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North
AQ76
KJ6
K7
A832
South
J32
AQ843
QJ6
75
W
N
E
S
1
1
P
2
P
2
P
4
P
P
P

The N-S hands are original, as they came up in the Turkish Open Winter Teams Championship in 2018; however, E-W hands and the bidding have been altered to make the problem catchy.

4 is an easy contract when s are 3-2. Finding no reason in ducking, we put up A (East giving K) and cash K and J. W follows with 2 to the first round, but discards 4 on the second. Bad news! The contract is in big jeopardy.

Or perhaps good news! If s were 3-2, it would be an even board; however, now there may be a chance to win 10 imps.

Since E opened the bidding, she/he must have all the missing honors. If E also has four or more s, when she/he gains the lead in  for the first time, (s)he cashes a  and plays another one forcing us to ruff in hand. Now, in case we clean out trumps, we should not let E in again; otherwise (s)he scores their last .

Is there a way to overcome this problem, so that 1-4  break turns out to be good news?

North
AQ76
KJ6
K7
A832
South
J32
AQ843
QJ6
75
W
N
E
S
1
1
P
2
P
2
P
4
P
P
P

Let’s consider E’s likely distributions, neglecting a singleton or .

E may have 3-4-3-3, 2-4-3-4, 3-4-2-4, 2-4-2-5.

• If E has three s (i.e. 3-4-3-3), no problem, the contract is a make with any line.
• What to do if E has 2-4-3-4?

LINE-1: In this case, we plan to discard a on the third , therefore play K at the fourth trick. E wins A, cashes a and plays a . We ruff in hand, cash QJ discarding our last in dummy, and play a . E wins K; if (s)he plays their last , we ruff in dummy getting home.

• What if E has 3-4-2-4?

LINE-2: This requires a subtler approach. At trick four, the only card leading to success is 7. If E hops up with A, cashes a and plays another ; we ruff in hand, cash two rounds of s (pitching a from dummy on the last one), go to K and exit with the . E is endplayed.

• If E has 2-4-2-5, no line seems to work.

Which line to select?

At trick four, we need to make the critical decision as to which to play: K or 7?

North
AQ76
KJ6
K7
A832
South
J32
AQ843
QJ6
75
W
N
E
S
1
1
P
2
P
2
P
4
P
P
P

Even though LINE-2 is more elegant, E will hold 2-4-3-4 considerably more often than 3-4-2-4, because there are fewer empty spaces in E.

A suit-break calculator (like that of Pavlicek’s) gives 41.96 % for 5-3  (3 in E) and 27.97 % for 6-2  (2 in E) when W has 9 and E has 5 vacant places (assuming E has four s in addition to four s).

Did we gain the 10 imps?

Sorry, no result, because the original E-W hands didn't give rise to any trouble.

NOTE: I made use of the Deep Finesse to affirm the analysis.

EDIT: "trick three" was corrected to "trick four".