(;GM[1]FF[4]CA[UTF-8]AP[CGoban:3]ST[2]
RU[Japanese]SZ[13]KM[0.00]
AW[fa][ja][fb][jb][fc][jc][dd][ed][jd][de][ke][df][kf][dg][ig][jg][kg][dh][ih][ci][di][fi][gi][hi][cj][dj][ej][ck][dk][ek]AB[ea][ka][eb][kb][cc][dc][ec][kc][cd][kd][ld][ce][je][le][cf][lf][cg][lg][bh][ch][hh][jh][kh][lh][bi][ii][ji][bj][fj][gj][hj][bk][fk][bl][cl][dl][el][fl]LB[ie:A][hg:B]C[The following problem is adapted from a recent game of mine. 

This is an extract from a 19x19 game, and not a complete 13x13 board position.

It is Black to play. White could have played a defensive move
to fix the bad aji as the dame were being filled, but then he
would have lost by half a point (the game was played under
Japanese rules). Therefore any gain counts as a success 
for Black: capturing some stones, seki or ko.

This problem is rather difficult: you probably need to be a
high dan player to solve it in your head. However, I suggest
that you try for a while, and then explore variations on a board
before looking at the answers: there are many beautiful
tesujis, snapbacks, and connect-and-dies hidden within as
rewards for the determined traveller.

A few hints:
1) Add a second stone and sacrifice both.
2) My solution relied on the dango in the lower left not being 
an eye, but someone else found another solution that works regardless.

Incidentally, this demonstrates an aspect of Japanese rules I 
prefer over AGA. It is much more satisfying for a game to be 
decided by a problem like this than the randomness of dame parity.]VW[]
(;B[ie]CR[fd]LB[hb:B][hc:A]C[If Black is to accomplish anything, it should be clear it will 
be by utilising his two cutting stones, White's marked cut and 
trying to capture White's four stones in the top right. The first 
move to consider would be simply extending as below, but
once 
White defends at 2, Black can try capturing White's 4
stones 
at the top (for example with A), but nothing quite
works (White 
answers at B).]
;W[hg]CR[fd]LB[hb:B][hc:A])
(;B[hg]
;W[if]LB[id:C][he:A][ie:B]
(;B[he]LB[ie:A][hf:B]C[If Black is to be successful in his assault on the top, he will 
need more than a single stone in sente to help him. Therefore he pulls out his other cutting stone first, intending
to sacrifice 
it to get additional forcing moves before attacking
the top. 
White has to make the empty triangle at 2: if she plays atari 
on the other cutting stone Black gets a snapback. Next 3
makes 
a net from which White has two ways to escape.]
(;W[ie]
;B[id]LB[ha:B][ia:C][ic:A][hd:B]C[If White captures the single stone, Black cuts again aiming to 
squeeze White. If White simply captures the single stone to
avoid 
snapback, Black squeezes and then pushes down at
A, and White is 
captured (if White jumps to B, Black throws in
at C).]
(;W[ic]
;B[hd]
;W[jf]
;B[hf]LB[ga:H][ha:F][ia:C][gb:E][hb:B][ib:D][gc:G][hc:A][fd:I]C[White might try putting in the atari of 6 first, but Black still 
squeezes and with A to I, White is on a one-way street to death.])
(;W[hd]C[After Black's cut of 5, White might try breaking out with two 
ataris; Black doesn't save the second stone but instead
catches 
White in a lovely connect-and-die.]
;B[ic]LB[ib:B][hf:A]
(;W[hf]
;B[ib]
;W[jf]
;B[ia])
(;W[ib]
;B[hc]
;W[hb]C[Given that the above succeeds for Black, White can try as
below, 
which looks better for a while as White gets to
connect along 
the top...]
;B[gd]
;W[gc]
;B[hd]
;W[jf]C[...and then captures to avoid snapback. However, Black
again 
squeezes, and then cuts. White has to live at 18 (or
black plays 
there and White dies very quickly) which allows
Black to play 
19 to take away White's eyes of the group which
he has cut off 
in sente. Black doesn't have an eye either,
though, so this is 
a seki (verification left to the reader). This
is why I mentioned 
earlier that White's dango in the lower left
was relevant: if 
she had an eye then Black would be dead.]
;B[hf]
;W[je]
;B[fd]
;W[ha]
;B[fh])))
(;W[hf]C[So from these variations, we can conclude that 4 capturing
the 
single stone fails, so instead must be as below. Black
keeps 
up getting as many forcing moves as possible by
blocking at 5 
(White can't answer at 7 or else Black plays one
above for the 
now familiar snapback). White captures 2
stones, but we can now 
see that Black has vastly improved
upon the first diagram as 
he has two extra stones at 3 and 5
as support for the attack 
on the top.]
;B[gf]
;W[gg]
;B[ie]
;W[gh]CR[fd]LB[hb:C][hc:A][ic:B]
(;B[ic]C[So now Black has to try to capture the 4 White stones, or at 
least harass them to reduce their eyespace and liberties so that 
the marked cut then works. Moves at A or B were my first ideas, 
but neither of these come to much when White connects at C in 
the previous diagram (10 in the diagram below), for example as 
below (once black connects at a, white takes a liberty at b and 
black can't atari due to his lack of an eye whilst white does 
have one).]
;W[hb]
;B[ib]
;W[ia]
;B[id]
;W[ha]
;B[fd]
;W[gd]
;B[fe]
;W[gc])
(;B[hb]LB[ha:B][hc:A]C["Your opponent's key point is your key point" suggests  black 
9 at C (10) from the previous diagram. If White tries to cut 
this off, Black succeeds in getting a ko.]
(;W[hc]
;B[ic]
;W[id]
;B[hd]
;W[ib]
;B[gc])
(;W[ha]C[As White can't connect on top, he connects underneath. 11
threatens 
to throw in and capture in a connect-and-die. White
can defend 
against this by connecting to either side, so
Black must be able 
to win against both. The variations are
similar; to stop this 
rather long solution getting longer I will
only show one.]
;B[ic]
;W[ia]
;B[fd]LB[hc:B][gd:A]
(;W[gd]
;B[fe]LB[gc:A][ge:B]
(;W[gc]
;B[ge]
;W[id]
;B[hd]
;W[hc]
;B[ga]C[Having squished White against the edge without an eye,
Black 
cuts, and if White connects with 16, Black gets an
exquisite 
double snapback (my favourite variation of this
problem).])
(;W[ge]
;B[gc]
;W[ff]
;B[ga]C[And if white cuts for 16, Black cuts and throws in.

