Join Bridge Winners
Computer Bridge: Counting Winners versus NT IV
(Page of 10)

(These "Computer Bridge" articles present techniques that might enable a computer-bridgeplayer to emulate certain qualities of human players, such as drawing inferences from actions taken or not taken, or imagining how a layout might be perceived by another player.)

I. Introduction

The article "Computer Bridge: Counting Winners versus NT II" (https://bridgewinners.com/article/view/computer-bridge-counting-winners-versus-nt-ii/, May 27, 2019, hereafter referred to as "Part II of this series") discussed how a defending computer-bridgeplayer (often called a "bridge robot," and which I have been calling, "Robot") might identify layouts where Declarer would already have enough winners on power to fulfill the contract in the declaring side's first possession and the defenders could neither cash out on their first possession nor strand the contract-satisfying winner. Such layouts would be deemed "hopeless" by Robot and dismissed from further consideration.

The article "Computer Bridge: Counting Winners versus NT III" (https://bridgewinners.com/article/view/computer-bridge-counting-winners-versus-nt-iii/, June 13, 2019, hereafter referred to as "Part III of this series") discussed how Robot might identify layouts where Declarer would imminently discover enough winners on power to fulfill the contract in the declaring side's first possession and the defenders could again not prevent it. Such layouts would also be deemed "hopeless" by Robot.

These two articles used the card-combination catalog to identify ways for either side to generate winners, as discussed in "Computer Bridge: Card Combinations in Notrump" (https://bridgewinners.com/article/view/computer-bridge-card-combinations-in-notrump/, Jan. 15, 2019).

This article considers how Robot could identify layouts in which no matter how the cards yet to be seen by Declarer might be allocated, the declaring side's cards would present at least one sure way for the declaring side to establish winners on power and use them to fulfill the contract in its second or third possession.

For layouts that can provide winners to the defenders, Declarer will be assumed to select cards that according to the card-combination catalog would prevent or have the best chance of preventing the defenders from establishing enough winners to set the contract, as discussed in "Computer Bridge: Card Combinations in Notrump II" (https://bridgewinners.com/article/view/computer-bridge-card-combinations-in-notrump-ii/, June 27, 2019). The notion "prevent or have the best chance of preventing" assumes that at least one plan for Declarer to handle the suit would never yield a worse result than some other plan. If every plan would be inferior on at least one legal layout, then Robot would reserve judgement on how Declarer would handle the suit.

Robot has been assuming that Declarer does not use inferences from the defenders' actions to plan the play. Many layouts can nonetheless be identified as "hopeless" (for the defenders) and therefore dismissed. Once this has been done, Robot would need to start considering Declarer's inferences in order to resolve which of the remaining layouts are "hopeless."

This article introduces two exceptions.

  1. When Robot is imagining how the play might unfold, Robot will assume the following: Declarer will infer that during the defenders' first possession, any card played by Third Hand that outranks the cards played thus far to the trick is the lowest of its meldoid. For example, if Third Hand plays a king and it outranks the first two cards played to the trick, Declarer is presumed to infer that Third Hand lacks the queen.
  2. When the opening lead is an eight or lower, Robot will assume that Declarer will correctly infer whether the lead is (a) highest from four or fewer cards, (b) fourth-best from fewer than six cards, or (c) fourth-best from at least six cards.

 

To qualify the word "consistent," the terms "Robot-consistent" and "Declarer-consistent" will indicate whether a given layout's being deemed "consistent" is based on Robot's inferences or on the inferences Robot assumes Declarer would make, respectively.

II. Terminology

Some of these Computer Bridge articles use the following homegrown terms:

  • Possession—one or more consecutive tricks in which the same side leads
  • Series—one or more consecutive tricks in which the same side leads a given suit
  • Meldoid—set of two or more equals in a given suit in a given hand, or a single card in a given hand that has no equals in that suit in that hand

I have adopted from other bridge communities the following term:

  • Court Card—ace, king, queen, or jack

III. Hand Types for Partner that Did Not Produce "Hopeless" Layouts in Part III of This Series

Parts II and III of this series considered the following deal:

West
Dummy
Q75
K5
AK1062
952
Robot
A82
42
987
KJ1073
South
W
N
E
S
P
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
4
5
1

In Part II of this series, Robot had inferred that the following spade holdings for Partner would be consistent with the auction and play up to Robot's turn and would incorporate the highest available spots up to the third-highest card in the suit (based on the rationale that if Declarer would have enough winners even if denied these spots, Robot would be justified in dismissing other layouts involving the same distribution and allocation of court cards):

  • KJ94x, K1094x, J10x4x, 109x4x
  • J10x4, 109x4

In accordance with the second "exception" given on page 1, Robot will assume that Declarer will interpret the 4 as fourth-best from four or five pieces.

The following hand types for Partner (with Declarer's implied holding in parentheses) emerged in Part III of this series after Robot applied criteria to exclude "hopeless" layouts. These hand types will be tested against the criteria in the present article.

KJ94x

5=5=3=0:

  • KJ94x Q109xx xxx void (10x AJxx QJ AQxxx)
  • KJ94x J109xx Jxx void (10x AQxx Qx AQxxx)
  • KJ94x 1098xx Qxx void (10x AQJx Jx AQxxx)

5=5=2=1:

  • KJ94x Q109xx xx x (10x AJxx QJx AQxx)
  • KJ94x 1098xx Qx x (10x AQJx Jxx AQxx)

5=5=1=2:

  • KJ94x Q109xx x xx (10x AJxx QJxx AQx)

5=5=0=3:

  • KJ94x Q109xx void xxx (10x AJxx QJxxx AQ)

5=4=3=1:

  • KJ94x Q109x xxx x (10x AJxxx QJ AQxx)
  • KJ94x J109x Jxx x (10x AQxxx Qx AQxx)
  • KJ94x 1098x Qxx x (10x AQJxx Jx AQxx)

5=4=2=2:

  • KJ94x Q109x xx xx (10x AJxxx QJx AQx)
  • KJ94x 1098x Qx xx (10x AQJxx Jxx AQx)

5=4=1=3:

  • KJ94x Q109x x xxx (10x AJxxx QJxx AQ)

K1094x

5=5=3=0:

  • K1094x QJ10xx xxx void (Jx A9xx QJ AQxxx)
  • K1094x Q109xx Jxx void (Jx AJxx Qx AQxxx)
  • K1094x J109xx Qxx void (Jx AQxx Jx AQxxx)
  • K1094x 1098xx QJx void (Jx AQJx xx AQxxx)

5=5=2=1:

  • K1094x QJ10xx xx x (Jx A9xx QJx AQxx)
  • K1094x Q109xx Jx x (Jx AJxx Qxx AQxx)
  • K1094x J109xx Qx x (Jx AQxx Jxx AQxx)
  • K1094x 1098xx QJ x (Jx AQJx xxx AQxx)

5=5=1=2:

  • K1094x QJ10xx x xx (Jx A9xx QJxx AQx)
  • K1094x Q109xx J xx (Jx AJxx Qxxx AQx)

5=5=0=3:

  • K1094x QJ10xx void xxx (Jx A9xx QJxxx AQ)

5=4=3=1:

  • K1094x QJ10x xxx x (Jx A9xxx QJ AQxx)
  • K1094x Q109x Jxx x (Jx AJxxx Qx AQxx)
  • K1094x J109x Qxx x (Jx AQxxx Jx AQxx)
  • K1094x 1098x QJx x (Jx AQJxx xx AQxx)

5=4=2=2:

  • K1094x QJ10x xx xx (Jx A9xxx QJx AQx)
  • K1094x Q109x Jx xx (Jx AJxxx Qxx AQx)
  • K1094x J109x Qx xx (Jx AQxxx Jxx AQx)
  • K1094x 1098x QJ xx (Jx AQJxx xxx AQx)

5=4=1=3:

  • K1094x QJ10x x xxx (Jx A9xxx QJxx AQ)
  • K1094x Q109x J xxx (Jx AJxxx Qxxx AQ)

J10x4x

5=5=2=1:

  • J10x4x J10xxx Qx Q (K9 AQ9x Jxx Axxx)
  • J10x4x 109xxx QJ Q (K9 AQJx xxx Axxx)

5=5=1=2:

  • J10x4x J10xxx x Ax (K9 AQ9x QJxx Qxx)
  • J10x4x 109xxx J Ax (K9 AQJx Qxxx Qxx)

5=5=0=3:

  • J10x4x J10xxx void Axx (K9 AQ9x QJxxx Qx)

5=4=3=1:

  • J10x4x AJ10x xxx x (K9 Q9xxx QJ AQxx)
  • J10x4x A109x Jxx x (K9 QJxxx Qx AQxx)
  • J10x4x QJ10x Qxx x (K9 A9xxx Jx AQxx)
  • J10x4x Q109x QJx x (K9 AJxxx xx AQxx)
  • J10x4x Q109x Jxx Q (K9 AJxxx Qx Axxx)
  • J10x4x J109x Qxx Q (K9 AQxxx Jx Axxx)

5=4=2=2:

  • J10x4x AJ10x xx xx (K9 Q9xxx QJx AQx)
  • J10x4x A109x Jx xx (K9 QJxxx Qxx AQx)
  • J10x4x QJ10x Qx xx (K9 A9xxx Jxx AQx)
  • J10x4x Q109x QJ xx (K9 AJxxx xxx AQx)
  • J10x4x J109x Qx Qx (K9 AQxxx Jxx Axx)
  • J10x4x J109x xx Ax (K9 AQxxx QJx Qxx)
  • J10x4x 1098x Jx Ax (K9 AQJxx Qxx Qxx)

5=4=1=3:

  • J10x4x AJ10x x xxx (K9 Q9xxx QJxx AQ)
  • J10x4x A109x J xxx (K9 QJxxx Qxxx AQ)
  • J10x4x J109x x Axx (K9 AQxxx QJxx Qx)
  • J10x4x 1098x J Axx (K9 AQJxx Qxxx Qx)

109x4x

5=5=2=1:

  • 109x4x 109xxx Qx A (KJ AQJx Jxx Qxxx)

5=5=1=2:

  • 109x4x 109xxx Q Ax (KJ AQJx Jxxx Qxx)
  • 109x4x 109xxx x AQ (KJ AQJx QJxx xxx)

5=5=0=3:

  • 109x4x 109xxx void AQx (KJ AQJx QJxxx xx)

