From the lead, South has Jxxx clubs.  In principle, a player who leads from Kx against a partscore contract normally has - or ought to have - a trump control.  Otherwise the lead is too risky.  After the lead you have 3 clubs and partners assumed spade winner.  The ♦A makes 5.  There isn't really room for him to hold the ♦K or ♥A, but you could play for him to hold ♦xxx. If you continue with a third and fourth round of clubs, partner can pitch 2 diamonds.  He can then win the ♠A and obtain a diamond ruff instead of the club ruff that he was originally expecting.

Declarer might have queen doubleton in hearts so you play off the two top hearts and find to your dismay that West, who started with ♥Q85, fails to unblock the queen.  After cashing your 3 heart tricks, declarer is able to win any return and cash 9 tricks.  Could this disaster have been avoided or is it just one of those things?  A simple convention (commonsense play really) is 'unusual play calls for unusual play'.  Knowing that you want your partner to unblock if he holds Qxx, you should play off the Ace and King of hearts in that order - rather than the King and then the Ace.  There are many similar situations where this kind of play will help, for example when dummy has xxx and East leads the suit from AKJT

Consider the diamond suit, West would not underlead the Ace so East has AK.  Presumably he lacks the Queen else he would have played that card so West holds the Queen but not the Knave for he would not have led low from QJ.  The spade honours must be split - if East held both he would have had an opening bid.  Hence the likelihood is that West holds both the ♥Q and ♣Q else again East would have opened the bidding.

So win the ♣A and run the ♥6 at trick 3 in case west holds 4.  Then draw trumps and play clubs from the top - hoping to make 3 club tricks.  You hope the layout is as shown.

Note that the light take-out double backfires against a declarer who is listening to the bidding

This is not a complicated affair, but is an area where misunderstandings occur.  You must certainly switch to a diamond.  If you instead play a heart say, declarer will have enough entries to establish a tenth trick by means of a ruffing finesse against partner's queen of clubs.  Also, South is not that likely to hold the ♦K after he has opened 3♠.  At the table, West cashed the ♦A, East encouraging, and continued with the ♦Q.  After some thought, East overtook the Queen with the ♦K and returned another diamond for West to ruff!  East played west for a doubleton diamond and thought that instead of leading the Queen, West would have done better to continue with ♦2.  Then if East takes this for a doubleton and returns the suit, no harm is done.  However, might not East think that West holds 4 diamonds?  If that is the case, East might well reason that West has a singleton club and return a club for West to ruff.

The solution is for West to switch to the ♦Q at trick 2.  When this holds, he continues with Ace and another and the scope for misunderstandings is eliminated.

You have 7 tricks and need to develop 2 more from spades and diamonds.  Clearly if the diamonds break 3-2 there are no problems.  If they are 4-1, you need to tackle the suits in the right order to give yourself the best chance of success.  Start by playing a top diamond from dummy.  If an honour falls on your right, unblock the ♦9.  Now continue with a low diamond to your 8.  If West wins this (from an original holding of HHxx), he sets up a finesse for you to gather 4 diamond tricks, giving you at least 9 tricks in total.  He therefore ducks the diamond and your 8 wins.  Now you have 3 diamond tricks without losing one, so you can turn your attention to spades, making 4 tricks whenever the suit breaks no worse than 4-2.  You will then make 4/5 spades, 2 hearts and 3 diamonds.

On this hand you simply need to count your tricks.  You have 4 red winners and can make 6 trump tricks even if either player has 4 trumps in defence.  All you need to do is discard a heart on the third club trick.  You can win the next lead and play off a top spade in hand.  When you discover 4 spades with West, a spade finesse allows you to pick up the suit.  If you mistakenly ruff high in hand at trick 3, you can no longer cope with West holding 4 spades.  

You can count eight winners and another four if spades prove to be 3-3. You need to consider what might be done if the spades are not 3-3. You have an extra chance if West holds the ace-doubleton in spades. After drawing two rounds of trumps with the king and queen, lead the ♠2 from hand. If West rises with the ace, you have four spade tricks, so West follows small and dummy’s king wins the trick. Now a low spade is played from dummy to your jack and West’s ace.

This leaves West on lead with only clubs and hearts. If he exits with a club, you ruff in dummy and throw a heart from hand. After cashing the ♠Q, you throw a second heart from hand, ruff a spade to establish a spade trick in dummy. Then cross back to dummy with a diamond and discard your ♥ J on dummy’s spade winner.  A heart exit from West at the critical point gives you an even easier ride.

You have 3 tricks and if partner holds the ♣K he will surely make it.  The more interesting card is the ♥K.  If declarer holds ♥ AQ, a finesse will give him the contract.  Can you talk him out of this winning line?  One possibility for declarer is to play for a squeeze which would operate if you held ♥K as well as 4 clubs.  You might try nudging him in this direction by discarding ♥J.  This is a good try but there is a better play.  What about underruffing, like a man seeking to postpone the evil day?  When later you discard two hearts, declarer might assume you have left yourself with a singleton King.  This defensive idea is quite easy to remember: when you know that you must guard one suit, pretend that you hold a critical card in another suit.

