AE[da][ad]AW[eb][fb][gb][hb][bc][cc][dc][bd][ce][cf][df][ci]AB[bb][cb][db][ib][ac][ec][fc][hc][ic][kc][dd][de][fe]C[Status? ] ;B[]C[Black can make a direct ko where black takes first in the corner. White can make a favorable 3-step ko.] (;B[gc]C[DH: same shape as above.] ;W[ea] ;B[da] ;W[ba] ;B[ha] ;W[ab] ;B[aa]) (;B[ea]C[Black can also try this move, it's a bit tricky, white has to play very precisely. DH: this seems better to me when white wins the ko, especially as white needs 4 more moves to win the ko and capture B efficiently. If W only wants to end the semeai, they may be forced to end the ko with A16 rather than with C19, which makes several points difference.] (;W[ba]C[Probably the best move for white?] (;B[gc] (;W[ab]C[Probably the best way for white?] ;B[aa]C[Direct ko.]) (;W[fa]C[Mistake.] ;B[ca] (;W[aa] ;B[ha] ;W[da] ;B[ab] ;W[ad] ;B[ea]C[Unfavorable two-step ko for white.]) (;W[ad] ;B[ha] ;W[da] ;B[aa] ;W[ab] ;B[ea]C[Black wins in double ko.]))) (;B[ca]C[Probably a mistake?] ;W[ad] ;B[aa] ;W[ab]C[Not sure how to characterize this ko.])) (;W[fa]C[Mistake.] ;B[ba]C[DH: tesuji, but it would be a fail if W had an extra hane at J19 in which case B must hold on at D19.] ;W[da] ;B[ha] ;W[ab] ;B[gc]C[Black wins unconditionally.])) (;W[ea] ;B[da] ;W[ba] ;B[aa] ;W[ad]C[White can interpose a ko capture before playing here too.] ;B[ca] ;W[ab]C[DH: Three approach move ko in white's favour. These are the bad things that happen when you sacrifice a stone to throw into a ko yourself.])) (;W[cb]C[B1] ;AE[ad]AW[eb][fb][gb][hb][ib][bc][cc][dc][bd][ce][cf][df][ci]AB[bb][cb][db][jb][ac][ec][fc][gc][hc][ic][jc][lc][dd][de][fe]C[Status? ] ;B[]C[Black can make a direct ko where black takes first in the corner. White can make a favorable 3-step ko.] (;B[ia] ;W[ea] ;B[da] ;W[ba] ;B[ha] ;W[ab] ;B[aa]) (;B[ea]C[ DH: I think this is still a good move.] (;W[ba] ;B[ia] ;W[ab]C[This ko seems clearly better for B than the mainline since it is lighter.]) (;W[fa] ;B[ia] (;W[da] ;B[ca] ;W[aa] ;B[ha] ;W[ab]) (;W[ba] ;B[ca]C[B has other options but this is good] ;W[ad] ;B[ha] ;W[ab] ;B[aa] ;W[da] ;B[ac]))) (;W[ea] ;B[da] ;W[ba] ;B[aa] ;W[ad]C[White can interpose a ko capture before playing here too.] ;B[ca] ;W[ab]))) (;B[af]LB[ca:A1][da:A2][ea:A3]C[Three extending to the corner. Notes: Nothing unusual. Usually: 3 liberties inside the corner (plus however many outside).] (;W[ca]C[A1] ;AE[ca][af]AW[db][eb][fb][bc][cc][cd][ce][de][cg]AB[ab][bb][cb][gb][dc][ec][fc][gc][jc][dd]C[Status?] ;B[]C[White is dead.] ;W[ca] ;B[fa]) (;W[da]C[A2] ;AE[da][af]AW[db][eb][fb][bc][cc][cd][ce][de][cg]AB[ab][bb][cb][gb][dc][fc][gc][jc][dd][fe]C[Status?] ;B[]C[Black