As an aside, notice the importance of the marked Black
stone 
(at Black 3) taking a liberty which allows this sequence
to work. 
This is why the net of Black 3 way back at the start was key; 
if Black simply plays as below (variation at 3) then this
sequence 
doesn't work (of course the net does give White
the option of 
playing 4, breaking out above, but we've shown
Black wins that 
way too).]MA[gf]
;W[ee]C[To continue, White can't capture 19 or it's a snapback, so
Black 
is able to capture the White stones on the right (in
sente!).]
;B[gb]
;W[fd]))
(;W[hc]LB[gc:A][id:B]C[There are a few other resistances for White to try, such as
the 
fancy wedge (at 140) below. If Black falls for the bait and
ataris, 
White wins as Black only has one eye inside, and
White is safely 
connected.]
(;B[gc]
;W[gd]
;B[hd]
;W[fe])
(;B[id]C[Instead black must calmly connect. Even though it looks like 
White's wedge fixed the cutting point in sente, Black still can 
force a shortage of liberties by throwing-in there for another 
connect-and-die.]
;W[gd]
;B[fe]
;W[ge]
;B[gc]
;W[gb]
;B[ib]))))))
(;B[ie]C[(Variation for 3 referred to at the end of the snapback sequences 
- PAR).]
;W[hf]
;B[he]
;W[gf])
(;B[id]C[But that's not the end of it! When I posted this problem
online, 
someone found an alternative solution. It is better 
in that it 
works even if White has an eye in the lower left, 
and is elegant 
in its simplicity, but is worse in that its
denouement is not 
a double snapback! Rather than playing
the net for black 3, he 
takes a liberty as below in the key
shape point for setting up 
the squeeze.

If White cuts, Black pushes down and White has to capture
the 
2 stones somehow, such as 6. But Black continues
squeezing and 
then plays 11, making miai of A and B. White
4 at B ends up very 
similar, with black 11 being the key point
of creating miai to 
capture some stones. White 4 blocking at
black 5 is answered 
at B and ends up as the one-way road
to death of the 4th diagram.]
;W[ie]
;B[ic]LB[ib:B][gf:A]
(;W[gf]
;B[ib]
;W[jf]
;B[ia]
;W[je]
;B[gd]LB[fd:A][hf:B])
(;W[ib]C[White's strongest resistance is the rather silly-looking empty 
triangle of 6, but again A and B end up as miai. This variation 
reveals why this alternative black 3 is better than the net: 
with the net, Black had already made the exchange of 9 for
10 
before White played at 6, which meant white 6 was atari
and he 
was able to connect underneath with 7, and Black
only won because 
of the seki that resulted later. By keeping
the 9-10 exchange 
in reserve until it is needed, White is
denied this opportunity 
so is dead even with an eye instead
of a dango in the lower left.]
;B[hb]
;W[jf]
;B[he]
;W[hd]
;B[hc]
;W[hf]
;B[gd]LB[fd:B][gf:A]C[In conclusion, Black has many beautiful ways to win; I hope you 
enjoyed this problem as much as I did. I find it remarkable that 
such an exquisite problem arose naturally in a game (http://www.online-go.com/games/board.php?boardID=271713).]))))
