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Is 1D-2C by a non-passed hand GF?
In different partnerships the answer is yes and no...
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What is meant by 1D-2N by a non-passed hand?
Natural invitation, balanced, about 11-12 points.
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What do you respond to 1D with 3-3-3-4 and (say)11-12 points?
2NT
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What do you respond to 1D with 3-1-3-6 and 10 points?
2C if that's not GF, else 3C. If 2C is not GF you can use 3C as a diamond raise.
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Is 1D-2C-2N forcing?
You are playing 15-17 1N opener and open 1D with 4-4-3-2 and 12 points. Partner responds 2C. What is your rebid? If it is 2NT, what is your rebid with 14 points? (question assumes 2C response is not GF ... ignore if it is GF.)
I'm playing 14 - 16 NT to avoid this and other problems, but I'd say 3NT with hands that would accept an invitation. See the trick below to see what to do with 18-19 balanced hands in this case.
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Does 1D-2C-2H now promise that D suit is genuine, and does it show extras?
If 2C is GF the answer is yes (else just bid 2NT). If not, here's a trick I play (I know this complicates things).
1D - 2C - 2H =
a) GF club raise -> 1D - 2C - 2H - 3C is non-forcing

GF with 5D + 4H
c) Any strength, 4=4=4=1 / 4=4=5=0.
And if you play 15-17 NT, you can add:
d) 18 - 19 balanced
2S inquires and opener rebids:
2NT with a 3-suiter and a minimum
3C with GF club support
3D with a reverse with 0/1 club
3H with a reverse with 2 clubs
3S with a 3-suiter and a singleton club
3NT with 18 - 19 balanced
4C with a club void and 15+
Gerben