can kill. White can kill. ] (;B[ec] ;W[ca] ;B[fa]) (;W[ac] ;B[da] ;W[ba])) (;W[ea]C[A3] ;AE[ea][af]AW[db][eb][fb][bc][cc][cd][ce][de][cg]AB[ab][bb][cb][gb][dc][gc][jc][dd][ed][fd]C[Status?] ;B[]C[Black is dead.] ;B[da] ;W[ba] ;B[fa] ;W[ca])) (;B[ag]LB[ca:A1][da:A2][ea:A3][cb:B1]C[Four extending to the corner. Notes: The larger corner enables a threat to live that can be an extra effective liberty, if the corner player goes first. But if the opponent has enough space to make an eye too, it's a seki. Usually: 4 liberties inside the corner if attacker plays first. 5 liberties inside if the corner side plays first, except the last liberty only makes seki if the attacker can also make an eye. ] (;W[ca]C[A1] ;AE[ca][ag]AW[eb][fb][gb][hb][bc][cc][dc][cd][ce][de][cg][ci]AB[ab][bb][cb][db][ib][kb][ec][fc][gc][hc][ic][dd][ed][ee]C[Status?] ;B[]C[White is dead.] ;W[da] ;B[ha]) (;W[da]C[A2] ;AE[da][ag]AW[eb][fb][gb][hb][bc][cc][dc][cd][ce][de][cg][ci]AB[ab][bb][cb][db][ib][kb][ec][gc][hc][ic][dd][ed][ee]C[Status?] ;B[]C[Black can kill. White can kill.] (;B[fc]) (;W[ea] ;B[ha] ;W[ac] ;B[ga] ;W[da])) (;W[ea]C[A3] ;AE[ea][ag]AW[eb][fb][gb][hb][bc][cc][dc][cd][ce][de][cg][ci]AB[ab][bb][cb][db][ib][kb][ec][hc][ic][dd][ed][gd][ee]C[Status?] ;B[]C[Black can kill. (Extra liberty!) White can kill. DH: nice tesuji for B.] (;B[ba] ;W[da]C[DH: W must play here but then has no eyes. This costs 3 liberties at F19, then E19 connect and C19 to attack. Since playing B19 cost B a liberty from 5 to 4 even with the cost of a move, overall, with white one liberty ahead at the start, it changes to black winning by one liberty. ] ;B[ea] ;W[fa] ;B[ha]) (;W[ea])) (;W[cb]C[B1] ;AE[ag]AW[eb][fb][gb][hb][ib][bc][cc][dc][cd][ce][de][cg][ci]AB[ab][bb][cb][db][jb][lb][ec][gc][hc][ic][jc][dd][ed][ee]C[Status?] ;B[]C[Black can make a seki or an unfavorable 2-step ko. White can kill.] (;B[ba] ;W[da] ;B[ea] ;W[fa] ;B[ia] ;W[ha] ;B[ja]) (;B[ea] ;W[fa] ;B[ia] ;W[ac] ;B[fc] ;W[da] ;B[ca] ;W[ba] ;B[ea]) (;W[ea]))) (;B[ai]LB[ca:A1][cb:B1][db:B2][eb:B3]C[Mini L group Notes: The mini L group has 3 liberties but has the power of an eye and descent at C19, which is often an effective fourth liberty. Usually: 3 liberties inside + eye power liberty] (;W[ca]C[A1] ;AE[ca][ai]AW[db][dc][bd][cd][de][cf][ch][cj]AB[cb][fb][bc][cc][fc][dd][ed]C[Status?] ;B[]C[Black can kill. White can kill. (eye power liberty doesn't count here)] (;B[ac]) (;W[ba] ;B[ca] ;W[ac]) (;W[da]C[ DH: This is preferred locally to prevent F19 if there is no corner ko. This depends on if A16 is sente. It doesn't seem to be good here unless W is solid enough around B11 and