Early in our bridge careers, we noticed that something was wrong when the bidding went 1♥-1N-2♣-2♠. The 1N response apparently denied 4+ spades, so how could partner be bidding them now? As a result, good players decided that 2♠ was artificial. The usual approach was to play that it shows a “serious” raise of clubs, while a direct 3♣ is a “courtesy” raise. This is certainly better than having no understanding at all about 2♠, but it is not the best use for the bid.
The best use for the impossible 2♠ depends on what 1N is. In some partnerships, it is forcing; in others it is semi-forcing; and in still others it is not at all forcing. If it is forcing, then opener might fairly often have to rebid 2♣ on a doubleton (e.g., if he is 4=5=2=2). If it is semi-forcing, then it depends on what “semi-forcing” means. If it means that opener may not pass with 13+ hcp, then he will still have to bid 2♣ on a doubleton occasionally. If it is not at all forcing, then he will usually have real clubs when he bids 2♣ (although he might be 4=5=1=3).
Another consideration is what opener does with 3=5=3=2 hands when he doesn’t have the option systemically to pass. Nowadays, many partnerships take the sensible approach to bid 2♣ with this pattern, so that 1♥-1N-2♦ shows real diamonds (although opener could be 4=5=3=1). This is beneficial because it exchanges two ambiguous rebids for one fairly descriptive and one very ambiguous rebid (this also the basic concept behind the so-called "short club", in which the opening bid with 4=4=3=2 hands is 1♣ rather than 1♦).
We can boil all this down to two systemic situations: (1) opener might fairly often have only a club doubleton to bid 1♥-1N-2♣, and (2) opener will usually have real clubs when he bids this way. The optimal use for the impossible 2♠ is different for (1) and (2).
In situation (1), the impossible 2♠ should show a hand with 4 or 5 clubs, invitational values, and lacking the stoppers to bid 2N. Raising to 3♣ with such a hand is dangerous, because we might have only a 7-card fit (in the extreme, a 6-card fit). After this kind of 2♠ bid, opener can sign off in either 2N or 3♣; he can accept the invitation by rebidding 3N or even 5♣; if he happened to bid 2♣ with a non-minimum-range hand, hoping that responder wouldn't pass, he can bid something else, which is natural and game-forcing.
In situation (2), the impossible 2♠ should be a relay to 2N, after which responder can rebid three of a minor or 3♥. Direct rebids generally show weaker hands than indirect rebids. This is especially important when you play bergen raises or strong jump shifts. 1♥-1N-2♣-3♦ would show a weak jump shift in diamonds, while 1♥-1N-2♣-2♠-2N-3♦ would show very long diamonds and invitational values. You would still have your serious and courtesy raises of clubs. 1♥-1N-2♣-2♠-2N-3♥ would show a sound 3-card limit raise, while 1♥-1N-2♣-3♥ would show a light, often distributional, 3-card limit raise (e.g., x QJx Axxxxxx xx).
The impossible 2♠ arises in two other sequences: 1♥-1N-2♦-2♠ and 1♥-1N-2♥-2♠. These are situation (2) because opener’s length in the suit he rebids is pretty well defined. So 2♠ is a relay to 2N, with direct rebids at the 3-level being weaker than belated rebids. 1♥-1N-2♣-2♥-3♥ should be a 3-card limit raise, while 1♥-1N-2♣-2♥-2♠-2N-3♥ should be a 2-card invitation; this is consistent with the idea that direct 3-level bids are weaker than belated 3-level bids because with 3-card support, you need fewer hcp to invite than you do with only a doubleton in support.
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