5=4=3=1:

  • 109x4x AQ10x xxx x (KJ J9xxx QJ AQxx)
  • 109x4x AJ10x Jxx x (KJ Q9xxx Qx AQxx)
  • 109x4x A109x Qxx x (KJ QJxxx Jx AQxx)
  • 109x4x A109x xxx Q (KJ QJxxx QJ Axxx)
  • 109x4x QJ10x QJx x (KJ A9xxx xx AQxx)
  • 109x4x QJ10x Jxx Q (KJ A9xxx Qx Axxx)
  • 109x4x Q109x Qxx Q (KJ AJxxx Jx Axxx)
  • 109x4x Q109x xxx A (KJ AJxxx QJ Qxxx)
  • 109x4x J109x QJx Q (KJ AQxxx xx Axxx)
  • 109x4x J109x Jxx A (KJ AQxxx Qx Qxxx)
  • 109x4x 1098x Qxx A (KJ AQJxx Jx Qxxx)

5=4=2=2:

  • 109x4x AQ10x xx xx (KJ J9xxx QJx AQx)
  • 109x4x AJ10x Jx xx (KJ Q9xxx Qxx AQx)
  • 109x4x A109x Qx xx (KJ QJxxx Jxx AQx)
  • 109x4x A109x xx Qx (KJ QJxxx QJx Axx)
  • 109x4x QJ10x QJ xx (KJ A9xxx xxx AQx)
  • 109x4x Q109x Qx Qx (KJ AJxxx Jxx Axx)
  • 109x4x Q109x xx Ax (KJ AJxxx QJx Qxx)
  • 109x4x J109x QJ Qx (KJ AQxxx xxx Axx)
  • 109x4x J109x Jx Ax (KJ AQxxx Qxx Qxx)
  • 109x4x 1098x Qx Ax (KJ AQJxx Jxx Qxx)
  • 109x4x 1098x xx AQ (KJ AQJxx QJx xxx)

5=4=1=3:

  • 109x4x AQ10x x xxx (KJ J9xxx QJxx AQ)
  • 109x4x AJ10x J xxx (KJ Q9xxx Qxxx AQ)
  • 109x4x A109x Q xxx (KJ QJxxx Jxxx AQ)
  • 109x4x A109x x Qxx (KJ QJxxx QJxx Ax)
  • 109x4x Q109x x Axx (KJ AJxxx QJxx Qx)
  • 109x4x J109x J Axx (KJ AQxxx Qxxx Qx)
  • 109x4x 1098x Q Axx (KJ AQJxx Jxxx Qx)
  • 109x4x 1098x x AQx (KJ AQJxx QJxx xx)

J10x4

4=4=3=2:

  • J10x4 AJ10x xxx xx (K9x Q9xxx QJ AQx)
  • J10x4 A109x Jxx xx (K9x QJxxx Qx AQx)
  • J10x4 J10xx Qxx Qx (K9x AQ9xx Jx Axx)
  • J10x4 J10xx xxx Ax (K9x AQ9xx QJ Qxx)
  • J10x4 109xx Jxx Ax (K9x AQJxx Qx Qxx)

4=4=2=3:

  • J10x4 AJ10x xx xxx (K9x Q9xxx QJx AQ)
  • J10x4 A109x Jx xxx (K9x QJxxx Qxx AQ)
  • J10x4 J10xx Qx Qxx (K9x AQ9xx Jxx Ax)
  • J10x4 J10xx xx Axx (K9x AQ9xx QJx Qx)
  • J10x4 109xx Jx Axx (K9x AQJxx Qxx Qx)

109x4

4=4=3=2:

  • 109x4 AQ10x xxx xx (KJx J9xxx QJ AQx)
  • 109x4 AJ10x Jxx xx (KJx Q9xxx Qx AQx)
  • 109x4 A109x Qxx xx (KJx QJxxx Jx AQx)
  • 109x4 A109x xxx Qx (KJx QJxxx QJ Axx)
  • 109x4 109xx Qxx Ax (KJx AQJxx Jx Qxx)
  • 109x4 109xx xxx AQ (KJx AQJxx QJ xxx)

4=4=2=3:

  • 109x4 AQ10x xx xxx (KJx J9xxx QJx AQ)
  • 109x4 AJ10x Jx xxx (KJx Q9xxx Qxx AQ)
  • 109x4 A109x Qx xxx (KJx QJxxx Jxx AQ)
  • 109x4 A109x xx Qxx (KJx QJxxx QJx Ax)
  • 109x4 109xx Qx Axx (KJx AQJxx Jxx Qx)
  • 109x4 109xx xx AQx (KJx AQJxx QJx xx)

IV. How to Identify Layouts Where the Defenders Cannot Prevent the Contract's Being Fulfilled in the Declaring Side's Second or Third Possession as a Result of Declarer's Estabiishing Winners on Power

Robot would use the following procedure to determine which of the layouts on page 3 were "hopeless" and which were "not hopeless":

  1. If a layout would permit the defenders to set the contract on their first possession (whether by force or by giving Declarer a guess), it would be considered "not hopeless." (This step also appeared in Parts II and III of this series.) The remaining layouts would be considered in the next step.
  2. For each remaining layout, if (a) there is a Declarer-consistent way to have dealt the defenders' unseen cards so that the contract could be set on their second possession if they obtain the lead in a suit where they hold the master (that is, the contract might be set even if Declarer's attempt to generate winners doesn't lose to a defender's original nonmaster); and (b) it would not allow the contract to be fulfilled on power in the declaring side's first possession, then the layout would be considered "not hopeless." (As noted earlier, layouts where the defenders could not prevent the contract from being fulfilled on the declaring side's first possession on power through an "option-preserving" line of play would have been excluded as "hopeless" in Part II or Part III of this series.) The remaining layouts would be considered in the next step.
  3. For each remaining layout, if (a) Declarer could take enough winners on power to fulfill the contract in the declaring side's second possession, irrespective of the layout of the cards not yet revealed to Declarer; (b) establishing the contract-fulfilling winner would not enable the defenders to set the contract in their second possession; and (c) the defenders would be unable to strand the contract-fulfilling winner before the declaring side's second possession, the layout would be considered "hopeless." The remaining layouts would be considered in the next step.
  4. For each remaining layout, if the declaring side's holdings (taken one suit at a time, without regard to entries) would provide no way to establish the contract-fulfilling winner by force (irrespective of the layout of the cards not yet seen by Declarer) in time to take it in the declaring side's third possession (even if Declarer were assumed to have ample controls and transportation), then the layout would be considered "not hopeless." The remaining layouts (which would be those where Declarer might be threatening to fulfill the contract in the declaring side's third possession) would be considered in the next step.
  5. For each remaining layout, if (a) there is a Declarer-consistent way to have dealt the defenders' unseen cards so that the contract could be set in their third possession (whether by force or by giving Declarer a guess); and (b) it would not allow the contract to be fulfilled on power in the declaring side's first or second possession, then the layout would be considered "not hopeless." Or if there is a Declarer-consistent way for the defenders to prevent the contract-fulfilling winner from being taken in any of the declaring side's first three possessions, then the layout would be considered "not hopeless." The remaining layouts would be considered "hopeless" because the contract would inevitably be fulfilled in or before the declaring side's third possession.

V. Identification of Layouts that Qualify as "Not Hopeless" in Step 1 (the Defenders Could Set the Contract in Their First Possession)

Step 1 will now be applied to the hand types for Partner given on page 3. The only layouts where the defenders could set the contract in their first possession would be those in which West holds AQx, Axx, AQ, or Ax. The last of these cases is included because it would set the contract if Declarer misguessed to play the queen after Robot took the A at Trick 1 and shifted to a secondary club honor or a low club.

Because taking the A and shifting to the K at Trick 2 would preclude taking advantage of Declarer's misguess where Partner holds Ax, Robot's examination of the layouts enumerated on page 4 would consider a shift only to a secondary club honor or a low club (both of which work also when Partner holds AQx, Axx, or AQ), even though shifting to the K would avoid blockage when Partner holds the stiff Q.

In the following, regular font indicates that the implied layout would give the defenders a chance to set in its first possession and therefore would qualify as "not hopeless." Italic font means that the layout wouldn't give the defenders a chance to set in its first possession and therefore would need to be further examined on the next page.

KJ94x

5=5=3=0:

  • KJ94x Q109xx xxx void (10x AJxx QJ AQxxx)
  • KJ94x J109xx Jxx void (10x AQxx Qx AQxxx)
  • KJ94x 1098xx Qxx void (10x AQJx Jx AQxxx)

5=5=2=1:

  • KJ94x Q109xx xx x (10x AJxx QJx AQxx)
  • KJ94x 1098xx Qx x (10x AQJx Jxx AQxx)

5=5=1=2:

  • KJ94x Q109xx x xx (10x AJxx QJxx AQx)

5=5=0=3:

  • KJ94x Q109xx void xxx (10x AJxx QJxxx AQ)

5=4=3=1:

  • KJ94x Q109x xxx x (10x AJxxx QJ AQxx)
  • KJ94x J109x Jxx x (10x AQxxx Qx AQxx)
  • KJ94x 1098x Qxx x (10x AQJxx Jx AQxx)

5=4=2=2:

  • KJ94x Q109x xx xx (10x AJxxx QJx AQx)
  • KJ94x 1098x Qx xx (10x AQJxx Jxx AQx)

5=4=1=3:

  • KJ94x Q109x x xxx (10x AJxxx QJxx AQ)

K1094x

5=5=3=0:

  • K1094x QJ10xx xxx void (Jx A9xx QJ AQxxx)
  • K1094x Q109xx Jxx void (Jx AJxx Qx AQxxx)
  • K1094x J109xx Qxx void (Jx AQxx Jx AQxxx)
  • K1094x 1098xx QJx void (Jx AQJx xx AQxxx)

5=5=2=1:

  • K1094x QJ10xx xx x (Jx A9xx QJx AQxx)
  • K1094x Q109xx Jx x (Jx AJxx Qxx AQxx)
  • K1094x J109xx Qx x (Jx AQxx Jxx AQxx)
  • K1094x 1098xx QJ x (Jx AQJx xxx AQxx)

5=5=1=2:

  • K1094x QJ10xx x xx (Jx A9xx QJxx AQx)
  • K1094x Q109xx J xx (Jx AJxx Qxxx AQx)

5=5=0=3:

