Problems about ko threats

[antonTobi]

I like problems that ask you to squeeze out the maximum number of ko threats from a position.

Here’s a fun one that came up in a game I played at the club yesterday:

I’m white and we are currently fighting a half-point ko elsewhere on the board. How many ko threats can I get from the upper left corner?

[shamisen]

I count 3 if Black responds optimally:

C17, then C18 if Black B17, then A16 if Black B18

C17, then A17 if Black C18, then A16 if Black B17, then A16 again

By the way, why is there a White stone in the bottom right corner?

[Atorrante]

@shamisen

By the way, why is there a White stone in the bottom right corner?

it is a 19x19 board

[uzaya]

I assume that stone is part of a bigger group.

I actually looked at that and although C17 is the obvious starter, is it a ko threat? wC17 T- B17 B18- A17 A18 or wC17 T- C18 B17 doesnt seem like leading anything to me.
Maybe wC17 T- wB18? it is hard to read for me (and in game I can see myself just responding with B18 rather than tenuki.
My inutition is maybe wC17 B18- A17 B17 is the variation? so 2 ko threats?
It cannot be less than 2 really.

[antonTobi]

Black can in fact ignore C17:

Although this result is a ko, it is an indirect ko which white would need multiple additional ko threats to be able to win. So C17 could be a valid threat in some board situations, but there is a cleaner solution which guarantees white more than 1 threat regardless of the board situation!

[shinuito]

I’m thinking A17 and then depending on the answer. If A16, then C17 might be a real next ko threat or if B17 then maybe I have A16 twice more ?

[antonTobi]

No need for the “might” and “maybe”, that’s exactly right

Diagrams

This is the corrrect first move:

In the above variations white gets 3 ko threats. The strongest response for black is actually to connect solidly, leaving only 2 ko threats total:

(B18 next is not a threat for white)

Final verdict: white has 2 ko threats.

[antonTobi]

Some easier problems today! How many ko threats does black have in each of these common shapes?
(assume the stones extending outside the diagrams are independently alive)

a)

b)

c)

[uzaya]

Attempt

a) 2 (c2,d1)
b) 3 (c2, d1, a1)
c) 4 (c2, c1, b1, c2 or c2, d2, b1, c2)