These hands are from Adventures in Bridge, September 27, 2018.
South (Dummy): ♠K1072 ♥KJ8 ♦K543 ♣Q10
East: ♠Q93 ♥752 ♦A976 ♣952
You passed as East and South opened 1♦. Partner bid 1♥ and North bid 1NT (8-10 in competition). You raised partner to 2♥ and South's 2NT end the auction.
You led the ♥7, Declarer played ♥8 from Dummy. Partner produced the ♥9 and Declarer took the ♥A. Now Declarer played the ♦Q. Don't even think about it! Play that the ♦A and put the ♥5 on the table. It is the duty of the weaker partner to use his/her entry (usually only one) as soon as possible in order to return partner's suit in no trump, before her entries are removed. Partner has overcalled a weak heart suit (apparently ♥Q109xx), so she must have good stuff outside. She probably has both black aces, but she cannot attack hearts from her side of the table. If you grab the ♦A and play another heart, you set the contract. If you duck twice in diamonds, Declarer will switch to clubs and your partner will be end played, allowing declarer to make.
East (Dummy): ♠K954 ♥Q762 ♦6 ♣QJ65
North: ♠1076 ♥983 ♦84 ♣AK873
The opponents are playing the hand in 3♥ with your partner having opened and rebid diamonds.
You lead the ♣A. Partner gives you the ♣4 (standard signals) and Declarer plays the ♣2. So, you duly switch to the ♦8. Partner takes the ♦A (denying the ♦K) and cashes the ♥A, which you know is singleton from the auction because West is known to have five hearts from the bidding. Now partner plays the ♣10 to you and Declarer follows with the ♣9. What is going on?
Partner is drawing you a roadmap. She asked you NOT to continue clubs, so you switched. She told you there is no future in diamonds because she does not have the King and Dummy has a singleton. There is clearly no future in clubs. She must be desperate for you to play a spade from your side of the table. If you put a spade on the table, partner will score her ♠AQ tenace, and the opponents will be down two for the magic +200.
West: ♠A6 ♥AK1095 ♦9872 ♣A8
East: ♠Q105 ♥6 ♦AKJ3 ♣K10943
You are declaring 3NT with the ♠2 lead which runs to your ♠10. You cash the ♦A and the singleton ♦Q falls on your right, so ♦10xxx was on your left.
You need to tackle clubs next. Your ♣10 and the ♣9 are important cards and even the ♣8 turns out to make a difference. So does the goal, i.e. whether you want to maximize your chance of four tricks, for example to make your game at teams, or maximize your average number of tricks. First, consider ♣Ax instead of ♣A8. If you cash the ♣A and the ♣K, you will pick up at least four clubs tricks if (1) if clubs divide 3-3 (36%) or (2) clubs are 4-2—a more common occurrence (48%)—and one of the opponents has a doubleton honor (60% = 9 out of 15 doubletons). Your overall chance is 36% + 60% × 48% = 65%. The alternative is to finesse the ♣9 on the second round, and play the ♣K next. This hurts when you lose to LHO's doubleton club honor (12.92%) but gains when RHO has ♣QJxx (9.69%) or ♣QJxxx (4.84%), gaining 14.53% of the time, making it the very slight favorite in isolation. But even the barest hint that LHO has more clubs than RHO will make the first line the favorite. Given that LHO has made a fourth best lead in spades and has a singleton diamond, it is extremely likely his shape is 4=4=1=4, so a 4=2 club split seems probable. Also even without the 4=4=1=4 shape inference, one must consider whether RHO will have the intestinal fortitude to duck with either ♣QJxx or ♣QJxxx. If you think there is even a small chance the RHO will split honors, then the first line is again favored.
The ♣8 changes things because it allows 5-1 breaks to be handled. Running the ♣8 on the second round guarantees three tricks, and maximizes your average number of tricks. Eating the ♣8 with the ♣K destroys this 5-1 break gain, but maximizes your chance for four tricks (note: if LHO has drops an honor under the ♣A, run the ♣8). Being able to pick up stiff honors on either side improves the 65% line above by 4.8% to bring it up to 69.44% for four tricks.
Now consider the ♥AK109x holding in dummy but suppose you had a low doubleton opposite, instead of stiff, to make it interesting. The practical percentage play is to finesse twice. Compare the alternative lines. If you cash the ♥AK immediately, you gain two tricks in the rare case of ♥QJ offside (1.61%), and one trick in the case of ♥QJx offside (7.11% = (4/20 tripletons) × 35.5%), but loses two extra tricks to ♥QJxx onside (9.69%), for a loss of 0.09 tricks on average (2 × 9.69% − 7.11% − 2 × 1.61%).
Curiously though when you look up this card combination in J.M. Roudinesco's The Dictionary of Suit Combinations or play it out with the SuitPlay program, the best line is to finesse once and finesse again only if the first finesse loses or an honor drops on your left. In other words if the first finesse succeeds, play off the ♥AK. Technically this is correct but it assumes RHO will play low with a doubleton or tripleton honor on the first round. That is not a crazy defensive play but it requires a strong or creative defender. Allowing for the typical spectrum of defenders, you are better off with the double finesse in the absence of other distributional information.