D13.] ;B[ad] ;W[ba] ;B[bf] ;W[ca] ;B[ae] ;W[ce] ;B[bg] ;W[cg]C[It seems B is still dead here but the aji is pretty bad])) (;W[cb]C[B1] ;AE[ai]AW[db][eb][fb][dc][bd][cd][de][cf][ch][cj]AB[cb][gb][bc][cc][ec][fc][gc][jc][dd][ed]C[Status?] ;B[]C[White is dead. DH: The difference is that W is unable to play on the mutual liberty at D19 without losing a liberty. This is because of the empty triangle with E18.] (;W[ac] ;B[ba]) (;W[ba] ;B[ca] ;W[ac] ;B[ab] ;W[ad] ;B[fa])) (;W[db]C[B2] ;AE[ai]AW[eb][fb][dc][bd][cd][de][cf][ch][cj]AB[cb][gb][bc][cc][ec][gc][jc][dd][ed][ge]C[Status?] ;B[]C[Black can kill. White can kill. ] (;B[ac]) (;W[ba] ;B[ca] ;W[ac])) (;W[eb]C[B3] ;AE[ai]AW[db][fb][gb][dc][bd][cd][de][cf][ch][cj]AB[cb][hb][bc][cc][ec][gc][hc][jc][dd][ed]C[Status?] ;B[]C[Black is dead.] (;B[ba] ;W[ab] ;B[da] ;W[ac] ;B[fc] ;W[bb]) (;B[ad] ;W[ab]C[Kills without ko?]))) (;B[aj]LB[ca:A1][db:B2][eb:B3]C[Mini L group with hane. Notes: The hane is quite useful, worth an extra liberty if the corner player goes first, and also turning many cases where the attacker would kill the corner into ko. Usually: 3 liberties inside + eye power liberty + ko resistance if attacker goes first. 4 liberties inside + eye power liberty if corner side goes first. ] (;W[ca]C[A1] ;AE[ca][aj]AW[db][dc][bd][cd][be][cf][df][ch]AB[cb][fb][bc][cc][fc][ad][dd][de][fe]C[Status?] ;B[]C[Black can kill. White can make a direct ko, black taking first in the corner. (eye power liberty doesn't count here)] (;B[ac]) (;W[ba] ;B[ca] ;W[da] (;B[bb] ;W[ab] ;B[aa]) (;B[ab]C[Also possible depending on the situation.] ;W[ae] ;B[ec] ;W[ac]C[White takes first in the corner, but black has less aji around E16.]))) (;W[db]C[B2] ;AE[aj]AW[eb][fb][gb][dc][bd][cd][be][cf][df][ch]AB[cb][hb][bc][cc][ec][fc][gc][hc][kc][ad][dd][de][fe]C[Status?] ;B[]C[Black can kill. White can make a direct ko, black taking first in the corner.] (;B[ac] ;W[ca]C[ DH: note that the ko aji doesn't work in positions like this because B is alive.] ;B[ba] ;W[ea] ;B[ab]) (;W[ba] ;B[ca] ;W[ae] ;B[ga] ;W[ac] ;B[ab] ;W[da]C[Clearly better than B18.] ;B[ad])) (;W[eb]C[B3] ;AE[aj]AW[db][fb][gb][dc][bd][cd][be][cf][df][ch]AB[cb][hb][bc][cc][ec][fc][hc][kc][ad][dd][hd][de][fe]C[Status? ] ;B[]C[Black can kill. (The A16 hane gains an extra liberty!) White can kill. (without ko?)] (;B[ba] ;W[ab] ;B[ga]C[DH: The B19 move is so magical here because not only does it force A18 (which isn't shorting the liberties of B17 yet), so it doesn't decrease B's liberties (from 3), but it adds one approach move at A15, and yet another at A16 (even after B18 atari, W is unable to play A19 due to self-atari). Hence, as long as B ensures