Computer Bridge: Card Combinations in Notrump
(Page of 10)

I. Introduction

This article presents thoughts on how a computer-bridgeplayer, often called a "bridge robot" (and which I will call, "Robot"), could discern the winner-establishment prospects for various single-suit layouts in notrump.

II. Usual Presentation of How to Handle a Card-combination

How to handle a given card-combination is often presented as a procedure. For example, with AQJTxxx opposite xxxx, x < 9, there would be three ways to potentially establish the suit without loss of a trick:

A. Lead an x from the short holding, and if

A1. Second Seat plays the king, win the ace from the long holding; or

A2. Second Seat plays the nine, cover with the ten from the long holding.

B. Lead an x from the short holding, and if

B1. Second Seat plays the king, win the ace from the long holding; or

B2. Second Seat plays the nine, win the ace from the long holding.

C. Lead the ace from the long holding.

III. Providing Robot Information on How to Handle Card-combinations

To enable Robot to discern the trick-establishment prospects for the various single-suit layouts that are consistent with the opponents' actions, Robot will have access to a card-combination catalog that will indicate how various ways of handling single-dummy, single-suit layouts would fare for each potential distribution of the hidden cards.

To avoid this catalog's containing layouts that can be made identical by rotation, Seat A will designate the holding that contains the leading side's highest card in the particular layout. That hand's partner will be designated Seat C. Seat B will be over Seat A, and Seat D will be under Seat A.

Below are some of the distributions applicable to the single-suit layout and plans presented on the previous page.

Single-suit plan A, part 1 (lead an x from the short holding, and if Second Seat plays the king, win the ace from the long holding) would yield the following "single-trick scenario" (with bold indicating each card played to this trick):

AQJTxxx (Seat A)

K (Seat D)                          9 (Seat B)

Single-suit plan A, part 2 (lead an x from the short holding, and if Second Seat plays the nine, cover with the ten from the long holding) would yield the following single-trick scenarios:

AQJTxxx

9                                           K

AQJTxxx

K9                                    (void)

Single-suit plan B, part 1 (lead an x from the short holding, and if Second Seat plays the king, win the ace from the long holding) would yield the following single-trick scenario:

AQJTxxx

K                                          9

Single-suit plan B, part 2 (lead an x from the short holding, and if Second Seat plays the nine, win the ace from the long holding) would yield the following single-trick scenarios:

AQJTxxx

K9                                     (void)

AQJTxxx

9                                       K

Single-suit plan C (lead the ace from the long holding) would yield the following successful single-trick scenarios:

K                                          9

xxxx

9                                           K

xxxx

IV. Characteristics of a Single-trick Scenario

A single-trick scenario has many characteristics that the card-combination catalog would need to store.

The following single-trick scenario exhausts both of the enemy's cards and thereby turns every card of the leading pair's longer holding into a winner (which I will call, "immediate winners" because the side with the winners is on lead).

AQJTxxx

K                                    9

In the following single-trick scenario, Seat A's ten is an original nonmaster, yet wins the trick for the leading side.

AQJTxxx

K9                              (void)

The next single-trick scenario is almost a hybrid of the above two. Seat B's king, an original nonmaster, wins the trick for the nonleading side. But, the remaining five original nonmasters of Seat A become winners (which I will call, "pending winners" because the side with the winners is not on lead).

AQJTxxx

9                                    K

The following single-trick scenario transforms the king into a pending winner for pair B/D.

AQJTxxx

K9                                (void)

These examples suggest the following characteristics of a single-trick scenario:

• Suit layout (for all four seats)
• Which seat wins the trick
• Number of immediate winners created for the pair that wins this trick
• Number of immediate winners created for each player of the pair that wins this trick
• Number of pending winners created for the pair that does not win this trick
• Number of pending winners created for each player of the pair that does not win this trick
• Whether the trick was won by an original nonmaster of the leading side
• Whether the trick was won by an original nonmaster of the nonleading side

V. Single-trick Scenarios Involving Fewer than 13 Cards

The single-trick scenario

AJTxx

Kxx                                    Qx

results in

AJxx

Kx                                     x

xx

The card-combination catalog would contain the various single-trick scenarios that stem from this layout. It would also contain layouts in which due to discards, the number of cards in the suit that were played to previous tricks was not a multiple of four.

When contemplating the play, Robot would assemble the relevant card-combinations, the single-trick scenarios stemming from them, and their "offspring" scenarios into a tree whose paths would represent multi-trick scenarios, usually involving more than one suit.

VI. Excluding Redundant Layouts from the Card-combination Catalog

The position resulting from the single-trick scenario on the previous page. that is,

AJxx

Kx                          x

xx

is equivalent to that reached by the single-trick scenario

AJTxx

Qxx                        Kx

namely,

AJxx

Qx                          x

xx

To enable the card-combination catalog to contain one item that reflects both layouts, its layouts that involve fewer than 13 cards will contain no gaps between explicit ranks. For example, in the layout where Seat D remains with Kx, Seat A's jack will be promoted to the queen, yielding

AQxx

Kx                           x

xx

This is also produced by taking the single-trick scenario that left Seat D with Qx, and promoting the queen to the king and Seat A's jack to the queen.