  • K1094x QJ10xx void xxx (Jx A9xx QJxxx AQ)

5=4=3=1:

  • K1094x QJ10x xxx x (Jx A9xxx QJ AQxx)
  • K1094x Q109x Jxx x (Jx AJxxx Qx AQxx)
  • K1094x J109x Qxx x (Jx AQxxx Jx AQxx)
  • K1094x 1098x QJx x (Jx AQJxx xx AQxx)

5=4=2=2:

  • K1094x QJ10x xx xx (Jx A9xxx QJx AQx)
  • K1094x Q109x Jx xx (Jx AJxxx Qxx AQx)
  • K1094x J109x Qx xx (Jx AQxxx Jxx AQx)
  • K1094x 1098x QJ xx (Jx AQJxx xxx AQx)

5=4=1=3:

  • K1094x QJ10x x xxx (Jx A9xxx QJxx AQ)
  • K1094x Q109x J xxx (Jx AJxxx Qxxx AQ)

J10x4x

5=5=2=1:

  • J10x4x J10xxx Qx Q (K9 AQ9x Jxx Axxx)
  • J10x4x 109xxx QJ Q (K9 AQJx xxx Axxx)

5=5=1=2:

  • J10x4x J10xxx x Ax (K9 AQ9x QJxx Qxx)
  • J10x4x 109xxx J Ax (K9 AQJx Qxxx Qxx)

5=5=0=3:

  • J10x4x J10xxx void Axx (K9 AQ9x QJxxx Qx)

5=4=3=1:

  • J10x4x AJ10x xxx x (K9 Q9xxx QJ AQxx)
  • J10x4x A109x Jxx x (K9 QJxxx Qx AQxx)
  • J10x4x QJ10x Qxx x (K9 A9xxx Jx AQxx)
  • J10x4x Q109x QJx x (K9 AJxxx xx AQxx)
  • J10x4x Q109x Jxx Q (K9 AJxxx Qx Axxx)
  • J10x4x J109x Qxx Q (K9 AQxxx Jx Axxx)

5=4=2=2:

  • J10x4x AJ10x xx xx (K9 Q9xxx QJx AQx)
  • J10x4x A109x Jx xx (K9 QJxxx Qxx AQx)
  • J10x4x QJ10x Qx xx (K9 A9xxx Jxx AQx)
  • J10x4x Q109x QJ xx (K9 AJxxx xxx AQx)
  • J10x4x J109x Qx Qx (K9 AQxxx Jxx Axx)
  • J10x4x J109x xx Ax (K9 AQxxx QJx Qxx)
  • J10x4x 1098x Jx Ax (K9 AQJxx Qxx Qxx)

5=4=1=3:

  • J10x4x AJ10x x xxx (K9 Q9xxx QJxx AQ)
  • J10x4x A109x J xxx (K9 QJxxx Qxxx AQ)
  • J10x4x J109x x Axx (K9 AQxxx QJxx Qx)
  • J10x4x 1098x J Axx (K9 AQJxx Qxxx Qx)

109x4x

5=5=2=1:

  • 109x4x 109xxx Qx A (KJ AQJx Jxx Qxxx)

5=5=1=2:

  • 109x4x 109xxx Q Ax (KJ AQJx Jxxx Qxx)
  • 109x4x 109xxx x AQ (KJ AQJx QJxx xxx)

5=5=0=3:

  • 109x4x 109xxx void AQx (KJ AQJx QJxxx xx)

5=4=3=1:

  • 109x4x AQ10x xxx x (KJ J9xxx QJ AQxx)
  • 109x4x AJ10x Jxx x (KJ Q9xxx Qx AQxx)
  • 109x4x A109x Qxx x (KJ QJxxx Jx AQxx)
  • 109x4x A109x xxx Q (KJ QJxxx QJ Axxx)
  • 109x4x QJ10x QJx x (KJ A9xxx xx AQxx)
  • 109x4x QJ10x Jxx Q (KJ A9xxx Qx Axxx)
  • 109x4x Q109x Qxx Q (KJ AJxxx Jx Axxx)
  • 109x4x Q109x xxx A (KJ AJxxx QJ Qxxx)
  • 109x4x J109x QJx Q (KJ AQxxx xx Axxx)
  • 109x4x J109x Jxx A (KJ AQxxx Qx Qxxx)
  • 109x4x 1098x Qxx A (KJ AQJxx Jx Qxxx)

5=4=2=2:

  • 109x4x AQ10x xx xx (KJ J9xxx QJx AQx)
  • 109x4x AJ10x Jx xx (KJ Q9xxx Qxx AQx)
  • 109x4x A109x Qx xx (KJ QJxxx Jxx AQx)
  • 109x4x A109x xx Qx (KJ QJxxx QJx Axx)
  • 109x4x QJ10x QJ xx(KJ A9xxx xxx AQx)
  • 109x4x Q109x Qx Qx (KJ AJxxx Jxx Axx)
  • 109x4x Q109x xx Ax (KJ AJxxx QJx Qxx)
  • 109x4x J109x QJ Qx (KJ AQxxx xxx Axx)
  • 109x4x J109x Jx Ax (KJ AQxxx Qxx Qxx)
  • 109x4x 1098x Qx Ax (KJ AQJxx Jxx Qxx)
  • 109x4x 1098x xx AQ (KJ AQJxx QJx xxx)

5=4=1=3:

  • 109x4x AQ10x x xxx (KJ J9xxx QJxx AQ)
  • 109x4x AJ10x J xxx (KJ Q9xxx Qxxx AQ)
  • 109x4x A109x Q xxx (KJ QJxxx Jxxx AQ)
  • 109x4x A109x x Qxx (KJ QJxxx QJxx Ax)
  • 109x4x Q109x x Axx (KJ AJxxx QJxx Qx)
  • 109x4x J109x J Axx (KJ AQxxx Qxxx Qx)
  • 109x4x 1098x Q Axx (KJ AQJxx Jxxx Qx)
  • 109x4x 1098x x AQx (KJ AQJxx QJxx xx)

J10x4

4=4=3=2:

  • J10x4 AJ10x xxx xx (K9x Q9xxx QJ AQx)
  • J10x4 A109x Jxx xx (K9x QJxxx Qx AQx)
  • J10x4 J10xx Qxx Qx (K9x AQ9xx Jx Axx)
  • J10x4 J10xx xxx Ax (K9x AQ9xx QJ Qxx)
  • J10x4 109xx Jxx Ax (K9x AQJxx Qx Qxx)

4=4=2=3:

  • J10x4 AJ10x xx xxx (K9x Q9xxx QJx AQ)
  • J10x4 A109x Jx xxx (K9x QJxxx Qxx AQ)
  • J10x4 J10xx Qx Qxx (K9x AQ9xx Jxx Ax)
  • J10x4 J10xx xx Axx (K9x AQ9xx QJx Qx)
  • J10x4 109xx Jx Axx (K9x AQJxx Qxx Qx)

109x4

4=4=3=2:

  • 109x4 AQ10x xxx xx (KJx J9xxx QJ AQx)
  • 109x4 AJ10x Jxx xx (KJx Q9xxx Qx AQx)
  • 109x4 A109x Qxx xx (KJx QJxxx Jx AQx)
  • 109x4 A109x xxx Qx (KJx QJxxx QJ Axx)
  • 109x4 109xx Qxx Ax (KJx AQJxx Jx Qxx)
  • 109x4 109xx xxx AQ (KJx AQJxx QJ xxx)

4=4=2=3:

  • 109x4 AQ10x xx xxx (KJx J9xxx QJx AQ)
  • 109x4 AJ10x Jx xxx (KJx Q9xxx Qxx AQ)
  • 109x4 A109x Qx xxx (KJx QJxxx Jxx AQ)
  • 109x4 A109x xx Qxx (KJx QJxxx QJx Ax)
  • 109x4 109xx Qx Axx (KJx AQJxx Jxx Qx)
  • 109x4 109xx xx AQx (KJx AQJxx QJx xx)

VI. Identification of Layouts that Qualify as "Not Hopeless" in Step 2 (the Defenders Could Set the Contract in Their Second Possession)

Step 2 will now be applied to the hand types that weren't identified as "not hopeless" in Step 1.

In the following list of holdings for West, regular font preceded by a single n-dash means that the layout was identified as "not hopeless" in Step 1 and therefore had appeared in regular font on page 5. Regular font without a leading n-dash means that the layout allows the defenders to threaten to set the contract in their second possession (possibly through Declarer's misguessing) without allowing the contract to be fulfilled on power in the declaring side's first possession. Italic font means that the layout wouldn't give the defenders a chance to set the contract in either of their first two possessions and would need to be further examined on the next page.

KJ94x

5=5=3=0:

  • KJ94x Q109xx xxx void (10x AJxx QJ AQxxx)
  • KJ94x J109xx Jxx void (10x AQxx Qx AQxxx)
  • KJ94x 1098xx Qxx void (10x AQJx Jx AQxxx)

In the above three layouts, Declarer would interpret West's 4 as fourth-best from at most five cards. Therefore, Declarer would infer the defenders' ducking a spade in their first possession as establishing at most four threatened spade winners, which would not set the contract.

In those layouts, if East were to take the A and shift to a club, Declarer's playing the A or a x would limit the defenders' club winners to two irrespective of the full layout. This precludes the defenders from threatening to set the contract in their second possession.

5=5=2=1:

  • KJ94x Q109xx xx x (10x AJxx QJx AQxx)
  • KJ94x 1098xx Qx x (10x AQJx Jxx AQxx)

5=5=1=2:

  • KJ94x Q109xx x xx (10x AJxx QJxx AQx)

In the layout above, if East rises at Trick 1 and shifts to the club jack, Declarer could not always avoid losing more than two club tricks in the defenders' first two possessions. Playing the queen could be essential if West started with K10xxx, but doing so could be bad if West started with Kx.

5=5=0=3:

  • KJ94x Q109xx void xxx (10x AJxx QJxxx AQ)

In the layout above, if East rises at Trick 1 and shifts to a club, playing the ace has to be considered because West might have started with Kx.

5=4=3=1:

  • KJ94x Q109x xxx x (10x AJxxx QJ AQxx)
  • KJ94x J109x Jxx x (10x AQxxx Qx AQxx)
  • KJ94x 1098x Qxx x (10x AQJxx Jx AQxx)

In the third layout above, it is assumed that Declarer will test hearts because (1) doing so does not require expending the declaring side's last master in the suit, and (2) the suit offers no finesse.

