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5, natural and quantitative. Partner will be far from perfect over that, but it's a lot better than me simply guessing what to do on my own. Between the other options, I won't double and risk partner passing, I can't cuebid since I play that as Michaels, and 4 right now is too big of an underbid for me.
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5, natural and quantitative.

I thought 5H was the "voluntary bid of 5M", asking partner for controls in some key suit (spades in this case)?

 

For the problem, does it still make sense to play partner for a random 7 count here? Slam looks decent opposite that; as long as I'm going there, may as well use exclusion to explore for a grand.

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5, natural and quantitative.

I thought 5H was the "voluntary bid of 5M", asking partner for controls in some key suit (spades in this case)?

That's pretty much useless in real life, particularly in competition. You will get probably hundreds of hands in your life that look like 4 bids but just too good. You will get 0 or 1 hands in your life that have 11 tricks outside spades and 2 small spades when your opponent comes in first with a spade bid.

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There's an argument for treating the "stopper asking" cue bid as any solid suit (including a major), to help handle strong 1-suiters. Especially with hearts (over 2) you only get 2 levels of strength with 3 and 4 (vs 3 levels with spades over 2). The obvious advantage is that stronger hands are likely to be able to out in 4, which is a little lower than a quantitative 5 invite :(.
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I was gonna ask what's wrong with double and then 3 but then I realized I might not get a chance to bid 3. 3 sounds reasonable, if partner shows a stopper or believes me to have two suits I'll just keep bidding hearts and he should get the picture.
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depends how you play 3S

Undiscussed :)

 

But I suppose you could assume it would be taken as a stopper ask.

I would start with 3S then.

This was the route I took. Partner now bid 3NT, and I bid 4, which turned out best when partner held KQxxx Q9 x xxxxx and I got the J lead. Who would you play to have the A for the overtrick?

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