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Computer Bridge: Single-suit Plans in Notrump
(Page of 11)

I. Introduction

This article presents thoughts on how a computer-bridgeplayer, often called a "bridge robot" (and which I will call, "Robot"), could after seeing Dummy but before calling for a card in a notrump contract, use the card-combination catalog (see https://bridgewinners.com/article/view/computer-bridge-card-combinations-in-notrump/) and the opening-lead catalog (see https://bridgewinners.com/article/view/computer-bridge-opening-lead-vs-notrump/) to identify single-suit plans that might help Declarer make the contract and those that might help set the contract. These could then be synthesized into a plan for Declarer that begins with the first play from Dummy (which synthesis is beyond the scope of this article).

II. Summary of How Robot Would Identify Single-suit Plans that Might Be Relevant

The method discussed here can be summarized as follows:

  • For a given holding implied by the opening lead, identify how the defenders' cards in the other three suits could be distributed so as not to give LHO a more desirable opening lead (and therefore would be consistent with the holding from which the opening lead was presumably made).
  • Count the number of each side's masters and determine the number of additional winners needed to make the contract and the number of additional defensive winners needed to set the contract.
  • For each side, use the card-combination catalog to identify single-suit plans that match the declaring side's hands, whose defender layouts are consistent with the inferences, and that in the abstract can provide at least one winner to that side and not too many to the other side. (These single-suit plans might encompass one trick or multiple tricks.)

III. Determining Which Spots Held by the Defenders Could in the Abstract Take Tricks

Imagine Robot as Declarer in the following setting:

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

This leaves the defenders the following cards:

  • Spades: 9 8 7 6 5 4 3 2, which are effectively eight x's.
  • Hearts: 10 9 8 7 6 4 3 2, which again are effectively eight x's.
  • Diamonds: K Q J 4 3, which are effectively K, Q, J, and two x's (that is, no distribution of the defenders' diamonds would enable the 4 or 3 to possibly take a trick without Declarer's conscious cooperation).
  • Clubs: A K 9 8 7, whose spots need to be preserved to reflect promotion and trick-taking potential.

IV. Assumed Priorities in Leading against Notrump

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

According to the defenders' agreement (and taking into account the declaring side's cards), the opening lead of the heart eight would suggest the highest of three spots or the higher of two spots. The cards held by the defenders might have been distributed so as to present more attractive opening leads than the one chosen. The opening-lead catalog will help identify such holdings, which Robot can then infer are not held by the opening leader.

Assume that for auctions (such as this one) in which no one has shown or requested the lead of a particular suit, the opening-lead catalog ranks the various leads based on the priorities recommended by Richard Pavlicek (at http://www.rpbridge.net/4g00.htm), the top five of which are

  1. Five-card or longer suit
  2. Safe honor sequence (which, based on Richard's examples, I understand to mean a holding headed by at least three equals, or a doubleton consisting of two equals, and I'll assume that either must contain at least one card higher than the nine and that AK-tight would be an exception)
  3. Four cards with neither ace nor king
  4. Worthless three cards
  5. Worthless doubleton

 

Within a category, Richard suggests choosing a major suit over a minor suit.

V. Applying Assumed Opening-lead Priorities to the Defenders' Cards

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

Robot would first consider the possibility that the opening lead is from a worthless tripleton.

Robot would now identify holdings made up of the defenders' non-heart cards, that would fall into one of Richard's top three and thereby be deemed not held by West.

1. Five-card or longer suit

  • Spades (x < 10): xxxxxxxx, xxxxxxx, xxxxxx, xxxxx
  • Diamonds (x in {3, 4}): KQJxx
  • Clubs: AK987

2. Safe honor sequence (excluding any in #1)

  • Spades: None
  • Diamonds (x in {3, 4}): KQ, QJ, KQJ, KQJx
  • Clubs: None

3. Four cards with neither ace nor king (excluding any in #2)

  • Spades (x < 10): xxxx
  • Diamonds (x in {3, 4}): QJxx
  • Clubs: None

This would leave West with the following holdings that would not take priority over three worthless:

  • Spades (x < 10): x, xx, xxx
  • Diamonds (x in {3, 4}): K, Q, J, x, KJ, Kx, Qx, Jx, xx, KQx, KJx, QJx, KQxx, KJxx
  • Clubs (x in {7, 8, 9}): A, K, x, AK, Ax, Kx, xx, AKx, Axx, Kxx, xxx, AKxx, Axxx, Kxxx

Adding West's inferred three cards in hearts to the maximum length among each other suit's not higher-priority holdings gives 14, so West's inferred distribution would be 3=3=4=4 minus one (except for hearts), that is, 3=3=4=3, 3=3=3=4, or 2=3=4=4. The opponents have agreed that acting over 1N with any of these shapes would require more than the 12 HCP available to West (if West held the K, Q, A, and K). Therefore, Robot will infer that any of the not higher-priority holdings (given just above) that are consistent with at least one of LHO's inferred shapes would be available to complete West's hand.

Returning to the possibility that West led from a doubleton, that would imply at most two spades (otherwise, having longer spades than hearts would have given priority to the former). But combining at most four cards in the majors with at most eight cards in the minors would not sum to 13. Therefore, Robot can infer that West led from three hearts rather than two.

VI. Taking Inventory of Each Side's Masters

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

Robot would now count each side's winners in the form of original (that is, dealt) masters.

The declaring side has the following numbers of original masters in each suit:

  • Spades: 3 (Original masters are counted only to the extent that they can take tricks, that is, up to the length of the side's longer holding.)
  • Hearts: 3
  • Diamonds: 1
  • Clubs: 0

They sum to 7, requiring that the declaring side create two winners to make the contract. (The adequacy of entries to the original masters would also need to be considered but is beyond the scope of this article.)

The defending side has the following numbers of original masters in each suit:

  • Spades: 0
  • Hearts: 0
  • Diamonds: 0
  • Clubs: 2

They sum to 2, requiring that the defending side create more than two winners to defeat the contract. (Although not applicable to this deal, Robot would ideally consider whether the count of the defenders' original masters might be constrained by the length of that side's longer holding.)

VII. Identifying Single-suit Plans that in the Abstract Could Benefit the Declaring Side

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

Robot would use each of the declaring side's holdings in a given suit to search the card-combination catalog for any single-suit plans that in the abstract (that is, without regard to any inferences drawn on this particular deal) could generate at least one winner for the declaring side, without yielding enough defensive winners to set the contract. A single-suit plan could consist of one trick or several tricks.

Here are some of the single-suit plans that would qualify:

  • Spades: None (being that the declaring side holds at least as many masters as its longer holding does cards)
  • Hearts: None (for the same reason)
  • Diamonds: Several, including (informally stated) lead the first two rounds from Dummy and duck in the closed hand unless RHO plays an honor (in which case, Declarer rises and continues the suit). This could generate two winners while turning two of the defenders' original nonmasters into winners (whether by finesse or by establishment).
  • Clubs: Several, including (informally stated) lead an honor on each of three rounds, which works for 3-2 breaks. (Although not the best single-suit plan in the abstract, it would be included in case the entry-situation or probable layouts precluded those that in the abstract would be superior.)

VIII. Identify Single-suit Plans that from Inferences Could Benefit the Declaring Side

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

For each single-suit plan identified on the previous page, Robot would use the card-combination catalog to determine how it would fare in each full single-suit layout that's consistent with the defenders' actions thus far. "How it would fare" would include number of winners generated for the declaring side, number of defensive winners generated, number of tempi expended (that is, number of times the defenders would gain the lead), and where the lead would need to be from on a given round of the suit. The following gives some of the information that Robot would generate.