5=4=2=2:

  • KJ94x Q109x xx xx (10x AJxxx QJx AQx)
  • KJ94x 1098x Qx xx (10x AQJxx Jxx AQx)

5=4=1=3:

  • KJ94x Q109x x xxx (10x AJxxx QJxx AQ)

K1094x

5=5=3=0:

  • K1094x QJ10xx xxx void (Jx A9xx QJ AQxxx)
  • K1094x Q109xx Jxx void (Jx AJxx Qx AQxxx)
  • K1094x J109xx Qxx void (Jx AQxx Jx AQxxx)
  • K1094x 1098xx QJx void (Jx AQJx xx AQxxx)

5=5=2=1:

  • K1094x QJ10xx xx x (Jx A9xx QJx AQxx)
  • K1094x Q109xx Jx x (Jx AJxx Qxx AQxx)
  • K1094x J109xx Qx x (Jx AQxx Jxx AQxx)
  • K1094x 1098xx QJ x (Jx AQJx xxx AQxx)

5=5=1=2:

  • K1094x QJ10xx x xx (Jx A9xx QJxx AQx)
  • K1094x Q109xx J xx (Jx AJxx Qxxx AQx)

5=5=0=3:

  • K1094x QJ10xx void xxx (Jx A9xx QJxxx AQ)

5=4=3=1:

  • K1094x QJ10x xxx x (Jx A9xxx QJ AQxx)
  • K1094x Q109x Jxx x (Jx AJxxx Qx AQxx)
  • K1094x J109x Qxx x (Jx AQxxx Jx AQxx)
  • K1094x 1098x QJx x (Jx AQJxx xx AQxx)

5=4=2=2:

  • K1094x QJ10x xx xx (Jx A9xxx QJx AQx)
  • K1094x Q109x Jx xx (Jx AJxxx Qxx AQx)
  • K1094x J109x Qx xx (Jx AQxxx Jxx AQx)
  • K1094x 1098x QJ xx (Jx AQJxx xxx AQx)

5=4=1=3:

  • K1094x QJ10x x xxx (Jx A9xxx QJxx AQ)
  • K1094x Q109x J xxx (Jx AJxxx Qxxx AQ)

J10x4x

5=5=2=1:

  • J10x4x J10xxx Qx Q (K9 AQ9x Jxx Axxx)
  • J10x4x 109xxx QJ Q (K9 AQJx xxx Axxx)

In the first layout above, if East wins Trick 1 and returns a club, Declarer is assumed to duck, on the grounds that the declaring-side holds fewer masters in clubs than in any other suit. This would render the defenders unable to continue the suit, given that Robot would have dismissed shifting to the K (as discussed on page 5).

5=5=1=2:

  • J10x4x J10xxx x Ax (K9 AQ9x QJxx Qxx)
  • J10x4x 109xxx J Ax (K9 AQJx Qxxx Qxx)

5=5=0=3:

  • J10x4x J10xxx void Axx (K9 AQ9x QJxxx Qx)

5=4=3=1:

  • J10x4x AJ10x xxx x (K9 Q9xxx QJ AQxx)
  • J10x4x A109x Jxx x (K9 QJxxx Qx AQxx)
  • J10x4x QJ10x Qxx x (K9 A9xxx Jx AQxx)
  • J10x4x Q109x QJx x (K9 AJxxx xx AQxx)
  • J10x4x Q109x Jxx Q (K9 AJxxx Qx Axxx)
  • J10x4x J109x Qxx Q (K9 AQxxx Jx Axxx)

5=4=2=2:

  • J10x4x AJ10x xx xx (K9 Q9xxx QJx AQx)
  • J10x4x A109x Jx xx (K9 QJxxx Qxx AQx)
  • J10x4x QJ10x Qx xx (K9 A9xxx Jxx AQx)
  • J10x4x Q109x QJ xx (K9 AJxxx xxx AQx)
  • J10x4x J109x Qx Qx (K9 AQxxx Jxx Axx)
  • J10x4x J109x xx Ax (K9 AQxxx QJx Qxx)
  • J10x4x 1098x Jx Ax (K9 AQJxx Qxx Qxx)

5=4=1=3:

  • J10x4x AJ10x x xxx (K9 Q9xxx QJxx AQ)
  • J10x4x A109x J xxx (K9 QJxxx Qxxx AQ)
  • J10x4x J109x x Axx (K9 AQxxx QJxx Qx)
  • J10x4x 1098x J Axx (K9 AQJxx Qxxx Qx)

109x4x

5=5=2=1:

  • 109x4x 109xxx Qx A (KJ AQJx Jxx Qxxx)

In the layout above, there is no layout of the club suit where the defenders could establish a fourth winner without letting the Q become the contract-fulfilling winner. Therefore, the defenders are not threatening to set the contract in their second possession, even though the contract might be in jeopardy if Declarer were to lose a trick to an original nonmaster during the declaring side's first possession.

5=5=1=2:

  • 109x4x 109xxx Q Ax (KJ AQJx Jxxx Qxx)
  • 109x4x 109xxx x AQ (KJ AQJx QJxx xxx)

5=5=0=3:

  • 109x4x 109xxx void AQx (KJ AQJx QJxxx xx)

5=4=3=1:

  • 109x4x AQ10x xxx x (KJ J9xxx QJ AQxx)
  • 109x4x AJ10x Jxx x (KJ Q9xxx Qx AQxx)
  • 109x4x A109x Qxx x (KJ QJxxx Jx AQxx)
  • 109x4x A109x xxx Q (KJ QJxxx QJ Axxx)
  • 109x4x QJ10x QJx x (KJ A9xxx xx AQxx)
  • 109x4x QJ10x Jxx Q (KJ A9xxx Qx Axxx)
  • 109x4x Q109x Qxx Q (KJ AJxxx Jx Axxx)
  • 109x4x Q109x xxx A (KJ AJxxx QJ Qxxx)
  • 109x4x J109x QJx Q (KJ AQxxx xx Axxx)
  • 109x4x J109x Jxx A (KJ AQxxx Qx Qxxx)
  • 109x4x 1098x Qxx A (KJ AQJxx Jx Qxxx)

In the first layout above, some ways of allocating the cards not yet visible to Declarer would allow the defense to take one spade, two hearts, and two clubs before the declaring side's second possession.

5=4=2=2:

  • 109x4x AQ10x xx xx (KJ J9xxx QJx AQx)
  • 109x4x AJ10x Jx xx (KJ Q9xxx Qxx AQx)
  • 109x4x A109x Qx xx (KJ QJxxx Jxx AQx)
  • 109x4x A109x xx Qx (KJ QJxxx QJx Axx)
  • 109x4x QJ10x QJ xx (KJ A9xxx xxx AQx)
  • 109x4x Q109x Qx Qx (KJ AJxxx Jxx Axx)
  • 109x4x Q109x xx Ax (KJ AJxxx QJx Qxx)
  • 109x4x J109x QJ Qx (KJ AQxxx xxx Axx)
  • 109x4x J109x Jx Ax (KJ AQxxx Qxx Qxx)
  • 109x4x 1098x Qx Ax (KJ AQJxx Jxx Qxx)
  • 109x4x 1098x xx AQ (KJ AQJxx QJx xxx)

5=4=1=3:

  • 109x4x AQ10x x xxx (KJ J9xxx QJxx AQ)
  • 109x4x AJ10x J xxx (KJ Q9xxx Qxxx AQ)
  • 109x4x A109x Q xxx (KJ QJxxx Jxxx AQ)
  • 109x4x A109x x Qxx (KJ QJxxx QJxx Ax)
  • 109x4x Q109x x Axx (KJ AJxxx QJxx Qx)
  • 109x4x J109x J Axx (KJ AQxxx Qxxx Qx)
  • 109x4x 1098x Q Axx (KJ AQJxx Jxxx Qx)
  • 109x4x 1098x x AQx (KJ AQJxx QJxx xx)

J10x4

4=4=3=2:

  • J10x4 AJ10x xxx xx (K9x Q9xxx QJ AQx)
  • J10x4 A109x Jxx xx (K9x QJxxx Qx AQx)
  • J10x4 J10xx Qxx Qx (K9x AQ9xx Jx Axx)
  • J10x4 J10xx xxx Ax (K9x AQ9xx QJ Qxx)
  • J10x4 109xx Jxx Ax (K9x AQJxx Qx Qxx)

4=4=2=3:

  • J10x4 AJ10x xx xxx (K9x Q9xxx QJx AQ)
  • J10x4 A109x Jx xxx (K9x QJxxx Qxx AQ)
  • J10x4 J10xx Qx Qxx (K9x AQ9xx Jxx Ax)
  • J10x4 J10xx xx Axx (K9x AQ9xx QJx Qx)
  • J10x4 109xx Jx Axx (K9x AQJxx Qxx Qx)

109x4

4=4=3=2:

  • 109x4 AQ10x xxx xx (KJx J9xxx QJ AQx)
  • 109x4 AJ10x Jxx xx (KJx Q9xxx Qx AQx)
  • 109x4 A109x Qxx xx (KJx QJxxx Jx AQx)
  • 109x4 A109x xxx Qx (KJx QJxxx QJ Axx)
  • 109x4 109xx Qxx Ax (KJx AQJxx Jx Qxx)
  • 109x4 109xx xxx AQ (KJx AQJxx QJ xxx)

4=4=2=3:

  • 109x4 AQ10x xx xxx (KJx J9xxx QJx AQ)
  • 109x4 AJ10x Jx xxx (KJx Q9xxx Qxx AQ)
  • 109x4 A109x Qx xxx (KJx QJxxx Jxx AQ)
  • 109x4 A109x xx Qxx (KJx QJxxx QJx Ax)
  • 109x4 109xx Qx Axx (KJx AQJxx Jxx Qx)
  • 109x4 109xx xx AQx (KJx AQJxx QJx xx)

VII. Identification of Layouts that Qualify as "Hopeless" in Step 3 ((a) Declarer Could Take Enough Winners on Power to Fulfill the Contract in the Declaring Side's Second Possession, Irrespective of the Layout of the Cards Not Yet Revealed to Declarer; (b) Establishing the Contract-fulfilling Winner Would Not Enable the Defenders to Take the Contract-setting Winner in Their Second Possession; and (c) the Defenders Would Be Unable to Strand the Contract-fulfilling Winner Before the Declaring Side's Second Possession)

The hand types that were not designated as "not hopeless" in Step 1 or 2 will now be examined for whether the contract can with certainty be fulfilled in the declaring side's second possession.

