Best line in 3NT

[JavaBean]

In a declarer-play book I've been working through, I found the following. I don't think the suggested line is the best one, but I'm wondering if there's something I'm missing.

 

[hv=pc=n&s=sa8haq63dj65ca532&n=s42hk84dat42cq764&d=s&v=0&b=11&a=1n3sdp3nppp]266|200[/hv]

 

He doesn't mention vulnerability, but it's IMPs so presumably the goal is to make your contract. West leads ♠K.

[Mbodell]

If I play W for 7 spades to the KQJ and either nothing outside or a diamond honor outside don't I duck the first round, win the second, and play 4 rounds of hearts (K,A,Q,x and pitching a club). If the hearts were 3-3 now a diamond to the T. And if they weren't presumably East is on lead and we'll duck a club to the Q and continue with the A and another club. Or if E exits diamonds after the 4th heart, duck a diamond to the T (or A over an honor) and then continue with the diamond A and another (or low to the J if we captured an honor with our A on the first round). If W has only one heart I think I try diamond to the T.

 

Judging from the book in question and the hour of the night my line is almost certainly not right.

[JavaBean]

If I play W for 7 spades to the KQJ and either nothing outside or a diamond honor outside don't I duck the first round, win the second, and play 4 rounds of hearts (K,A,Q,x and pitching a club). If the hearts were 3-3 now a diamond to the T. And if they weren't presumably East is on lead and we'll duck a club to the Q and continue with the A and another club. Or if E exits diamonds after the 4th heart, duck a diamond to the T (or A over an honor) and then continue with the diamond A and another (or low to the J if we captured an honor with our A on the first round). If W has only one heart I think I try diamond to the T.

 

Judging from the book in question and the hour of the night my line is almost certainly not right.

 

Hearts are 3-3, and after low to the ♦10, east exits in diamonds. Jack or small?

 

FWIW your line is the same as mine, so far. I don't think I cashed the fourth heart yet, but it shouldn't matter.

[Mbodell]

Hearts are 3-3, and after low to the ♦10, east exits in diamonds. Jack or small?

 

FWIW your line is the same as mine, so far. I don't think I cashed the fourth heart yet, but it shouldn't matter.

 

I originally wanted to say small because Hx opposite Hxxx is more likely than xx opposite HHxx. But it could also be x opposite HHxxx in which case J lets us win 2 ♦, 4♥, 1♠, 2♣. I'll try small. Unlucky?

[manudude03]

Agree with MBodell, except I now play the jack when the suit is led back. If diamonds are 3-3, Q9x on the left, nothing we can do really. If it gets covered, we have to hope the ♦9 isn't on our left. Cash the ♣A (incase of stiff K) and exit a diamond. Down 4 presumably :).

 

If the jack holds and west discards, we have to hope East has the ♣K and exit a diamond.

[Mbodell]

It is possible that it is too late for me to think about any of this, but I think J has to be right now. Even if you play low and it wins (I.e., an honor appears) I think we've lost because now we'll cross to the ♦J and can't throw East in in diamonds. Which means we'll have to duck a club which only works if West's spot is smaller than all of East's. So I think you have to play the J, if it holds with west following next play the ♣a, ♦a, and exit a ♦ and we are home. If it doesn't hold we win the A and exit a diamond but we are still dead as we are only going to win 8 tricks. If it holds with West pitching now we cash the ♦A and exit with a ♦ without cashing the ♣A since East will win 3 ♦ to go with the 1♠ we can't afford to cash the ♣A.

[inquiry]

We have incomplete info.

 

We "know" that West is 7=3 in the majors, so has three minor suit cards. The most likely distribution is for West to have one club and two diamonds, second most likely is two diamonds and one club. If West has two diamonds, the correct play, as noted above is low after the diamond ten loses as there are 10 ways West could have honor small in diamonds versus only six ways he could have small-small in diamonds.

 

The "solution" to this question is figure out the club distribution before you play on diamonds. On the last heart you pitch a diamond from dummy, what does East play? If it is diamond, expect diamonds to be 1=5 or 2=4, if it is a club, expect clubs to be 1= 4 or 0=5. The best line at this point is one that will help you figure out clubs before you play on diamonds. To do so, you can play on clubs now. On a club discard, lead a low club. If West plays the lowest outstanding club, play low. If West shows out, play low. If West plays any club but the lowest outstanding club, play the queen. Now you have an accurate diamond count (at least in theory). If West has a club, he would be 7=3=2=1, if he lacks a club, he would be 7=3=3=0. I think this is a 100% line after a club discard.