white can't approach from the D19 direction, B has 5 liberties in the corner. When B plays first at the top, black wins.]) (;W[ab]C[Kills without ko? DH: I think so yes.] ;B[da] ;W[ba]C[The unique way for white to kill without ko? Very fancy sequence.]) (;W[ba]C[This way black can make a two-step ko.] ;B[ca] (;W[da] ;B[ab] ;W[ae] ;B[gc] ;W[ac]) (;W[ab]C[Doesn't prevent ko.] ;B[bb] ;W[ae] ;B[aa] ;W[ac] ;B[ad] ;W[ac] ;B[ga] ;W[ad] ;B[fa] ;W[ab] ;B[gc]C[Direct ko.])))) (;B[ak]LB[ca:A1][cb:B1][db:B2][eb:B3][dc:C2]C[Mini L group with outside leg. Notes: Similar to the mini L group. It has 4 liberties, plus the power of an eye and a descent at C19, which is often an effective fifth liberty. It also has some ko resources beyond that. Usually: 4 liberties inside + eye power liberty + occasional ko resource if corner side goes first. ] (;W[ca]C[A1] ;AE[ca][ak]AW[db][dc][cd][dd][be][bf][cf][ch][cj]AB[cb][fb][bc][cc][fc][bd][fd][ce][de][ee]C[Status? ] ;B[]C[Black can kill. White can kill. (eye power liberty doesn't count here) DH: Basically it is a small eye and the D19 move doesn't cost W any liberties (it is not much different to an external liberty)] (;B[ad]C[DH: Black wins the capture race by 2 liberties due to D19 hane connect losing W a liberty.]) (;W[da] ;B[ea] ;W[ad] ;B[ba] ;W[ab]) (;W[ba]C[Doesn't work here due to F18.] ;B[da] ;W[ca] ;B[ea] (;W[ad] ;B[ab] ;W[ae] ;B[ac]C[DH: very nice! false seki.]) (;W[ab] ;B[bb]))) (;W[cb]C[B1] ;AE[ak]AW[db][eb][fb][gb][dc][cd][be][ce][de][cg][ci]AB[cb][hb][bc][cc][ec][fc][gc][hc][jc][bd][dd][ed]C[Status? ] ;B[]C[White is dead.] ;W[ba] ;B[ca] ;W[ad] ;B[ac]) (;W[db]C[B2] ;AE[ak]AW[eb][fb][gb][dc][cd][be][ce][de][cg][ci]AB[cb][hb][bc][cc][ec][gc][hc][jc][bd][dd][ed]C[Status? ] ;B[]C[Black can kill. White can kill.] (;B[fc]) (;B[ab]) (;W[ba] ;B[ca] ;W[ad])) (;W[eb]C[B3] ;AE[ak]AW[db][fb][gb][dc][cd][be][ce][de][cg][ci]AB[cb][hb][bc][cc][ec][hc][jc][bd][dd][ed][fd][gd]C[Status?] ;B[]C[Black can make a direct ko. White can kill. DH: I disagree. Black can only make one move approach ko.] (;B[ba]C[DH: It would be nicer if B had A15, but this is still a tesuji. After all, if W gets this with both 2-1 points, then B has no approach liberties.] ;W[ab] ;B[da]C[DH: lightvector claims that the extra A16 liberty means B has time to do this. However W is three moves to atari (A16, A17, B18) while B is three moves to connection underneath (G19, F19, E19) and W plays first. So W has the advantage here. If one of the liberties at G19 was moved to a G17 style completely external liberty, then I would accept lightvector's conclusion.] (;W[ea]C[DH: fairly inevitable but not necessary to play