In the following list of holdings for West, regular font preceded by two n-dashes means that the layout was designated "not hopeless" in Step 1 and therefore had first appeared in regular font on page 5. Regular font preceded by a single n-dash means that the layout was designated "not hopeless" in Step 2 and therefore had appeared in regular font on page 6. Bold italic means that the layout is being designated "hopeless" in Step 3 (the current step). Regular italic means that the layout wouldn't provide a risk-free way to fulfill the contract on power in the declaring side's second possession and would need to be further examined on the next page.

KJ94x

5=5=3=0:

  • KJ94x Q109xx xxx void (10x AJxx QJ AQxxx)
  • KJ94x J109xx Jxx void (10x AQxx Qx AQxxx)
  • KJ94x 1098xx Qxx void (10x AQJx Jx AQxxx)

In the first layout above, if Robot takes the A and shifts to a heart (for example), Declarer could rise with the A, unblock the diamonds, and lead toward the Q to establish the contract-fulfilling winner (being that Robot is expecting Declarer to place the spade king with West).

5=5=2=1:

  • KJ94x Q109xx xx x (10x AJxx QJx AQxx)
  • KJ94x 1098xx Qx x (10x AQJx Jxx AQxx)

5=5=1=2:

  • KJ94x Q109xx x xx (10x AJxx QJxx AQx)

5=5=0=3:

  • KJ94x Q109xx void xxx (10x AJxx QJxxx AQ)

5=4=3=1:

  • KJ94x Q109x xxx x (10x AJxxx QJ AQxx)
  • KJ94x J109x Jxx x (10x AQxxx Qx AQxx)
  • KJ94x 1098x Qxx x (10x AQJxx Jx AQxx)

In the third layout above, if Robot were to take the A and shift to a club, Robot would expect Declarer to rise with the A, cash the K and A to determine that the suit is running, and lead toward the Q to establish the contract-fulfilling winner.

5=4=2=2:

  • KJ94x Q109x xx xx (10x AJxxx QJx AQx)
  • KJ94x 1098x Qx xx (10x AQJxx Jxx AQx)

5=4=1=3:

  • KJ94x Q109x x xxx (10x AJxxx QJxx AQ)

K1094x

5=5=3=0:

  • K1094x QJ10xx xxx void (Jx A9xx QJ AQxxx)
  • K1094x Q109xx Jxx void (Jx AJxx Qx AQxxx)
  • K1094x J109xx Qxx void (Jx AQxx Jx AQxxx)
  • K1094x 1098xx QJx void (Jx AQJx xx AQxxx)

5=5=2=1:

  • K1094x QJ10xx xx x (Jx A9xx QJx AQxx)
  • K1094x Q109xx Jx x (Jx AJxx Qxx AQxx)
  • K1094x J109xx Qx x (Jx AQxx Jxx AQxx)
  • K1094x 1098xx QJ x (Jx AQJx xxx AQxx)

5=5=1=2:

  • K1094x QJ10xx x xx (Jx A9xx QJxx AQx)
  • K1094x Q109xx J xx (Jx AJxx Qxxx AQx)

5=5=0=3:

  • K1094x QJ10xx void xxx (Jx A9xx QJxxx AQ)

5=4=3=1:

  • K1094x QJ10x xxx x (Jx A9xxx QJ AQxx)
  • K1094x Q109x Jxx x (Jx AJxxx Qx AQxx)
  • K1094x J109x Qxx x (Jx AQxxx Jx AQxx)
  • K1094x 1098x QJx x (Jx AQJxx xx AQxx)

5=4=2=2:

  • K1094x QJ10x xx xx (Jx A9xxx QJx AQx)
  • K1094x Q109x Jx xx (Jx AJxxx Qxx AQx)
  • K1094x J109x Qx xx (Jx AQxxx Jxx AQx)
  • K1094x 1098x QJ xx (Jx AQJxx xxx AQx)

5=4=1=3:

  • K1094x QJ10x x xxx (Jx A9xxx QJxx AQ)
  • K1094x Q109x J xxx (Jx AJxxx Qxxx AQ)

J10x4x

5=5=2=1:

  • J10x4x J10xxx Qx Q (K9 AQ9x Jxx Axxx)
  • J10x4x 109xxx QJ Q (K9 AQJx xxx Axxx)

In the first layout above, if Robot were to duck the opening lead, Declarer would on some full layouts need to allow the Q and the A to take tricks, which would defer taking the contract-fulfilling winner until the declaring side's third possession. (Although this succeeds, this step is not intended to discern that.)

5=5=1=2:

  • ––J10x4x J10xxx x Ax (K9 AQ9x QJxx Qxx)
  • ––J10x4x 109xxx J Ax (K9 AQJx Qxxx Qxx)

5=5=0=3:

  • ––J10x4x J10xxx void Axx (K9 AQ9x QJxxx Qx)

5=4=3=1:

  • J10x4x AJ10x xxx x (K9 Q9xxx QJ AQxx)
  • J10x4x A109x Jxx x (K9 QJxxx Qx AQxx)
  • J10x4x QJ10x Qxx x (K9 A9xxx Jx AQxx)
  • J10x4x Q109x QJx x (K9 AJxxx xx AQxx)
  • J10x4x Q109x Jxx Q (K9 AJxxx Qx Axxx)
  • J10x4x J109x Qxx Q (K9 AQxxx Jx Axxx)

5=4=2=2:

  • J10x4x AJ10x xx xx (K9 Q9xxx QJx AQx)
  • J10x4x A109x Jx xx (K9 QJxxx Qxx AQx)
  • J10x4x QJ10x Qx xx (K9 A9xxx Jxx AQx)
  • J10x4x Q109x QJ xx (K9 AJxxx xxx AQx)
  • J10x4x J109x Qx Qx (K9 AQxxx Jxx Axx)
  • ––J10x4x J109x xx Ax (K9 AQxxx QJx Qxx)
  • ––J10x4x 1098x Jx Ax (K9 AQJxx Qxx Qxx)

5=4=1=3:

  • J10x4x AJ10x x xxx (K9 Q9xxx QJxx AQ)
  • J10x4x A109x J xxx (K9 QJxxx Qxxx AQ)
  • ––J10x4x J109x x Axx (K9 AQxxx QJxx Qx)
  • ––J10x4x 1098x J Axx (K9 AQJxx Qxxx Qx)

109x4x

5=5=2=1:

  • 109x4x 109xxx Qx A (KJ AQJx Jxx Qxxx)

5=5=1=2:

  • ––109x4x 109xxx Q Ax (KJ AQJx Jxxx Qxx)
  • ––109x4x 109xxx x AQ (KJ AQJx QJxx xxx)

5=5=0=3:

  • ––109x4x 109xxx void AQx (KJ AQJx QJxxx xx)

5=4=3=1:

  • 109x4x AQ10x xxx x (KJ J9xxx QJ AQxx)
  • 109x4x AJ10x Jxx x (KJ Q9xxx Qx AQxx)
  • 109x4x A109x Qxx x (KJ QJxxx Jx AQxx)
  • 109x4x A109x xxx Q (KJ QJxxx QJ Axxx)
  • 109x4x QJ10x QJx x (KJ A9xxx xx AQxx)
  • 109x4x QJ10x Jxx Q (KJ A9xxx Qx Axxx)
  • 109x4x Q109x Qxx Q (KJ AJxxx Jx Axxx)
  • 109x4x Q109x xxx A (KJ AJxxx QJ Qxxx)
  • 109x4x J109x QJx Q (KJ AQxxx xx Axxx)
  • 109x4x J109x Jxx A (KJ AQxxx Qx Qxxx)
  • 109x4x 1098x Qxx A (KJ AQJxx Jx Qxxx)

5=4=2=2:

  • 109x4x AQ10x xx xx (KJ J9xxx QJx AQx)
  • 109x4x AJ10x Jx xx (KJ Q9xxx Qxx AQx)
  • 109x4x A109x Qx xx (KJ QJxxx Jxx AQx)
  • 109x4x A109x xx Qx (KJ QJxxx QJx Axx)
  • 109x4x QJ10x QJ xx (KJ A9xxx xxx AQx)
  • 109x4x Q109x Qx Qx (KJ AJxxx Jxx Axx)
  • ––109x4x Q109x xx Ax (KJ AJxxx QJx Qxx)
  • 109x4x J109x QJ Qx (KJ AQxxx xxx Axx)
  • ––109x4x J109x Jx Ax (KJ AQxxx Qxx Qxx)
  • ––109x4x 1098x Qx Ax (KJ AQJxx Jxx Qxx)
  • ––109x4x 1098x xx AQ (KJ AQJxx QJx xxx)

5=4=1=3:

  • 109x4x AQ10x x xxx (KJ J9xxx QJxx AQ)
  • 109x4x AJ10x J xxx (KJ Q9xxx Qxxx AQ)
  • 109x4x A109x Q xxx (KJ QJxxx Jxxx AQ)
  • 109x4x A109x x Qxx (KJ QJxxx QJxx Ax)
  • ––109x4x Q109x x Axx (KJ AJxxx QJxx Qx)
  • ––109x4x J109x J Axx (KJ AQxxx Qxxx Qx)
  • ––109x4x 1098x Q Axx (KJ AQJxx Jxxx Qx)
  • ––109x4x 1098x x AQx (KJ AQJxx QJxx xx)

J10x4

4=4=3=2:

  • J10x4 AJ10x xxx xx (K9x Q9xxx QJ AQx)
  • J10x4 A109x Jxx xx (K9x QJxxx Qx AQx)
  • J10x4 J10xx Qxx Qx (K9x AQ9xx Jx Axx)
  • ––J10x4 J10xx xxx Ax (K9x AQ9xx QJ Qxx)
  • ––J10x4 109xx Jxx Ax (K9x AQJxx Qx Qxx)

4=4=2=3:

  • J10x4 AJ10x xx xxx (K9x Q9xxx QJx AQ)
  • J10x4 A109x Jx xxx (K9x QJxxx Qxx AQ)
  • J10x4 J10xx Qx Qxx (K9x AQ9xx Jxx Ax)
  • ––J10x4 J10xx xx Axx (K9x AQ9xx QJx Qx)
  • ––J10x4 109xx Jx Axx (K9x AQJxx Qxx Qx)

