This is problem 9:
| ♠ | 5 2 |
| ♥ | A K 10 7 |
| ♦ | K 8 6 |
| ♣ | A K 4 2 |
| ♠ | K 6 |
| ♥ | Q J 5 3 |
| ♦ | Q 7 5 |
| ♣ | Q J 9 3 |
West opens 1♠, North doubles, East passes, and you wind up in 4♥. Opening lead is the ♣8 with runs to the ten and your jack. Everyone follows to the first round of trumps. Now what? [You may want to stop here and think about it.]
As the solution points out, if trumps split you can duck a spade from both hands and later exit the ♠K to force West to lead away from the ♦A. (Presuming that West has both Aces for his opening bid.)
What if trumps are 1=4, though? [Again, consider stopping to consider.]
Now you run trumps and clubs coming down to:
| ♠ | 5 2 |
| ♥ | |
| ♦ | K 8 6 |
| ♣ | |
| ♠ | K 6 |
| ♥ | |
| ♦ | Q 7 5 |
| ♣ |
You have 2 options here: if West has 2 spades, duck a spade and again he'll have to lead away from ♦A. If he has 3, then you can lead a diamond to the K and duck one on the way back. Bourke says if you're unsure of West's distribution, you should play East to have the most even possible distribution in spades and diamonds, or exactly one more spade. This is good general advice, but ignores an advantage to the 2nd line: if West started with 2 or more of the jack, ten or 9 of diamonds then a diamond to the K and covering East's jack or ten on the way back will work even if you misguess his shape.
If West started with 10 spades & diamonds, this advice is good enough. If he pitches 2 diamonds, then playing diamonds is guaranteed to work (assuming he didn't open 1S on 4=1=6=2). If he only pitches 1 diamond, then the key cases (for RHO's initial holdings in spades=diamonds) are 2=4 and 3=3. By my math (similar to below but not shown) this is 1:4. Playing diamonds will work in the first case or half the 2nd case (again see below), for 60%, so play spades and make it 80% of the time.
But, say West started with 3 clubs and so 9 spades and diamonds. Again, if he pitches 2 diamonds you can't go wrong. But if he pitches 1, the relevant cases (again for East, though you could also think of these as diamond layouts) are 3=4 and 4=3. This time, the odds (ignoring the auction) are 1:2 (I'm placing West with 2 Aces and the SQ but allowing all the other spades & diamonds to be distributed randomly). Bourke's advice is to go with 4=3 and so play on spades. But, playing on diamonds will work in the 3=4 case and half the 4=3 cases and so it's essentially a toss-up. (I get "half" since each defender has 3 non-Ace-diamonds and West needs a minority of 3 middle diamonds to defeat you with perfect defense).
So, this isn't really an error. However, I do think that going into more depth suggests that the alternative line (when West starts with 3 clubs) is slightly better. When you consider other possible inferences, everything points to the alternative: Perhaps East would have acted over 1♠-X with 4=4=3=2? Perhaps with AJxx or ATxx in diamonds, West will fail to unblock the low honor?
With no Jacks, West might not have opened (e.g. if he follows the "rule of 20"), or he might be more likely to open a 6133 hand than a 5143 hand, and/or a hand with good diamond spots (the 10-9).
What if we assume a religious rule of 20 West and place him with 1 or both key Jacks? Since I've come this far I'll try to work it out:
1. (40% of the time) West has SJ, East has DJ. Diamonds started 4=3 vs 3=4 is 2:1 (2=4 spade spots half as likely as 3=3 while diamonds-below-J are 3-2 in either case). Ducking a spade works 2/3, playing on diamonds works 1.6/3 (1/3 + original AT9x) + as much as another .6 vig if West might fail to unblock from ATxx.
2. (60% of the time) West has DJ, SJ in either hand. 4=3 vs 3=4 is again 2:1 (7 spades 3=4 or 4=3 equally likely while 5 diamonds twice as likely to be 2=3 as 1=4). Here, the recommended line is a 2/3 shot, while the alternative is 2.4/3. The possible vig of an unblock failure gets it to 100%.
Ignoring the vig, this makes the recommended line 2/3 and the alternative 2.08/3, for a small advantage. More intuitively, the lines started as a tossup, but West should be a bit more likely to have the DJ than open spaces suggests and that helps the alternative line by making it more likely that West can't (or won't) be able to get off lead even when he shows up with 4 diamonds.
Finally, if West has both Jacks but no SQ another diamond spot could be important to whether he opens. This again points to the alternative line.
Nice post.
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