 

What if East discards a diamond? The solution is a lot trickier. I think you will be back to a guess. If West is 2=1 (most likely distribution), you will have to go with the percentage and duck the diamond return. If he is 1=2 you will have to play the diamond jack. East wins an honor and returns a diamond. What are the odds?

 

If West held just one diamond, it was low. There were C(4,1) or 4 ways he could hold a small diamond. If he held two diamonds, there are C(4,2) chances it was low (6 ways) but C(5,3) or 10 ways he could have held Hx. I am not sure if the calculations should reflect the low diamond East returns or the high diamond East won with. So I will leave those numbers. Since East threw a diamond, we assume he did not hold five clubs, so west will not be 3=0.\\

 

Playing Jack on East's diamond return wins when West held singleton small diamond (4 chances) or diamond low-low (6 chances). Playing low wins if West held high-low (10 chances). Are all these chances equally likely? The 2=4 diamond splits, yes. However, is West's minor suit holding 1=2 and 2=1 equally likely? The answer here is no. The 2=1 is more likely than the 1=2. So the chances for singleton small diamond is less likely than for any doubleton diamond. Without calculating the odds, this differences pushes me to play LOW on the diamond return. Still, it is almost essentially a guess (1=2 compared to 2=1 is about 4 to 5 ratio). I usually get these things wrong the first time, but that is my logic. 1) play on clubs if east discards a club, 2) play West for High-low doubleton diamond if East discards a diamond.

[Mbodell]

Playing Jack on East's diamond return wins when West held singleton small diamond (4 chances) or diamond low-low (6 chances). Playing low wins if West held high-low (10 chances). Are all these chances equally likely? The 2=4 diamond splits, yes. However, is West's minor suit holding 1=2 and 2=1 equally likely? The answer here is no. The 2=1 is more likely than the 1=2. So the chances for singleton small diamond is less likely than for any doubleton diamond. Without calculating the odds, this differences pushes me to play LOW on the diamond return. Still, it is almost essentially a guess (1=2 compared to 2=1 is about 4 to 5 ratio). I usually get these things wrong the first time, but that is my logic. 1) play on clubs if east discards a club, 2) play West for High-low doubleton diamond if East discards a diamond.

 

Say West has Hx of diamonds and you play low. How do you make your contract? You've ducked a spade (1 loser), and then won the A. You've won 3 or 4 rounds of hearts, you've lost the ♦T to East (2 losers). If you play low and win the W ♦H with the A now what? If you play ♦J then ♣A then duck a ♣, East now wins a low ♣, K♣, 4th♦ and you are down. If you play ♦J and then duck a ♣, East can duck unless West has the smallest spot and now West runs spades.

 

Am I missing something, or does this mean we have to play the ♦J on the return.

 

Also, on the last heart (if we cash it) don't we have to pitch a club, not a diamond. We want to use the diamond to throw East in and/or as a possible winning trick if diamonds split 3-3.

[inquiry]

Say West has Hx of diamonds and you play low. How do you make your contract? You've ducked a spade (1 loser), and then won the A. You've won 3 or 4 rounds of hearts, you've lost the ♦T to East (2 losers). If you play low and win the W ♦H with the A now what? If you play ♦J then ♣A then duck a ♣, East now wins a low ♣, K♣, 4th♦ and you are down. If you play ♦J and then duck a ♣, East can duck unless West has the smallest spot and now West runs spades.

 

Am I missing something, or does this mean we have to play the ♦J on the return.

 

Also, on the last heart (if we cash it) don't we have to pitch a club, not a diamond. We want to use the diamond to throw East in and/or as a possible winning trick if diamonds split 3-3.

 

You can discard a club or a diamond on the heart, it makes no difference. There maybe a psychological advantage to the club, as East with 4-4 in the minors might discard a club, which makes our life easier. But lets stick with the diamond discard for a moment, and ASSUME diamonds are 3-3 (your reason for keeping the 4th diamond). What will East discard? The assumption is a club. Now you lead a low club (as I stated) and West shows out. You duck in dummy in this ending (it doesn't matter if West holds a high diamond or not. Here I give him one) with East to lead, yoiu have lost two tricks and can lose two more... in ohter words you need four tricks.

 

[hv=pc=n&s=shdj65ca53&w=sjt9hdq73c&n=shdat4cq76&e=shdk82ckjt]399|300|A club allows you to win two clubs and then endplay East in clubs for a diamond lead. A high diamond from east, you duck (in case West has a high diamond). A low diamond, you can run to your AT, winning when West plays a high diamond. Now a diamond towards you hand, East wins and exits a diamond, but then you duck a second club to East.

