Whenever you open 2NT, overcall 2NT, or open a strong 2 and rebid 2NT, it is useful to have some effective system to find the right contract. The bidding is already uncomfortably high, so some artificiality is useful in order to find the right contracts.
A good basic system: 3 is five card stayman, 3 and 3 are transfers, and 3 shows the minors. 4 and 4 are single-suited slam tries with that suit. Over the 3 bid, opener shows his 5 card major, or else bids 3, over which responder bids his four card major. Some examples:
1. AQxx Kx AKxx Axx opposite Kxxx J10xx Qxx xx. 2NT-3-3-3-3-4.
2. AKxxx Kx AKx Kxx opposite Qxx Qxx xx AQxxx. 2NT-3-3-4-4-4-5-6. After hearing that partner has five spades, cuebid 4 to show slam interest.
3. With QJx xxx xxxx Kxx, simply raise to 3NT.
4. QJxxx x Kxxxx xx. 2NT-3-3-3NT. Show partner your spades and then offer him a choice between 3NT and 4. Do not bid 4 - whilst this would be natural, it should also show a slam try as you should not try to play in 5 rather than 3NT.
5. Kxx Axx Kxx Qxxx. Raise to 4NT, invitational to 6NT.
6. AQx KQxx Kxx Ax opposite xx J10xx AQxx xxx. After opponents open a weak two in spades it proceeds (2)-2NT-3-3-3-4-4. The 2NT bid here shows about 16-18 points, then the four card heart fit is found. Note also that the overcaller cuebids 4 on the way, since he is suitable for slam so he must tell his partner in case he is too.
A problem: You may notice with this system that with a single-suited major slam try you have something of a problem. You can transfer to the major, but what then? The only way to show the hand is to invent a second suit to show the slam interest, and then later pull back to the major. Even worse is that if it starts 2NT-3-3-4-4 you do not even know whether partner is cuebidding for spades or for the clubs you don't have! A good solution to this problem is to play all four level bids as single-suited slam tries - but for the suit two up! Thus 4 is a slam try in spades (over which partner can sign off with 4, or show interest with 4). Similarly, a 4 response is showing a slam try in clubs.
Describing and right-siding: There are two aims over 2NT - to be able to show all the hand types, and to right-side the contracts. Most people will be content with the above system, but below we give some fine-tuning to achieve those ends. In my opinion it is the best of many versions on the market, although what follows is somewhat simplified!
The big difference is over the 3 bid. 3 and 3NT responses are used to show 5 spades and hearts respectively. 3 denies a four or five card major, and 3 shows at least one four card major. Over 3, responder bids the major he hasn't got with one of them, 3NT with neither, 4 with both and a slam try, 4 with both without a slam try. 4 and 4 are also used to deny a major and show a five card club/diamond suit respectively.
Other possible auctions include:
1. 2NT-3-3-3-3NT-4. This shows 4 spades and 5 clubs, with slam interest (and opener is known to have four hearts). If instead responder had bid 4, that would have been 4 spades and 4 clubs (he has already denied hearts so it cannot be natural, and hearts and clubs are 'linked' suits).
2. 2NT-3-3-3-3NT-4. 3 denies a major, and 3 over this shows the troublesome hand with five spades and four hearts. Over 3NT, which denies a fit, partner can now show a three (possibly 4) card diamond suit, with a slam try, leaving opener in a good position to judge if slam is on. Alternatively responder could have rebid 4 to show a 5-5 majors slam try.
3. 2NT-3-3NT-4. The 3 bid shows at least 44 minors, and now responder shows a fifth club. With 55 minors he would instead bid 4 or 4 to show his shortage.
4. 2NT-3-3-4. Opener has shown five spades, and responder makes a general slam try with 4, agreeing spades. If instead responder had a balanced hand with five diamonds or clubs, he would now bid 4 or 4 respectively. Or if responder had a strong hand with 4-4 in two suits would now invite slam with 4NT, since opener is unlikely to have a second four card suit - if he has, he can still bid it later.
Now consider what hands types responder might have. Without a slam try he simply looks for a major suit fit and otherwise plays in 3NT. He finds all these via a transfer or 3. With a slam try, then if only four card suits he starts off with 3 with a major, or 3 with both minors. With a five card major he starts off with a transfer and then invites with 4NT if balanced, or bids a 4 card suit (with 5 spades and 4 hearts he starts off with 3 as above). With a 5 card minor he starts with 3, then either shows the minor immediately if balanced, or shows his major, and then the minor later. With a 6 card suit he shows it at the four level immediately. Everything covered!
Another key feature of the system is that whenever possible retransfers apply to enable opener to play the contract. For example:
A. 2NT-3-4-4. The 4 bid was showing a good hand with spade support. Responder is not interested and retransfers with 4. If instead it went 2NT-3-4-4, this would show a heart cue and non-forcing. (If responder wants to go beyond 4 with a heart cue, he starts with the retransfer, and then bids on over 4.)
B. 2NT-3-3-3-4-4. The 3 bid shows four hearts, and opener agrees this suit and shows slam interest by bidding 4. His partner is not interested, but rather than playing the contract by the weak hand, transfers it to partner by bidding 4.
C. 2NT-4-4-4NT-5-6-6. Responder shows a single-suited heart slam try, and when his partner shows interest, uses his favourite version of RKCB. He then leaps to 6 to allow the strong hand to play it. In this case 6 would also have been to play, so responder can choose which way to play the hand.