  • Diamonds: The analysis on page 5 implies that LHO holds three or four diamonds (if KQxx or KJxx). Therefore, leading (or leading to) the diamond ace on the first round and continuing diamonds should ultimately generate two winners for the declaring side and two defensive winners. Against KQxx or KJxx with LHO where that player wins the second round but ducks the third round, this would entail the declaring side's leading round 1 from either hand, round 3 from either hand (after regaining the lead), round 4 from the closed hand (after crossing), and round 5 from the closed hand (after regaining the lead). Round 2 is omitted here because it would entail leading from the same hand that won round 1, whereas this discussion is meant to help determine the declaring side's demand for outside entries.
  • Clubs: The analysis on page 5 implies that LHO holds three or four clubs. Therefore, Robot should refrain from expending Dummy's queen or jack on the first round, in case it is necessary to subsequently lead twice through LHO. The most entry-economical line would be to lead the first round toward the queen, and if it loses to LHO, lead twice from the closed hand (after regaining the lead, and the third-round lead-constraint assumes that RHO shows out on the second round) and ultimately cash the fourth round from either hand.

IX. Identify Single-suit Plans that Could Benefit the Defending Side

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

Robot would search the card-combination catalog for layouts/play-plans that match the declaring side's hands and that could produce winners for the defenders. (This is without regard to inferences drawn in the current deal.) For diamonds, matching layouts would have the holdings depicted below for Seat A and Seat C and would have Seat D or Seat B on lead. (The question marks below indicate criteria that aren't specified.)

              A76xx (Seat A)

? (Seat D)                   ? (Seat B, leader)

                 T98 (Seat C)

 

                 A76xx (Seat A)

? (Seat D, leader)           ? (Seat B)

                 T98 (Seat C)

The query with the criteria depicted in the first layout would return all possible distributions of the unspecified five cards (K, Q, J, x, x) for pair B/D except where Seat B is void. The query based on the second layout would be analogous except that it would instead exclude where Seat D is void.

Among the single-trick scenarios based on the first layout would be the following:

                 A76xx (Seat A)

J (Seat D)                       KQxx (Seat B, leader)

                  T98 (Seat C)

There would also be a single-trick scenario that differs from the above only in that Seat A would win the ace instead of ducking.

For each single-suit plan, Robot would obtain (among other things) from the card-combination database the number of winners generated for the defending side, number of winners generated for the declaring side, number of tempi expended (that is, number of times the defending side would lose the lead), and where the lead would need to be from on a given round of the suit.

Here are some of the single-suit plans that would qualify:

  • Spades: If 4-4, it would take the defenders three leads (from either hand) to establish one winner, and one more (again from either hand) to cash it. If 3-5, the same number of tempi would be required to establish any winners, but the long hand would need to get in to cash the two long-card winners. (Other distributions of the spade suit would be noted but are omitted here.)
  • Hearts: Similar to spades except that the defense is one tempo closer to establishing its trick(s).
  • Diamonds: Barring blockage, the defenders can establish two winners (or take one by finesse and establish another) during one "possession" (a term borrowed from American football, for a series of tricks in which the same side leads). Taking both tricks with defenders' original nonmasters would establish two winners for the declaring side.
  • Clubs: If 5-0, the defender holding the suit can establish a winner by leading it twice, but at the cost of establishing two winners for the declaring side. The defender with the club winner would need to regain the lead to cash it.

X. Identify Single-suit Plans that from Inferences Could Benefit the Defending Side

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

Robot would now exclude from the single-suit plans identified on the previous page, any whose leading side's holdings are inconsistent with the inferences drawn thus far (as reflected in the analysis on page 5).

  • Spades: The analysis on page 5 implies that the suit is 3=5 (for which the single-suit plan was discussed on page 9) or 2=6.
  • Hearts: The analysis on page 5 implies that the suit is 3=5 (for which the single-suit plan was discussed on page 9).
  • Diamonds: The defenders can establish two winners (or take one by finesse and establish another) in a single possession (barring blockage), as was the case on the previous page.
  • Clubs: A 5-0 break would be inconsistent with the analysis on page 5.

XI. Using the Retrieved Single-suit Plans

Robot would use the single-suit plans from pages 8 and 10 (including the winners created for each side, entries required, tempi required, and other information) to construct scenarios involving both sides and thereby determine the series of single-suit plans that, given the layouts consistent with the inferences thus far, would provide the best chance of making the contract. This would drive Robot's decision on what to play from Dummy at Trick 1. How Robot would accomplish this is a topic for future writings.

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