immediately. ] ;B[ga] (;W[ad] ;B[gc] ;W[ac] ;B[fc] ;W[ca]) (;W[bb] ;B[gc] ;W[ca])) (;W[ad]C[ DH: It is more professional style to take profit first] ;B[ac] ;W[ae]C[Still a one move approach ko which white takes first.])) (;W[ba])) (;W[dc]C[C2] ;AE[ak]AW[db][eb][fb][gb][hb][cd][be][ce][de][cg][ci]AB[cb][ib][bc][cc][ec][fc][gc][hc][ic][kc][bd][dd][ed]C[Status? ] ;B[]C[Black can kill. White can kill.] (;B[ha]) (;B[ab]C[DH: This seems simply worse than H19 as A15 is still sente (after forcing with B19, A16), but B hasn't yet shorted the liberties of H18 yet (just incase the J17 gets into trouble).]) (;W[ba]))) (;B[al]LB[ca:A1][da:A2][ea:A3][db:B2][eb:B3]C[Mini L group with inside leg. Notes: The corner side often has 5 effective liberties due to eye power and/or throwin tactics. Also, in almost the same way as four-in-the-corner, it gains one more liberty when playing first due to a throwin tactic, where the opponent also making an eye results in seki. Usually: ~5 liberties inside when the attacker plays first. ~6 liberties inside when the corner side plays first, except the last liberty only makes seki if the attacker can also make an eye. ] (;W[ca]C[A1] ;AE[ca][al]AW[eb][fb][gb][hb][dc][bd][cd][dd][ed][cf][ch]AB[cb][db][ib][bc][cc][ec][fc][gc][hc][ic][lc][fd]C[Status?] ;B[]C[White is dead.] (;W[da] ;B[ba]C[DH: An important move to prevent ko. B19 and D19 were miai. Note that if W had A16, then white could play at B19 directly to set up a ko but without A16, B has a golden chicken situation.] ;W[ab] ;B[ac] ;W[ea] ;B[ha]) (;W[ac] ;B[ab] ;W[ba] ;B[ad] ;W[da] ;B[ca] ;W[ae] ;B[ha])) (;W[da]C[A2] ;AE[da][al]AW[eb][fb][gb][hb][dc][bd][cd][dd][ed][cf][ch]AB[cb][db][ib][bc][cc][ec][fc][hc][ic][lc][fd][fe]C[Status?] ;B[]C[Black can kill. White can kill.] (;B[ba]C[DH: Good move here when W can't easily connect to the D19 shape point.] ;W[ab] ;B[ac] ;W[da] ;B[ea]) (;W[ab] ;B[ac] ;W[ea])) (;W[ea]C[A3] ;AE[ea][al]AW[eb][fb][gb][hb][dc][bd][cd][dd][ed][cf][ch]AB[cb][db][ib][bc][cc][ec][fc][ic][lc][fd][hd][fe]C[Status?] ;B[]C[Black can kill. White can kill.] (;B[ba] ;W[ab] ;B[ac] ;W[da] ;B[ea] ;W[fa] ;B[ha]) (;W[ab] ;B[ac] ;W[ea])) (;W[db]C[B2] ;AE[al]AW[eb][fb][gb][hb][ib][dc][bd][cd][dd][ed][cf][ch]AB[cb][db][jb][bc][cc][ec][fc][gc][hc][ic][jc][lc][fd]C[Status?] ;B[]C[Black can kill. White can kill.] (;B[da]C[Unconditional kill.] (;W[ba] (;B[ia] ;W[ac] ;B[ab] ;W[ad] ;B[ha]) (;B[ab]C[ DH: too many liberties.])) (;W[ab]C[ This is the correct shape point locally, but W still loses by a liberty, so the game variation is better to prevent B pushing at A16.])) (;B[ba]C[Probably bad but maybe