Quote
play with pickup pd, you hold:
S: XX
H: KJ9X
D: T87XX
C: AX
W N E S
P
P 1D P 1H
P 2S P 3D
P 3N P 4C
P 4N P P
P
Pd has:
S: AKQX
H: X
D: AKJXX
C: KXX
Also, pls mark these three bids after 4c
a) 4d

4s
c) 4N
Playing with a pickup partner everything 4NT is blackwood. The reason is that a) they will think it is even if they don't normally play it as blackwood on any specific sequence, and

try to keep the bidding simple with pick up partners. Last night I held AT8x AQ9x void KJ7xx
Bidding--- 1C-1D-1H-1S-3S-4N (clearly blackwood), did I bid the "correct 5NT (two and a void?), of course not, not with a pickup partner, I showed two and we played the cold 6NT versus the sure down one 6S and it was matchpoints in case you could squeek out 6S.
In your auction, perhaps you might want to play 4NT is sure I reverse, and sure you made a slam try with 4C, but I have a mini-reverse (say 17), so proceed at your own risk. But there is another problem here with a pick up partner. Your partner in all likelyhood meant 4C as GERBER and your 4NT showed 3 aces (out of 4, out of 5?). Maybe he was thinking 3 is not enough out of five and he stops.
As for ranking the bids...
4NT seems safe as partner will never pass this as it is "blackwood"... whoops, a slip of the ship :-)
4D is certainly ok, as 3D followed by 4C is slam try. You seem very happy to move towards slam with your hand, 4D to see what partner has in addition to his announced club ace.
4S is not so useful to a partner who might be looking at a Q high heart suit, surely he would no longer want to cooperate if you bypass a 4H cue here.
The main problems with each of these bids is that an unknown pickup partner might shock you and pass. Can you imagine bidding only 4D and having partner pass, or worse yet 4H to hear the same (GACK), or in this case your 4NT blackwood getting passed.
Seeling all the hands, it is easy to bid 6D, but this is what south should probably bid anyway. He is staring at a great diamond suit, a no loser spade suit opposite a known diamnd fit, a max of one heart loser, and clearly no club losers after the cue-bid. Leave science here for the experienced partnerships. Your partner invited you to slam and you hold a lot better hand than you could have for your reverse and then retreat to 3NT. Blast away to 6D... avoid random pick partner confusion when the hand is so clear cut.