[pescetom]

You hold ♦J985 in hand and ♦AQ63 in dummy and need four tricks from the suit.

What is the best way to tackle this, with no prior knowledge about where the remaining cards may lie?

[jdiana]

This is very similar to hand #5 in Root's How to Play a Bridge Hand (p. 7). "The only chance to win four tricks is to find West with a doubleton king. So lead low from your hand and finesse the queen. If the queen wins, cash the ace and hope for the king to drop. Note that leading toward the jack is a hopeless play."

[DavidKok]

The presence of the 9 and 8 allows you to pick up certain other distributions as well. I'm not sure what the optimal play is, but my thoughts are:

With the king offside we can only take four tricks if it is bare, we play the ace, drop the king and then run the 9 (or 8) finessing against the Txxx. This line loses against most other distributions, so I am prepared to pay out if the king is offside. So, focusing on king onside (by assumption), the question now turns to the remaining diamonds. There's 16 distributions to consider for the remaining four diamonds. Placing AQ63 in the North hand and J985 in the South hand, if we decide to finesse the queen first round we always lose to KTx(xx) with West (7 distributions)- they will duck first round, and then retain KT(xx) over our jack. Then there are 3+3 mirrored distributions - Kx with West, Txx with East versus Kxx with West, Tx with East - and after a successful finesse we have to choose which to pick up (playing West for a doubleton: cash the ace. Playing East for a doubleton: run the jack, pinning the ten). Regardless of our further play we can therefore never pick up more than 6 out of 16 distributions. Running the nine first round has the same chance of winning, winning if West has K, KT, KTx or KT42 (but not if West has KT7x, where after covering with the ten the K7x sitting over our J85 will establish a trick).
I can't see any other play that improves on this - running the jack is poor, slamming the ace is poor and therefore we should start by either running the 9 or taking the finesse, both of which have the same success rate. Maybe one or the other is slightly indicated by opponents' length in the other suits.

[Stephen Tu]

There are a ton of equivalently successful lines, in terms of picking up 4 tricks, according to suitplay. Basically you can play for Kx onside or Tx offside, or you can can cater to KTx/KTxx onside (not KT7x), and you can either pick up stiff K onside (don't run J first) or stiff T offside (run J first, or low to Q first).
But for cases where you don't pick up 4 tricks, the safest way is low to Q, then run the 9 if the Q won and the K/t don't appear second round (don't lose 2nd trick to KTxx onside). If the Q lost, play ace and still pick up KTxx off for one loser. Other lines while equally successful at picking up 4 tricks, are less consistent at taking at least 3.

[pescetom]

Thanks to all who replied.

It was MP, so yes taking at least 3 tricks was also important as Stephen Tu considers.
In real life there was of course some information about the probable layout, suggesting the K onside in a probable three card suit: but I didn't want to bias the discussion with this.
IIRC, I played 5,7,Q,4 and then 9,T,A with RHO showing out.
In any case one down like everyone else, but awareness that I hadn't really nailed down the probabilities of all possible options.