109x4

4=4=3=2:

  • 109x4 AQ10x xxx xx (KJx J9xxx QJ AQx)
  • 109x4 AJ10x Jxx xx (KJx Q9xxx Qx AQx)
  • 109x4 A109x Qxx xx (KJx QJxxx Jx AQx)
  • 109x4 A109x xxx Qx (KJx QJxxx QJ Axx)
  • ––109x4 109xx Qxx Ax (KJx AQJxx Jx Qxx)
  • ––109x4 109xx xxx AQ (KJx AQJxx QJ xxx)

4=4=2=3:

  • 109x4 AQ10x xx xxx (KJx J9xxx QJx AQ)
  • 109x4 AJ10x Jx xxx (KJx Q9xxx Qxx AQ)
  • 109x4 A109x Qx xxx (KJx QJxxx Jxx AQ)
  • 109x4 A109x xx Qxx (KJx QJxxx QJx Ax)
  • ––109x4 109xx Qx Axx (KJx AQJxx Jxx Qx)
  • ––109x4 109xx xx AQx (KJx AQJxx QJx xx)

VIII. Identification of Layouts that Qualify as "Not Hopeless" in Step 4 (the Contract-fulfilling Winner Could Not Be Established by Force in Time to Be Taken in the Declaring Side's Third Possession, Even If Declarer Were Assumed to Have Ample Controls and Transportation)

Step 4, which will identify layouts where at least one allocation of cards not yet revealed to Declarer would preclude taking the contract-fulfilling winner in the declaring side's third possession, will now be applied to the hand types that are yet to be designated "hopeless" or "not hopeless." Hand types found to be "hopeless" in Step 3 are omitted.

In the following list of holdings for West, regular font preceded by two n-dashes means that the layout was designated "not hopeless" in Step 1 and therefore had first appeared in regular font on page 5. Regular font preceded by a single n-dash means that the layout was designated "not hopeless" in Step 2 and therefore had appeared in regular font on page 6. Regular font preceded by no n-dash means that the layout is being designated "not hopeless" in Step 4 (the current step). Italic font means that the declaring side's holdings, taken independently, provide at least one way to establish by force the contract-fulfilling winner in time to take it in that side's third possession; such holdings would need to be further examined on the next page.

KJ94x

5=5=3=0:

  • KJ94x J109xx Jxx void (10x AQxx Qx AQxxx)
  • KJ94x 1098xx Qxx void (10x AQJx Jx AQxxx)

5=5=2=1:

  • KJ94x 1098xx Qx x (10x AQJx Jxx AQxx)

5=5=1=2:

  • KJ94x Q109xx x xx (10x AJxx QJxx AQx)

5=5=0=3:

  • KJ94x Q109xx void xxx (10x AJxx QJxxx AQ)

5=4=3=1:

  • KJ94x J109x Jxx x (10x AQxxx Qx AQxx)

5=4=2=2:

  • KJ94x Q109x xx xx (10x AJxxx QJx AQx)
  • KJ94x 1098x Qx xx (10x AQJxx Jxx AQx)

5=4=1=3:

  • KJ94x Q109x x xxx (10x AJxxx QJxx AQ)

K1094x

5=5=3=0:

  • K1094x Q109xx Jxx void (Jx AJxx Qx AQxxx)
  • K1094x J109xx Qxx void (Jx AQxx Jx AQxxx)
  • K1094x 1098xx QJx void (Jx AQJx xx AQxxx)

5=5=2=1:

  • K1094x J109xx Qx x (Jx AQxx Jxx AQxx)
  • K1094x 1098xx QJ x (Jx AQJx xxx AQxx)

5=5=1=2:

  • K1094x QJ10xx x xx (Jx A9xx QJxx AQx)
  • K1094x Q109xx J xx (Jx AJxx Qxxx AQx)

5=5=0=3:

  • K1094x QJ10xx void xxx (Jx A9xx QJxxx AQ)

5=4=3=1:

  • K1094x Q109x Jxx x (Jx AJxxx Qx AQxx)
  • K1094x J109x Qxx x (Jx AQxxx Jx AQxx)

5=4=2=2:

  • K1094x QJ10x xx xx (Jx A9xxx QJx AQx)
  • K1094x Q109x Jx xx (Jx AJxxx Qxx AQx)
  • K1094x J109x Qx xx (Jx AQxxx Jxx AQx)
  • K1094x 1098x QJ xx (Jx AQJxx xxx AQx)

5=4=1=3:

  • K1094x QJ10x x xxx (Jx A9xxx QJxx AQ)
  • K1094x Q109x J xxx (Jx AJxxx Qxxx AQ)

J10x4x

5=5=2=1:

  • J10x4x J10xxx Qx Q (K9 AQ9x Jxx Axxx)

In the layout above, Declarer could (irrespective of the allocation of the cards yet to be revealed to Declarer) knock out the Q in the declaring side's first possession for one winner, knock out the A in that side's second possession for two spade winners total, and take the contract-fulfilling winner in that side's third possession.

5=5=1=2:

  • ––J10x4x J10xxx x Ax (K9 AQ9x QJxx Qxx)
  • ––J10x4x 109xxx J Ax (K9 AQJx Qxxx Qxx)

5=5=0=3:

  • ––J10x4x J10xxx void Axx (K9 AQ9x QJxxx Qx)

5=4=3=1:

  • J10x4x AJ10x xxx x (K9 Q9xxx QJ AQxx)
  • J10x4x A109x Jxx x (K9 QJxxx Qx AQxx)
  • J10x4x QJ10x Qxx x (K9 A9xxx Jx AQxx)
  • J10x4x Q109x QJx x (K9 AJxxx xx AQxx)
  • J10x4x Q109x Jxx Q (K9 AJxxx Qx Axxx)
  • J10x4x J109x Qxx Q (K9 AQxxx Jx Axxx)

In the first layout above, East's ducking at Trick 1 would give Declarer 1 spade, 5 diamonds, and 1 club. Declarer could knock out the A on the declaring side's first possession, knock out the A on the declaring side's second possession, and thus have the contract-fulfilling winner ready for that side's third possession. (The blockage in diamonds will be considered on the next page.) If East instead rose at Trick 1, this would establish two spade winners for Declarer and leave only one winner to be established. Therefore, in the remaining layouts in the above group, East will be assumed to duck at Trick 1.

In the second layout above, Declarer would "start" with 1 spade, 3 diamonds, and 1 club. Two winners could be created by knocking out the A, and another spade winner could be created by knocking out the A. But being able to create the ninth winner would not be guaranteed (for example, if LHO held A109xx and RHO held all five diamonds). Therefore, this layout is tagged "not hopeless."

In the third layout above, Declarer would start with 1 spade, 2 hearts, 2 diamonds, and 1 club. Only two additional winners can be guaranteed (by knocking out the A and Q). Therefore, this layout is tagged "not hopeless."

In the fourth layout above, Declarer would start with 1 spade, 2 hearts, 2 diamonds, and 1 club. An additional winner by force is available from spades, but that would bring the winner total to only seven. Therefore, this layout is tagged "not hopeless."

In the fifth layout above, Declarer would start with 1 spade, 2 hearts, 3 diamonds, and 1 club. An additional winner by force is available from spades, but that would bring the winner total to only eight. Therefore, this layout is tagged "not hopeless."

In the sixth layout above, Declarer would start with 1 spade, 3 hearts, 2 diamonds, and 1 club. Declarer could knock out the Q in the declaring side's first possession, knock out the A in the declaring side's second possession, and have the contract-fulfilling winner ready for that side's third possession.

5=4=2=2:

  • J10x4x AJ10x xx xx (K9 Q9xxx QJx AQx)
  • J10x4x A109x Jx xx (K9 QJxxx Qxx AQx)
  • J10x4x QJ10x Qx xx (K9 A9xxx Jxx AQx)
  • J10x4x Q109x QJ xx (K9 AJxxx xxx AQx)
  • J10x4x J109x Qx Qx (K9 AQxxx Jxx Axx)
  • ––J10x4x J109x xx Ax (K9 AQxxx QJx Qxx)
  • ––J10x4x 1098x Jx Ax (K9 AQJxx Qxx Qxx)

5=4=1=3:

  • J10x4x AJ10x x xxx (K9 Q9xxx QJxx AQ)
  • J10x4x A109x J xxx (K9 QJxxx Qxxx AQ)
  • ––J10x4x J109x x Axx (K9 AQxxx QJxx Qx)
  • ––J10x4x 1098x J Axx (K9 AQJxx Qxxx Qx)

109x4x

5=5=2=1:

  • 109x4x 109xxx Qx A (KJ AQJx Jxx Qxxx)

In the layout above, East's ducking at Trick 1 would give Declarer 1 spade, 4 hearts, and 2 diamonds. Declarer could in principle knock out the Q on the declaring side's first possession, knock out the A on the declaring side's second possession, and thus have the contract-fulfilling winner ready for that side's third possession.

5=5=1=2:

  • ––109x4x 109xxx Q Ax (KJ AQJx Jxxx Qxx)
  • ––109x4x 109xxx x AQ (KJ AQJx QJxx xxx)

5=5=0=3:

  • ––109x4x 109xxx void AQx (KJ AQJx QJxxx xx)

5=4=3=1:

  • 109x4x AQ10x xxx x (KJ J9xxx QJ AQxx)
  • 109x4x AJ10x Jxx x (KJ Q9xxx Qx AQxx)
  • 109x4x A109x Qxx x (KJ QJxxx Jx AQxx)
  • 109x4x QJ10x QJx x (KJ A9xxx xx AQxx)
  • 109x4x QJ10x Jxx Q (KJ A9xxx Qx Axxx)
  • 109x4x Q109x Qxx Q (KJ AJxxx Jx Axxx)
  • 109x4x J109x QJx Q (KJ AQxxx xx Axxx)
  • 109x4x J109x Jxx A (KJ AQxxx Qx Qxxx)

In the second through eighth layouts above, Declarer could not by force guarantee enough winners to fulfill the contract. Therefore, these layouts are tagged "not hopeless."

