Let's tackle the harder problem first.
It is not obvious, but Black has 1.25 eyes, 0.5 of an eye on the side and 0.75 of an eye in the corner, and has to make another 0.75 of an eye to live. She does this by making 1.5 eyes in the corner.
Stage 1 is to get
and 
in place in sente, threatening the 
stone.White cannot afford to play atari at 4.
Black now has and a half eyes in the corner. Added to the half eye on the side gives two eyes for life. (
could also be at "a".) Why one and a half eyes?
throws in at 1.White can hold Black to one eye in the corner in gote.
Black to play can make two eyes in the corner by oshitsubushi, also in gote.
In the book White played
. Black gets 5 points in the corner instead of 3, but still she has only one eye. 
throws in at 4. at 2 captures four stones.After White holds Black to one eye in the corner Black has to pick up half an eye on the side. She does this by an under the stones sacrifice, which is familiar to experienced tsumego solvers.
captures the stones to make the second eye.I don't know whether Kobayashi père composed this problem or whether it is a famous ancient problem, but it is pretty cool, isn't it?
Despite the slight flaw of . First Black encloses the White stone in the corner in sente, with a throw-in if White tries to connect. Then Black makes one and a half eyes with the 1-1 play, threatening to make two eyes with oshitsubushi. And finally Black plays an under the stones sacrifice to pick up half an eye on the side. Poetic! 
( Would be nice to see AlphaGo's numbers. 
, as the last big play of the opening. But that is lackadaisical thinking.
. Sato continued the attack with 
and 
, and still got the two space extension at 
. 
. It is a thick play that counters White's thickness. Not much happens on the top side for some time, now. 
in the bottom right corner and then, after 
, is satisfied to have prevented Black from making an ideal enclosure, and takes the big point at 
. Skillful play. 
, and then Retsugen jumps to 
, a big whole board move. 
and then takes kikashi in the corner before extending towards the center. 
is already connected back with his friendly stones LL.
, is big. After White's attach-and-extend Black takes kikashi with 
and 
, which prevents the White clamp at "a". Then 
jumps into the center.
, 
also works here:
... 
) before 
plays a two step hane.
descends to the first line, and 
connects.