possible.] ;W[ab] ;B[ac] ;W[da] ;B[ia] ;W[ha] ;B[ja] ;W[fa] (;B[ca]C[Favorable 2-step ko, black's corner is a bit bigger.]) (;B[ad]C[ DH: The only benefit I can see is gaining these 2 points in double sente.])) (;W[ab] ;B[ac] ;W[ea])) (;W[eb]C[B3] ;AE[al]AW[fb][gb][hb][ib][dc][bd][cd][dd][ed][cf][ch]AB[cb][db][jb][bc][cc][ec][fc][hc][ic][jc][lc][fd][fe]C[Status?] ;B[]C[Black can make a seki or an unfavorable two-step ko. White can kill. ] (;B[ba] ;W[ab] ;B[ac] ;W[da] ;B[ea] ;W[fa] ;B[ia] ;W[ha] ;B[ja] ;W[ad]C[Seki here.] ;B[ca]C[Black can continue to try to get everything in a difficult ko.] ;W[bb] ;B[ea]) (;W[ab] ;B[ac] ;W[ea]))) (;B[an]LB[ca:A1][da:A2][cb:B1][db:B2][eb:B3][fb:B4]C[L group, tall side outward. Notes: The corner side can spend sente to establish 7 eye liberties and a descent at D19. This usually gives E19 as an 8th liberty, plus the power of the descent itself is often an effective 9th liberty. Usually: 5 liberties inside when the attacker side plays first + unfavorable step ko resistance. 7 liberties inside + eye power liberty when the corner side plays first. First play makes a difference of a ~3 liberty swing!] (;W[ca]C[A1] ;AE[ca][an]AW[eb][fb][gb][hb][ec][bd][cd][dd][ee][cf][ch]AB[db][ib][kb][bc][cc][dc][fc][hc][ic][ed][fd]C[Status?] ;B[]C[Black can kill. White can make a favorable two-step ko.] (;B[ab]C[DH: Truly remarkable move. D19 is met by two eyes at B19 but B19 directly for a bulky five is terrible and B gets 8 liberties with D19. The 2-1 point is critical for both ensuring W doesn't make the 1-1 into an eye and also letting B threaten to make the 1-1 eye.]) (;W[ab]C[DH: some very nice tesujis here. After this B only has 4 liberties. Sometimes W can play B18 instead to be more forcing but B still has at least 4 liberties and a ko option. Still, in the simple variation, W can capture some corner stones with the B16 group which can help liberties or eyespace. At the least, W can get A16 in sente.] ;B[bb]C[DH: Cut off weak stone from the C18 point. Blocking at A17 instead would be futile and losing a liberty after C18, especially as W has eyespace potential in the corner.] ;W[ca] ;B[ac] ;W[da] ;B[ea] ;W[ba] ;B[ha] ;W[ad] ;B[aa] ;W[fa] ;B[gc] ;W[ab])) (;W[da]C[A2] ;AE[da][an]AW[eb][fb][gb][hb][ec][bd][cd][dd][ee][cf][ch]AB[db][ib][kb][bc][cc][dc][fc][ic][ed][fd][gd][hd]C[Status?] ;B[]C[Black can kill. White can make a favorable three-step ko.] (;B[ab]) (;W[ab] ;B[bb] ;W[ca] ;B[ac] ;W[da] ;B[ea] ;W[fa] ;B[ha] ;W[ba] ;B[aa] ;W[ad] ;B[hc] ;W[ab])) (;W[cb]C[B1] ;AE[cb][an]AW[eb][fb][gb][hb][ib][ec][bd][cd][dd][ee][cf][ch]AB[db][jb][lb][bc][cc][dc][fc][gc][hc][ic][jc][ed][fd]C[Status?] ;B[]C[Black can kill. White can make a