2

With the three black stones placed, I did the 3-3 invasion.

$$Wm1
$$ ----------
$$ ........|
$$ ...X....|
$$ ...X.1..|
$$ ....X...|
$$ ........|

I tried tried to force a KO and did get 2 moves elsewhere, but I would rather have lived.

$$Wm1
$$ ----------
$$ ........|
$$ ...X..5.|
$$ ...X.O.7|
$$ ....X236|
$$ ......4.|

Does 3-3 invasion work here?

3

Liam showed a good way to kill the invasion already, but you can be even more mean and play 2 at 4:

$$Wm1
$$ ----------
$$ ........|
$$ ...X....|
$$ ...X.1..|
$$ ....X...|
$$ ......2.|

If white then plays 3, you just draw back instead of playing below 1, and you avoid a cut at 4:

$$Wm1
$$ ----------
$$ .....c..|
$$ ...Xb.a.|
$$ ...X.1..|
$$ ....X.3.|
$$ .....42.|

Then, a, b and c are miai to kill.

The cut would not usually be dangerous, but it at least allows white to get 1 or 2 outside sente moves in the area, which is more than she should get - the original invasion at 3-3 was way too deep (the deepest possible would be white 1 at 2 in the above diagram, but that is a bit advanced, don't worry about it too much).

White may try to be sneaky and return to the variation with the cut present like this:

$$Wm1
$$ ----------
$$ ........|
$$ ...X....|
$$ ...X.1..|
$$ ....X...|
$$ .....32.|

In that case, you can easily play at 6 instead of a to gain more and avoid white's sente atari at 6. The white inside group is still dead as per what we saw above.

$$Wm1
$$ ----------
$$ ........|
$$ ...X....|
$$ ...X.1a.|
$$ ....X45.|
$$ .....32.|
$$ ......6.|
1

Black 6 should be at white 7. After this white dies uncondittionally.

https://online-go.com/demo/66482

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