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You have nowhere to go. My rules, although other partnerships might do things differently, is to bid a five card at the two level as a sign-off showing a weak hand. With the West hand shown you sometimes have to pass and give +380 to the opps. You can not win every hand. Sometimes the 1NT opener will not like to play in 1NTX so they will remove to their own 5 card suit. More poker than bridge.
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If 1NT is 15-17, the double is presumably 18+, so your side has half the pack. Against a weak NT the double is 15+ and that leaves more points for dummy; I'd probably pass at teams but 2 at pairs.
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Hi,

 

you should have a agreement, how weak / strong the X is.

My take is, that Capp. req. strong NT strength, i.e. (good) 15+.

A possible agreement is, to play the same structure after a X, like after a strong NT opening,

2C Stayman, 2D, 2H xfer.

This does not help you here, the best you can do with the given hand is pass.

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Presumably it is a penalty X of a weak NT.

Do penalty X of strong NT exist?

 

We would play weak natural 2 bids and GF 3 bids over that. Pass with any balanced hand.

 

But Souths silence needs to be queried.

Maybe nothing better to bid. With 7 opposite 12-14 one will often Pass.

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A penalty X of strong NT exists but is a dubious prospect, hence the tendency of conventions like Multilandy or Woolsey to use X for some more specific hand type instead.

But our beginners are still taught to play it as natural penalty (or slightly weaker meaning "I have a strong NT too").

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[hv=pc=n&w=st765h653d82cq963&e=s83hakqj742dK54c7&d=w&v=0&b=8&a=p1n(15-17)dp]266|200|How should West respond to East's double?[/hv]

 

Yes, there are better on average meanings for double against a strong NT than "penalty". But this looks very much like my most embarrassing result playing a Weak NT - because I, as South, with a 10-count, sent it back. To add insult to injury, opening leader lead fourth best. It works just as well against 16-opposite-7 as against 13-opposite-10.

 

Which is also why I strongly disapprove of a common idea around this area that you play "systems on when pulling 1NT-X". Never mind the fact that you can't play 2 or 2, what does partner's 3 mean in response to 2 "transfer to spades"? Also, never mind "endplayed at trick 1" is worth at least as much as "lead into the strong hand".

 

The best penalty doubles (and, frankly, the best doubles converted to penalty) of 1NT aren't based solely on high cards, but on tricks. Pass with confidence, signal honestly, and pay special attention to protecting whatever potential stopper you end up having. And if you concede -360 anyway, chalk it up and get on to the next board. It's not -500, after all.

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