If trumps are no worse than 3-1, you have a complete elimination play available.  Duck trick 1 and win the heart continuation.  Draw trumps in 3 rounds, ruff dummy's last heart and then play 3 rounds of diamonds.  The defense will then have to open the club suit or give you a ruff and discard.  However, when you play a trump to dummy at trick 3, East shows out, so you need to reconsider now that the trumps are 4-0.

The solution is a partial elimination.  Continue as before, drawing just 3 trumps and ruffing a heart, before playing 3 rounds of diamonds.  You are in luck as East has to win this trick and play a club. You duck this trick to West, who exits with his last trump.  Win this on the table and finesses the ♣10.  

To succeed when trumps are 4-0, you need the club honours split and East to hold the ♦Q.  Note that you have to draw 3 of West's trumps, else he could ruff the third diamond and exit with a trump, leaving you to play the club suit yourself.

Before playing to the first trick, you need to form a plan. You can count nine top tricks and you can add at least two more in clubs as long as the suit breaks reasonably favourably. Also, if trumps are no worse than 3-1, you  can use two diamond ruffs as entries to set up the clubs. All of this makes it imperative to win the first trick in dummy, so play dummy’s J♥ . Once East follows with the 6♥, you can put your plan into action.

At trick two, cash the A♣  and ruff a club low in hand. After ruffing a diamond with dummy’s 4♥ , cash the K♥  and ruff a club with the 10♥ . Next ruff a diamond with dummy’s Q♥ .

All that remains is to cross back to hand via a low spade to the ace to draw West’s last trump with the ace, discarding dummy’s remaining low spade. A spade to dummy’s king is followed by the K♣ , leaving dummy with the two established clubs to cash. 

The slam is no worse than the diamond finesse, and you may be able to duck a diamond to West in the endgame to ensure a diamond return to your AQ.  If your diamonds were as good as AQ9 you would have a certain endplay, but that is not the case here.  Still, you win the club lead and cash the ♠ K finding a 2-1 break. Now a club ruff, then ♥AK and a heart ruff high is followed by a trump to dummy.  Your plan is to ruff another heart and then  use dummy's last trump as an entry for the diamond finesse.  However, when you play the fourth round of hearts from the table, East shows out. Now all you need to do is discard a diamond.  West will win this trick but then must play a diamond into your AQ or concede a ruff and discard.

It looks like declarer is 2443.  To exit with a club is useless as declarer will cash 2 clubs and exit with a diamond, leaving you on play.  Whilst giving declarer a ruff and discard is normally bad, giving him 2 ruff and discards can be very effective.  When you win the first diamond, exit with a spade.  Repeat this procedure when you win the second diamond.  This removes declarer's last trump and you are now in the driving seat with a diamond trick and a long spade to cash.  Have a look at all four hands.  Could declarer have done better?

Yes he could - If declarer takes just 2 rounds of trumps and his 2 club tricks before exiting in diamonds, West's defence no longer works.  South takes the force in hand, discarding a club from dummy, and plays another diamond. Now a further spade from West allows South to ruff with his last trump, thus making 6 trumps, 2 spades and 2 clubs for his contract.

There are 4 top tricks in the black suits and you can probably count on 4 trump tricks in hand, so you need 2 ruffs in dummy.  Win the ♠A and play AK and ruff a club.  Return to hand with the ♠K and play another club.  If West follows, it is dangerous to ruff with the ♥J as this may be overruffed and you might still have another trump to lose.  It is better to discard the losing spade on the fourth club.  Let's say West continues with a trump.  You win in hand and then ruff a spade with ♥J.  You still have a trump and a diamond to lose, but that is all. 

It would be asking for trouble to play off the ♠AK.  If trumps are 4-1, you would then have to turn your attention to clubs and if the defender with ♣A has the long trumps, he will gleefully draw your trumps and the defence will have several hearts to cash.  Equally, ruffing a heart at trick 2 is no good as that  weakens your control of the heart situation.  You can afford to lose 2 trumps and a club, so win ♥A and cash just one top trump.  Then knock out the ♣A.  If an opponent takes the first club and forces dummy with a heart, then play a second club before taking the second top trump.  Now a 4-1 break is under control as you just ruff another heart and discard your last heart on a club.  Eventually, your diamond loser goes on the long club and you just lose at most 2 trumps and the ace of clubs.

Many declarers would look no further than running the ♦9 at trick 2.  In fact, this is a vastly inferior line, and will fail on this occassion as alook at all 4 hands will show.  The best line is to cross to dummy's ♦A and trick 2 and then play a club towards hand.  If East takes the ace, the club suit will provide enough tricks for the contract on a 3-2 break.  If the ♣K loses to West, he cannot profitably continue hearts, and you can then knock out the ♦K.  If the ♣K holds, then you revert to playing diamonds and must make 9 tricks.  The total chance of success with this line is 85%, a great improvement on the straight diamond finesse.