VII. Card Combinations that Cannot Create an Immediate Winner

The card-combination catalog will include layouts that can not create an immediate winner for either side. Here's an example.

Txx

xxxxxx                 QJ

However, the leading side to the next trick would create a winner if Seat A were to continue with the king. This would be depicted as

Qx

xxxxx                   K

VIII. Single-trick Scenarios that Would Be Omitted from the Card-combination Catalog

The following are two of the single-trick scenarios stemming from the layout introduced on page 2.

Scenario 1.

AQJTxxx

K9                       (void)

Scenario 2.

AQJTxxx

K9                        (void)

In the layout resulting from Scenario 1, Seat A would need to play its master (the ace) to draw the opponents' remaining card. In the layout resulting from Scenario 2, Seat A would need to play one of its master-equals (the queen, jack, or ten) to draw the opponents' remaining card. In either case, the resulting layout adjusted for promotion would be AKxxx opposite xx. Both scenarios have the same effect on creating winners and entries

The above indicates that having both Scenario 1 and Scenario 2 in the card-combination catalog would be superfluous. Being that Seat D's playing the king spares pair A/C a guess, Scenario 2 would be omitted from the card-combination catalog.

Seat D might have a legitimate reason (signaling, for example) for having played the king in Scenario 2. However, signallng and deception are not intended to be addressed by the card-combination catalog.

This catalog would also omit the layout resulting from Scenario 1, because layouts in which a given side holds only one card would be trivial to work out. For the same reason, the card-combination catalog would omit layouts in which the number of winners held by one side is greater than or equal to the number of unplayed cards of the other side. (The number of winners is constrained not to exceed the length of the side's longer holding, so that AQJ9 opposite KT would count as four winners and not six.)

IX. Summarizing the Scenarios that Can Result from a Given Card-combination

If the card-combination catalog were to concisely present the possible results of the various single-trick and multi-trick scenarios stemming from a given card-combination, this would often permit Robot to reject a single-suit plan or even a card-combination altogether, without having to examine its single-trick or multi-trick scenarios.

Page 4 listed some of the characteristics of a single-trick scenario. For the card-combination presented on pages 2 and 3 (AQJTxxx opposite xxxx, x < 9), the result of single-suit plan A, part 1 (lead an x from Seat C, and if Seat D plays the king, win the ace from Seat A) could be expressed as follows:

• The trick was won by Seat A.
• Six immediate winners were created for pair A/C.
• No pending winners were created for the pair that did not win this trick.
• The trick was not won by an original nonmaster.

The results of single-suit plan A, part 2 (lead an x from Seat C, and if Seat D plays the nine, cover with the ten from Seat A), where the ten loses to the king from Seat B, could be expressed as follows:

• The trick was won by Seat B.
• No immediate winners were created for pair B/D (being that it's exhausted in the suit).
• Five pending winners were created for the pair that did not win this trick.
• The trick was won by an original nonmaster (the king from Seat B).

The results of single-suit plan A, part 2 (lead an x from Seat C, and if Seat D plays the nine, cover with the ten from Seat A), where the ten holds, could be expressed as follows:

• The trick was won by Seat A.
• No immediate winners were created for pair A/C. (But next playing the ace will create five winners for pair A/C.)
• The trick was won by an original nonmaster (the ten from Seat A).

The results of what I'll call single-suit plan A, part 3 (lead an x from Seat C, and if Seat D discards, play the ten from Seat A) could be expressed as follows:

• The trick was won by Seat B.
• Five pending winners were created for pair A/C.
• The trick was won by an original nonmaster (the king from Seat B).

Therefore, single-suit plan A can be said to yield one of the following (stated informally):

• Six winners, without losing a trick to pair B/D or creating a winner for pair B/D.
• Five winners, losing one trick to an original nonmaster of Seat B.

So, if Robot needed five winners without losing the lead to Seat D, the above summary would indicate that single-suit plan A would work.

X. Use of the Card-combination Catalog on Defense

The card-combination catalog could be used on defense. Consider the following single-dummy layout:

Qxx (Dummy)

Suppose that the leader, Robot, needs three tricks from this holding. Robot could search the card-combination catalog for layouts where Seat A holds AJ9, Seat B holds Qxx, and there is a single-suit plan in which (1) the number of pair A/C's master cards, the number of pair A/C's winners by finesse, and the number of pair A/C's immediate winners sum to three; (2) pair B/D wins none of the first three tricks; and (3) the leader to a given trick is the same as the winner of the previous trick (that is, no outside entry is needed). This query should fetch the following single-trick scenarios (among others):

Qxx

Txxx

Qxx