I think this hand is a good example of bidding constructively, trying to decide how high to bid.
South’s first double shows a good opening bid. His 3♦ shows a hand simply too good to bid 2♦. This hand is close to a minimum for this sequence. Your partner has come to life with a cue bid. What does he have?
First he has diamond support. Otherwise he would either pass, bid a new suit or bid 3NT. What about values? He probably has one sure trick. Does three spades mean anything? True he could have simply bid 4♦, but he probably wanted to keep 3NT open in case you have a stopper. What should you bid? 4♦. This will be plenty high enough. Partner was catering to you having one of the following hands:
♠Ax ♥Kx ♦KQJxxx ♣AKx or
♠x ♥A10 ♦KQ9872 ♣AKQ7
Partner actually held ♠xx ♥J9xx ♦Axx ♣109xx and passed 4♦, which made on the nose.
Lessons to Learn
1. Remember to make decisions in the context of the auction. Here you are a minimum for your double followed by three diamond approach, which is why four diamonds is right.
2. Another way to look at it is that North has a maximum, considering his passes and two bidding opponents. His 3 spades, therefore, is a very good bid recognizing this point.
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