favorable two-step ko or seki.] (;B[ab]) (;W[ab] ;B[bb] ;W[ca] ;B[ac] ;W[da] ;B[ea] (;W[ba] ;B[ia] (;W[ad] ;B[aa] ;W[fa] ;B[ha] ;W[ab]C[Favorable two-step ko.]) (;W[ha] ;B[fa]C[White can't make an eye.] ;W[ja] ;B[aa] ;W[ga] ;B[ea] ;W[ab] ;B[ka] ;W[ad] ;B[aa])) (;W[fa] ;B[ba]C[Seki.]))) (;W[db]C[B2] ;AE[an]AW[eb][fb][gb][hb][ib][ec][bd][cd][dd][ee][cf][ch]AB[db][jb][lb][bc][cc][dc][fc][hc][ic][jc][ed][fd]C[Status?] ;B[]C[Black can kill. White can kill.] (;B[ab]) (;W[ab] ;B[bb] ;W[ca] ;B[ac] ;W[da] ;B[ea] ;W[fa] (;B[ia] ;W[ba] (;B[aa] ;W[ha] ;B[ja] ;W[ad]) (;B[ha] ;W[aa])) (;B[ba] ;W[ad] ;B[ia] ;W[ea] ;B[ha] ;W[cb]))) (;W[eb]C[B3] ;AE[an]AW[fb][gb][hb][ib][ec][bd][cd][dd][ee][cf][ch]AB[db][jb][lb][bc][cc][dc][fc][ic][jc][ed][fd][gd][hd]C[Status? ] ;B[]C[Black can kill. White can kill.] (;B[ab] ;W[ba] ;B[da] ;W[ad] ;B[ac]C[Black establishes 7 eye liberties and one shared liberty, plus eye power. White has only 7 liberties and therefore loses by two.] ;W[ja] ;B[ha] ;W[ea] ;B[ga]C[Not even multi-step ko for white.]) (;W[ab] ;B[bb] ;W[ca] ;B[ac] ;W[da] ;B[ea] ;W[fa] ;B[ia] (;W[ba]C[Probably best.] (;B[ha] ;W[aa]) (;B[aa] ;W[ha])) (;W[ha]C[White gets slightly more points but black gets a much stronger squeeze.] ;B[ba] ;W[ja] ;B[ka] ;W[ad] ;B[hc] ;W[ea] ;B[gc] ;W[cb]))) (;W[fb]C[B4] ;AE[an]AW[eb][gb][hb][ib][ec][bd][cd][dd][ee][cf][ch]AB[db][jb][lb][bc][cc][dc][fc][jc][ed][fd][gd][hd][id]C[Status? ] ;B[]C[Black can kill. White can kill.] (;B[ab] ;W[ba] ;B[da] ;W[ad] ;B[ac]C[Black establishes 7 eye liberties and one shared liberty, plus eye power. White has only 8 liberties and therefore loses by one.] ;W[ja] ;B[ha] ;W[ea] ;B[ga]C[Not even multi-step ko for white.]) (;W[ab] ;B[bb] ;W[ca] ;B[ac] ;W[da]))) (;B[ao]LB[ca:A1][da:A2][cb:B1][db:B2][fb:B4][cc:C1][dc:C2][fc:C4]C[L group, tall side inward. Notes: Similar to tall-side-outward when the corner side plays first, but when the attacker side plays first, sequences are pretty different. Usually: 5 liberties inside when the attacker side plays first + favorable step ko resistance. 7 liberties inside + eye power liberty when the corner side plays first (in extreme cases, opponent may get a virtually unwinnable step-ko). First play makes a difference of a ~2 liberty swing!] (;W[ca]C[A1] ;AE[ca][ao]AW[db][eb][fb][dc][dd][be][ce][df][bg][dg]AB[cb][gb][ib][cc][gc][bd][cd][ed][de][ee][ef]C[Status?] ;B[]C[Black can kill. White can make an unfavorable two-step ko.] (;B[ad]C[The greediest way to win, might not be best. DH: looks good to me.] ;W[ba] ;B[bb]) (;W[ba] ;B[bb] ;W[ac] ;B[ad] (;W[ab] ;B[ca] ;W[ae] ;B[aa]C[Two-step ko for white.]) (;W[ca]C[Also leads to two-step