 

[/hv]

 

Ok, having dealt with the fallicy of needing to save a diamond in case of a 3-3 diamond break, we can examine the earlier, point with West having High-low doubleton diamond. Here we assume East discarded a diamond, remember, if he discards a club, we play a small club -- NOT the diamond finesse)". That means in your analysis above he will not have a diamond to cash when he wins the low club. You said: "How do you make your contract? You've ducked a spade (1 loser), and then won the A. You've won 3 or 4 rounds of hearts, you've lost the ♦T to East (2 losers). If you play low and win the W ♦H with the A now what? If you play ♦J then ♣A then duck a ♣, East now wins a low ♣, K♣, 4th♦ and you are down. " You see, that "4th diamond" you have him cashing is not available if he discarded a diamond on the 4th heart. Of course if he was 4=4 in the minors and discarded a club, we play as above, planning on losing two clubs and a diamond while keeping West off the lead.

 

[hv=pc=n&s=shdj52ca532&w=sjt98hdk3cj&n=shdat4cq764&e=shdq987ckt98]399|300| The position after the discard on hearts will be this. When the ♦T loses to East (2nd lost trick) and he returns a low diamond, you play low and the win the king in dummy with the ace. Now you cash the last diamond -- extracting East's last diamond, and cash then duck a club. East can win two clubs but your club queen wins the last trick. So yo win 1S, 4H, 2D, 2C for 9 tricks. Of course, it is still somewhat of a guess when East returns a low diamond. Was West 7=3=1=2. (have to play the JACK), or was West 7=3=2=1 but wiht two low diamonds (have to play the jack). This is where trying to calculate the odds for the best line gets a little fuzzy, and which is why low club out if EAST discards one is better line, now the location of the second high diamond honor is not significant. [/hv]

[inquiry]

I am not sure what lesson the author of the book javabean was reading wanted to show with this hand, but I think I want to make a slight change to my analysis.

 

To make the hand, you NEED east to hold the club King and at least one of the top diamonds, so we might as well assign those cards to East. That leaves four "small" clubs and five "other diamonds" including the second high one. So after you discover West has 7♠ and 3♥, that means he has 3 potential slots for the these 9 cards, and East has six potential slots for them. Let's consider for a second, these nine cards as a single suit. How can they be split?

 

West can have C(9,3) different holdings, and East C(9,6) different holdings. Either way, this comes to 84 possible holdings where game can be made C(9,3) which is 84 equal possibilities. As a check, we can test the following... #3=0 means west has 3 diamonds and no clubs, etc.

 

#3=0 is C(5,3) * C(4,0) = 10 * 1 = 10 combinations

#2=1 is C(5,2) * C(4,1) = 10 * 4 = 40 combinations

#1=2 is C(5,1) * C(4,2) = 5 * 6 = 30 combinations

#0=3 is C(5,0) * C(4,2) = 1 * 4 = 4 combinations

Total combinations are 84.

 

Ok, we have eliminated some of these. When EAST doesn't discard a club, we can eliminate West is void in clubs (losing 10 of the 84) and when we lead a low diamond and West DOES NOT play a diamond honor, we lose five of the 30 combinations (the one with a stiff diamond honor) from the 2=1 split and all of the 0-3 splits. So the corrected numbers are now....

 

#3=0 is ZERO combinations

#2-1 is still 40 combinations

#1-2 is reduced to 25 combinations

#0-3 is ZERO combinations.

NEW total is 65

 

Of the 25 remaining #1=2 splits, you have to play a diamond when East wins the diamond honor and returns a diamond. Of the 40 splits with 2 diamonds and 1 club with west, we have to determine the frequency of Honor-low versus low-low. Remember, already of the 65 possible hands, 25 require the play of the jack on a low !D back. If the 10 to 6 for High-low still exist in diamonds (and I think it doesn;t) that would make 15 times playing the JACK is right and 25 times, playing low is right. But 15+25 (1=2, and low-low=stiff) is greater than 25. So it looks like playing the JACK is right after all.

 

The reason I said it doesn't is I already forced one high diamond in East, so there is only one high diamonds (unknown) left out. Perhaps it is best to use C(4:1) for the low diamond (high low) and C(4:2) for low-low. I am not sure, but if so, then it is wayyyyyyy more likely that the both high diamonds are with East. I think I might be overthinking this one now.