5=4=2=2:

  • 109x4x AQ10x xx xx (KJ J9xxx QJx AQx)
  • 109x4x AJ10x Jx xx (KJ Q9xxx Qxx AQx)
  • 109x4x A109x Qx xx (KJ QJxxx Jxx AQx)
  • 109x4x A109x xx Qx (KJ QJxxx QJx Axx)
  • 109x4x QJ10x QJ xx (KJ A9xxx xxx AQx)
  • 109x4x Q109x Qx Qx (KJ AJxxx Jxx Axx)
  • ––109x4x Q109x xx Ax (KJ AJxxx QJx Qxx)
  • 109x4x J109x QJ Qx (KJ AQxxx xxx Axx)
  • ––109x4x J109x Jx Ax (KJ AQxxx Qxx Qxx)
  • ––109x4x 1098x Qx Ax (KJ AQJxx Jxx Qxx)
  • ––109x4x 1098x xx AQ (KJ AQJxx QJx xxx)

5=4=1=3:

  • 109x4x AQ10x x xxx (KJ J9xxx QJxx AQ)
  • 109x4x AJ10x J xxx (KJ Q9xxx Qxxx AQ)
  • 109x4x A109x Q xxx (KJ QJxxx Jxxx AQ)
  • 109x4x A109x x Qxx (KJ QJxxx QJxx Ax)
  • ––109x4x Q109x x Axx (KJ AJxxx QJxx Qx)
  • ––109x4x J109x J Axx (KJ AQxxx Qxxx Qx)
  • ––109x4x 1098x Q Axx (KJ AQJxx Jxxx Qx)
  • ––109x4x 1098x x AQx (KJ AQJxx QJxx xx)

J10x4

4=4=3=2:

  • J10x4 AJ10x xxx xx (K9x Q9xxx QJ AQx)
  • J10x4 A109x Jxx xx (K9x QJxxx Qx AQx)
  • J10x4 J10xx Qxx Qx (K9x AQ9xx Jx Axx)
  • ––J10x4 J10xx xxx Ax (K9x AQ9xx QJ Qxx)
  • ––J10x4 109xx Jxx Ax (K9x AQJxx Qx Qxx)

4=4=2=3:

  • J10x4 AJ10x xx xxx (K9x Q9xxx QJx AQ)
  • J10x4 A109x Jx xxx (K9x QJxxx Qxx AQ)
  • J10x4 J10xx Qx Qxx (K9x AQ9xx Jxx Ax)
  • ––J10x4 J10xx xx Axx (K9x AQ9xx QJx Qx)
  • ––J10x4 109xx Jx Axx (K9x AQJxx Qxx Qx)

109x4

4=4=3=2:

  • 109x4 AQ10x xxx xx (KJx J9xxx QJ AQx)
  • 109x4 AJ10x Jxx xx (KJx Q9xxx Qx AQx)
  • 109x4 A109x Qxx xx (KJx QJxxx Jx AQx)
  • 109x4 A109x xxx Qx (KJx QJxxx QJ Axx)
  • ––109x4 109xx Qxx Ax (KJx AQJxx Jx Qxx)
  • ––109x4 109xx xxx AQ (KJx AQJxx QJ xxx)

4=4=2=3:

  • 109x4 AQ10x xx xxx (KJx J9xxx QJx AQ)
  • 109x4 AJ10x Jx xxx (KJx Q9xxx Qxx AQ)
  • 109x4 A109x Qx xxx (KJx QJxxx Jxx AQ)
  • 109x4 A109x xx Qxx (KJx QJxxx QJx Ax)
  • ––109x4 109xx Qx Axx (KJx AQJxx Jxx Qx)
  • ––109x4 109xx xx AQx (KJx AQJxx QJx xx)

IX. Identification of Layouts that Qualify as "Hopeless" in Step 5 (Declarer Would Be Threatening to Fulfill the Contract in the Declaring Side's Third Possession, and the Defenders Could Neither Set the Contract Before Then Nor Strand the Contract-fulfilling Winner)

Step 5 will now be applied to the hand types that are yet to be designated "hopeless" or "not hopeless."

In the following list of holdings for West, regular font preceded by three n-dashes means that the layout was designated "not hopeless" in Step 1 and therefore had first appeared in regular font on page 5. Regular font preceded by two n-dashes means that the layout was designated "not hopeless" in Step 2 and therefore had first appeared in regular font on page 6. Regular font preceded by a single n-dash means that the layout was designated "not hopeless" in Step 4 and therefore had appeared in regular font on page 8. Regular font preceded by no n-dash means that the layout is being designated "not hopeless" in Step 5 (the current step). Bold italic font means that the layout is being designated "hopeless" in Step 5 (the current step).

KJ94x

5=5=3=0:

  • KJ94x J109xx Jxx void (10x AQxx Qx AQxxx)
  • KJ94x 1098xx Qxx void (10x AQJx Jx AQxxx)

5=5=2=1:

  • KJ94x 1098xx Qx x (10x AQJx Jxx AQxx)

5=5=1=2:

  • ––KJ94x Q109xx x xx (10x AJxx QJxx AQx)

5=5=0=3:

  • ––KJ94x Q109xx void xxx (10x AJxx QJxxx AQ)

5=4=3=1:

  • KJ94x J109x Jxx x (10x AQxxx Qx AQxx)

5=4=2=2:

  • ––KJ94x Q109x xx xx (10x AJxxx QJx AQx)
  • ––KJ94x 1098x Qx xx (10x AQJxx Jxx AQx)

5=4=1=3:

  • ––KJ94x Q109x x xxx (10x AJxxx QJxx AQ)

K1094x

5=5=3=0:

  • K1094x Q109xx Jxx void (Jx AJxx Qx AQxxx)
  • K1094x J109xx Qxx void (Jx AQxx Jx AQxxx)
  • K1094x 1098xx QJx void (Jx AQJx xx AQxxx)

5=5=2=1:

  • K1094x J109xx Qx x (Jx AQxx Jxx AQxx)
  • K1094x 1098xx QJ x (Jx AQJx xxx AQxx)

5=5=1=2:

  • ––K1094x QJ10xx x xx (Jx A9xx QJxx AQx)
  • ––K1094x Q109xx J xx (Jx AJxx Qxxx AQx)

5=5=0=3:

  • ––K1094x QJ10xx void xxx (Jx A9xx QJxxx AQ)

5=4=3=1:

  • K1094x Q109x Jxx x (Jx AJxxx Qx AQxx)
  • K1094x J109x Qxx x (Jx AQxxx Jx AQxx)

5=4=2=2:

  • ––K1094x QJ10x xx xx (Jx A9xxx QJx AQx)
  • ––K1094x Q109x Jx xx (Jx AJxxx Qxx AQx)
  • ––K1094x J109x Qx xx (Jx AQxxx Jxx AQx)
  • ––K1094x 1098x QJ xx (Jx AQJxx xxx AQx)

5=4=1=3:

  • ––K1094x QJ10x x xxx (Jx A9xxx QJxx AQ)
  • ––K1094x Q109x J xxx (Jx AJxxx Qxxx AQ)

J10x4x

5=5=2=1:

  • J10x4x J10xxx Qx Q (K9 AQ9x Jxx Axxx)

In the layout above, Declarer intends to knock out the Q on the declaring side's first possession, knock out the A on that side's second possession, and take the contract-fulfilling winner in that side's third possession.

5=5=1=2:

  • –––J10x4x J10xxx x Ax (K9 AQ9x QJxx Qxx)
  • –––J10x4x 109xxx J Ax (K9 AQJx Qxxx Qxx)

5=5=0=3:

  • –––J10x4x J10xxx void Axx (K9 AQ9x QJxxx Qx)

5=4=3=1:

  • J10x4x AJ10x xxx x (K9 Q9xxx QJ AQxx)
  • J10x4x A109x Jxx x (K9 QJxxx Qx AQxx)
  • J10x4x QJ10x Qxx x (K9 A9xxx Jx AQxx)
  • J10x4x Q109x QJx x (K9 AJxxx xx AQxx)
  • J10x4x Q109x Jxx Q (K9 AJxxx Qx Axxx)
  • J10x4x J109x Qxx Q (K9 AQxxx Jx Axxx)

In the first layout above, if East held the A, then East's taking the A at Trick 1 could strand Dummy's spade winner unless Declarer overtook in diamonds (thereby jeopardizing a winner if the suit were to break 5-1 or 6-0).

5=4=2=2:

  • ––J10x4x AJ10x xx xx (K9 Q9xxx QJx AQx)
  • ––J10x4x A109x Jx xx (K9 QJxxx Qxx AQx)
  • ––J10x4x QJ10x Qx xx (K9 A9xxx Jxx AQx)
  • ––J10x4x Q109x QJ xx (K9 AJxxx xxx AQx)
  • ––J10x4x J109x Qx Qx (K9 AQxxx Jxx Axx)
  • –––J10x4x J109x xx Ax (K9 AQxxx QJx Qxx)
  • –––J10x4x 1098x Jx Ax (K9 AQJxx Qxx Qxx)

5=4=1=3:

  • ––J10x4x AJ10x x xxx (K9 Q9xxx QJxx AQ)
  • ––J10x4x A109x J xxx (K9 QJxxx Qxxx AQ)
  • –––J10x4x J109x x Axx (K9 AQxxx QJxx Qx)
  • –––J10x4x 1098x J Axx (K9 AQJxx Qxxx Qx)

109x4x

5=5=2=1:

  • 109x4x 109xxx Qx A (KJ AQJx Jxx Qxxx)

In the layout above, if East ducks the opening lead and Declarer were to lose a diamond to East, a club return might give the defenders the A, the Q, and three clubs.