ko?] ;B[da] (;W[ea]C[Doesn't work.] ;B[ab] ;W[ae] ;B[fc] ;W[da] ;B[ec]) (;W[ab]C[Tesuji?] ;B[aa] ;W[ba] ;B[ca] ;W[ae] ;B[aa]C[Two-step ko for white.])))) (;W[da]C[A2] ;AE[da][ao]AW[db][eb][fb][dc][dd][be][ce][df][bg][dg]AB[cb][gb][ib][cc][gc][bd][cd][fd][de][ee][ef]C[Status?] ;B[]C[Black can kill. White can kill.] (;B[ad]C[The greediest way to win, might not be best.] ;W[ba] ;B[bb] ;W[ca] ;B[da]) (;W[ba] ;B[bb] ;W[ac] ;B[ad] ;W[ca] ;B[da] ;W[ea] ;B[ab] ;W[ae] ;B[ed] ;W[da] ;B[fc] ;W[aa])) (;W[cb]C[B1] ;AE[ao]AW[db][eb][fb][gb][dc][dd][be][ce][df][bg][dg]AB[cb][hb][jb][cc][ec][gc][hc][bd][cd][ed][de][ee][ef]C[Status?] ;B[]C[Black can kill. White can make seki.] (;B[ba] ;W[ab] ;B[ad]) (;B[ad]C[Not so good now.] ;W[ba] ;B[bb] ;W[ca] ;B[da] ;W[ea] ;B[ga] ;W[ab]) (;W[ba] ;B[bb] ;W[ac] ;B[ad] ;W[ca] ;B[da] ;W[ea] ;B[ab] ;W[ga])) (;W[db]C[B2] ;AE[ao]AW[eb][fb][gb][dc][dd][be][ce][df][bg][dg]AB[cb][hb][jb][cc][ec][hc][bd][cd][ed][gd][de][ee][ef]C[Status?] ;B[]C[Black can kill. White can kill.] (;B[ba] ;W[ab] ;B[ad]) (;B[ad]C[Not so good.] ;W[ba] ;B[bb] ;W[ca] ;B[da] ;W[ea] ;B[ga] ;W[ab]) (;W[ba] ;B[bb] ;W[ac] ;B[ad] ;W[ca] ;B[da] ;W[ea] ;B[ab] ;W[ae] ;B[gc] ;W[da] ;B[ga] ;W[aa])) (;W[cc]C[C1] ;AE[ao]AW[db][eb][fb][gb][hb][dc][dd][be][ce][df][bg][dg]AB[cb][ib][kb][cc][ec][fc][gc][hc][ic][bd][cd][ed][de][ee][ef]C[Status?] ;B[]C[Black can kill. White can make a sente seki.] (;B[ba] ;W[ab] ;B[ad]) (;W[ba] ;B[bb] ;W[ac] ;B[ad] ;W[ca] ;B[da] ;W[ea] ;B[ab])) (;W[dc]C[C2] ;AE[ao]AW[db][eb][fb][gb][hb][dd][be][ce][df][bg][dg]AB[cb][ib][kb][cc][ec][gc][hc][ic][bd][cd][ed][de][ee][ef]C[Status?] ;B[]C[Black can kill. White can kill.] (;B[ba] ;W[ab] ;B[ad] ;W[da] ;B[ha] ;W[ca] ;B[bb] ;W[ac] ;B[ga] ;W[ae] ;B[fc]) (;W[ba] ;B[bb] ;W[ac] ;B[ad] ;W[ca] ;B[da] ;W[ea] ;B[ab] ;W[ae] ;B[fc] ;W[da] ;B[ha] ;W[aa])) (;W[fb]C[B4] ;AE[ao]AW[db][eb][gb][dc][dd][be][ce][df][bg][dg]AB[cb][hb][jb][cc][hc][bd][cd][fd][de][ee][he][ef]C[Status?] ;B[]C[Black can kill. White can kill.] (;B[ba] ;W[ab] ;B[ad] ;W[da] ;B[ca]C[Black wins by 1 liberty.] ;W[ha] ;B[fa]C[Not even multi-step ko for white.]) (;W[ba])) (;W[fc]C[C4] ;AE[fc][ao]AW[db][eb][fb][gb][hb][dc][dd][be][ce][df][bg][dg]AB[cb][ib][kb][cc][ec][ic][bd][cd][ed][gd][de][ee][ef]C[Status?] ;B[]C[Black can make an extremely favorable 3-step ko. White can kill.] (;B[ba] ;W[ab] ;B[ad] ;W[da] ;B[ca] ;W[ia] ;B[ja] ;W[ga] ;B[hc]C[(Black can take H19 between each of these moves to force white to spend a threat, so this is basically impossible for white, but in the interests of getting to the end of the variation...)] ;W[ae] ;B[gc] ;W[bb] ;B[fc] ;W[ac] ;B[fa] ;W[bc] ;B[aa] ;W[ab] ;B[ha]C[3-step ko.]) (;W[ba]))))