5=5=1=2:

  • –––109x4x 109xxx Q Ax (KJ AQJx Jxxx Qxx)
  • –––109x4x 109xxx x AQ (KJ AQJx QJxx xxx)

5=5=0=3:

  • –––109x4x 109xxx void AQx (KJ AQJx QJxxx xx)

5=4=3=1:

  • ––109x4x AQ10x xxx x (KJ J9xxx QJ AQxx)
  • 109x4x AJ10x Jxx x (KJ Q9xxx Qx AQxx)
  • 109x4x A109x Qxx x (KJ QJxxx Jx AQxx)
  • 109x4x QJ10x QJx x (KJ A9xxx xx AQxx)
  • 109x4x QJ10x Jxx Q (KJ A9xxx Qx Axxx)
  • 109x4x Q109x Qxx Q (KJ AJxxx Jx Axxx)
  • 109x4x J109x QJx Q (KJ AQxxx xx Axxx)
  • 109x4x J109x Jxx A (KJ AQxxx Qx Qxxx)

5=4=2=2:

  • ––109x4x AQ10x xx xx (KJ J9xxx QJx AQx)
  • ––109x4x AJ10x Jx xx (KJ Q9xxx Qxx AQx)
  • ––109x4x A109x Qx xx (KJ QJxxx Jxx AQx)
  • ––109x4x A109x xx Qx (KJ QJxxx QJx Axx)
  • ––109x4x QJ10x QJ xx (KJ A9xxx xxx AQx)
  • ––109x4x Q109x Qx Qx (KJ AJxxx Jxx Axx)
  • –––109x4x Q109x xx Ax (KJ AJxxx QJx Qxx)
  • ––109x4x J109x QJ Qx (KJ AQxxx xxx Axx)
  • –––109x4x J109x Jx Ax (KJ AQxxx Qxx Qxx)
  • –––109x4x 1098x Qx Ax (KJ AQJxx Jxx Qxx)
  • –––109x4x 1098x xx AQ (KJ AQJxx QJx xxx)

 

5=4=1=3:

  • ––109x4x AQ10x x xxx (KJ J9xxx QJxx AQ)
  • ––109x4x AJ10x J xxx (KJ Q9xxx Qxxx AQ)
  • ––109x4x A109x Q xxx (KJ QJxxx Jxxx AQ)
  • ––109x4x A109x x Qxx (KJ QJxxx QJxx Ax)
  • –––109x4x Q109x x Axx (KJ AJxxx QJxx Qx)
  • –––109x4x J109x J Axx (KJ AQxxx Qxxx Qx)
  • –––109x4x 1098x Q Axx (KJ AQJxx Jxxx Qx)
  • –––109x4x 1098x x AQx (KJ AQJxx QJxx xx)

J10x4

4=4=3=2:

  • ––J10x4 AJ10x xxx xx (K9x Q9xxx QJ AQx)
  • ––J10x4 A109x Jxx xx (K9x QJxxx Qx AQx)
  • ––J10x4 J10xx Qxx Qx (K9x AQ9xx Jx Axx)
  • –––J10x4 J10xx xxx Ax (K9x AQ9xx QJ Qxx)
  • –––J10x4 109xx Jxx Ax (K9x AQJxx Qx Qxx)

4=4=2=3:

  • ––J10x4 AJ10x xx xxx (K9x Q9xxx QJx AQ)
  • ––J10x4 A109x Jx xxx (K9x QJxxx Qxx AQ)
  • ––J10x4 J10xx Qx Qxx (K9x AQ9xx Jxx Ax)
  • –––J10x4 J10xx xx Axx (K9x AQ9xx QJx Qx)
  • –––J10x4 109xx Jx Axx (K9x AQJxx Qxx Qx)

109x4

4=4=3=2:

  • ––109x4 AQ10x xxx xx (KJx J9xxx QJ AQx)
  • ––109x4 AJ10x Jxx xx (KJx Q9xxx Qx AQx)
  • ––109x4 A109x Qxx xx (KJx QJxxx Jx AQx)
  • ––109x4 A109x xxx Qx (KJx QJxxx QJ Axx)
  • –––109x4 109xx Qxx Ax (KJx AQJxx Jx Qxx)
  • –––109x4 109xx xxx AQ (KJx AQJxx QJ xxx)

4=4=2=3:

  • ––109x4 AQ10x xx xxx (KJx J9xxx QJx AQ)
  • ––109x4 AJ10x Jx xxx (KJx Q9xxx Qxx AQ)
  • ––109x4 A109x Qx xxx (KJx QJxxx Jxx AQ)
  • ––109x4 A109x xx Qxx (KJx QJxxx QJx Ax)
  • –––109x4 109xx Qx Axx (KJx AQJxx Jxx Qx)
  • –––109x4 109xx xx AQx (KJx AQJxx QJx xx)

X. Roster of Layouts Determined as "Hopeless" in Part II, III, or IV of This Series

Below are the holdings for West that have been identified as "hopeless" in Part II, III, or IV of this series.

In the following list, bold italic font preceded by three n-dashes means that the layout was designated "hopeless" in Part II of this series. Bold italic font preceded by two n-dashes means that the layout was designated "hopeless" in Part III of this series. Bold italic font preceded by a single n-dash means that the layout was designated "hopeless" in Step 3 of the present article. Bold italic font preceded by no n-dash means that the layout was designated "hopeless" in Step 5 of the present article.

Robot will also treat as "hopeless" any layout that can be obtained by interchanging one or more of West's designated spots with one of Declarer's x's in that suit.

KJ94x

5=5=3=0:

  • KJ94x Q109xx xxx void (10x AJxx QJ AQxxx)

5=5=2=1:

  • KJ94x Q109xx xx x (10x AJxx QJx AQxx)
  • ––KJ94x J109xx Jx x (10x AQxx Qxx AQxx)
  • –––KJ94x 1098xx xx Q (10x AQJx QJx Axxx)

5=5=1=2:

  • ––KJ94x J109xx J xx (10x AQxx Qxxx AQx)
  • ––KJ94x 1098xx Q xx (10x AQJx Jxxx AQx)
  • –––KJ94x 1098xx x Qx (10x AQJx QJxx Axx)

5=5=0=3:

  • –––KJ94x 1098xx void Qxx (10x AQJx QJxxx Ax)

5=4=3=1:

  • KJ94x Q109x xxx x (10x AJxxx QJ AQxx)
  • KJ94x 1098x Qxx x (10x AQJxx Jx AQxx)
  • –––KJ94x 1098x xxx Q (10x AQJxx QJ Axxx)

5=4=2=2:

  • ––KJ94x J109x Jx xx (10x AQxxx Qxx AQx)
  • –––KJ94x 1098x xx Qx (10x AQJxx QJx Axx)

5=4=1=3:

  • ––KJ94x J109x J xxx (10x AQxxx Qxxx AQ)
  • ––KJ94x 1098x Q xxx (10x AQJxx Jxxx AQ)
  • –––KJ94x 1098x x Qxx (10x AQJxx QJxx Ax)

K1094x

5=5=3=0:

  • K1094x QJ10xx xxx void (Jx A9xx QJ AQxxx)

5=5=2=1:

  • K1094x QJ10xx xx x (Jx A9xx QJx AQxx)
  • K1094x Q109xx Jx x (Jx AJxx Qxx AQxx)
  • –––K1094x J109xx xx Q (Jx AQxx QJx Axxx)
  • ––K1094x 1098xx Jx Q (Jx AQJx Qxx Axxx)

5=5=1=2:

  • ––K1094x J109xx Q xx (Jx AQxx Jxxx AQx)
  • –––K1094x J109xx x Qx (Jx AQxx QJxx Axx)
  • ––K1094x 1098xx J Qx (Jx AQJx Qxxx Axx)

5=5=0=3:

  • –––K1094x J109xx void Qxx (Jx AQxx QJxxx Ax)

5=4=3=1:

  • K1094x QJ10x xxx x (Jx A9xxx QJ AQxx)
  • –––K1094x J109x xxx Q (Jx AQxxx QJ Axxx)
  • K1094x 1098x QJx x (Jx AQJxx xx AQxx)
  • ––K1094x 1098x Jxx Q (Jx AQJxx Qx Axxx)

5=4=2=2:

  • –––K1094x J109x xx Qx (Jx AQxxx QJx Axx)
  • ––K1094x 1098x Jx Qx (Jx AQJxx Qxx Axx)

5=4=1=3:

  • ––K1094x J109x Q xxx (Jx AQxxx Jxxx AQ)
  • –––K1094x J109x x Qxx (Jx AQxxx QJxx Ax)
  • ––K1094x 1098x J Qxx (Jx AQJxx Qxxx Ax)

J10x4x

5=5=2=1:

  • J10x4x J10xxx Qx Q (K9 AQ9x Jxx Axxx)
  • –––J10x4x J10xxx xx A (K9 AQ9x QJx Qxxx)
  • J10x4x 109xxx QJ Q (K9 AQJx xxx Axxx)
  • ––J10x4x 109xxx Jx A (K9 AQJx Qxx Qxxx)

5=5=1=2:

  • ––J10x4x J10xxx Q Qx (K9 AQ9x Jxxx Axx)

5=5=0=3:

5=4=3=1:

  • –––J10x4x QJ10x xxx Q (K9 A9xxx QJ Axxx)
  • J10x4x J109x Qxx Q (K9 AQxxx Jx Axxx)
  • –––J10x4x J109x xxx A (K9 AQxxx QJ Qxxx)
  • ––J10x4x 1098x QJx Q (K9 AQJxx xx Axxx)
  • ––J10x4x 1098x Jxx A (K9 AQJxx Qx Qxxx)

5=4=2=2:

  • –––J10x4x QJ10x xx Qx (K9 A9xxx QJx Axx)
  • ––J10x4x Q109x Jx Qx (K9 AJxxx Qxx Axx)
  • ––J10x4x 1098x QJ Qx (K9 AQJxx xxx Axx)

5=4=1=3:

  • ––J10x4x QJ10x Q xxx (K9 A9xxx Jxxx AQ)
  • –––J10x4x QJ10x x Qxx (K9 A9xxx QJxx Ax)
  • ––J10x4x Q109x J Qxx (K9 AJxxx Qxxx Ax)
  • ––J10x4x J109x Q Qxx (K9 AQxxx Jxxx Ax)

109x4x

5=5=2=1:

5=5=1=2:

5=5=0=3:

5=4=3=1:

  • 109x4x A109x xxx Q (KJ QJxxx QJ Axxx)
  • 109x4x Q109x xxx A (KJ AJxxx QJ Qxxx)
  • 109x4x 1098x Qxx A (KJ AQJxx Jx Qxxx)

5=4=2=2:

  • ––109x4x QJ10x Jx Qx (KJ A9xxx Qxx Axx)

5=4=1=3:

  • ––109x4x QJ10x J Qxx (KJ A9xxx Qxxx Ax)
  • ––109x4x Q109x Q Qxx (KJ AJxxx Jxxx Ax)

J10x4

4=4=3=2:

  • ––J10x4 109xx QJx Qx (K9x AQJxx xx Axx)

4=4=2=3:

  • ––J10x4 109xx QJ Qxx (K9x AQJxx xxx Ax)

109x4

4=4=3=2:

4=4=2=3:

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