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In four articles published so far about pattern´s play in bridge, I have received much criticism from experts who consider the distributional theory to be inaccurate and opposite to the principles of bridge logic. Such opinions have gone from a ban to make comments in a column of Bridge Winners, under the concept that it´s theory is not bridge, up to some funny affirmation that an unknown player should abstain of manifesting a different criterion if not shared with the experts. Perhaps after reading this analysis can change of opinion and give a more serious look to pattern´s play, beginning to use it as a tool when playing percentages in those hands in which there not exist clues on the actual distribution of the hidden hands.

Going back into the subject and speaking in mathematical terms, once accepted by experts the argument that in 75% of cases the pattern of a hidden hand coincides with the length of his first known suit, curiously the same did not happen in reference with the fall of the suits in affinity with the number of cards the player has in each, despite that the theory´s support is the same, something easy to verify. In the bridge game there are two types of distributional references that are directly related with the number 13, divided into four addends:

1. The distribution of a hand in which three of the four suits are found with the same type of count, even or odd, and the 4th suit with the opposed. And,

2. The fall of the cards in a suit, in which three players provide with a same type of number of cards, even or odd, and the 4th player with the opposed.

Concatenated with the first paragraph, and analyzing two seen hands is also possible to establish as certain a third distributional reference:

3. The sums of cards in the four suits between the player´s hand and dummy´s, let´s call it the Distributional Scheme of the player “DS26”, give as result two even numbers and two odd in the 75% of cases, or four figures of the same type, odd or even, in a 25%. In the first case the player can assume a certain Distributional Scheme “DS13” to one hidden hand with chances 3 to 1 in favor, while in the second can do the same with both hidden hands simultaneously, since they share the 4th suit in the same denomination.

Without intention of entering in new arguments that distort the concept of percentage play, I think necessary to clarify what is understood as such in the game of bridge: if a decision should be taken in a determined moment of card play, without clues about the distributions of the hidden hands, unless for reasons of playing safe against abnormal distributions, the solution should align assuming counts with the greater chances of presentation. When the problem occurs at the beginning of the card play, if we know with precision the fall of a first suit around the table, playing percentages and in accordance with the numerals 1 and 3 can assign DS13´s to one or both hidden hands that matches the greatest odds. Accordingly, reviewing the table of frequencies of the Encyclopedia of Bridge, allow me to enumerate again in approximate values, the better odds before passing to the examples:

• If the player has a void, 0 cards, assume distributions 0-6-4-3 or 0-5-4-4 (26% and 24% each).

• If having a singleton, 1 card, assume as more likely the 1-5-4-3 (40%).

• If the player has a doubleton, 2 cards, his probable distributions are 2-4-4-3 or 2-5-4-2 (26%).

• With three cards in a suit, assume distributions 3-4-3-3 or 3-5-3-2 (28% and 27% respectively).

• With four cards in the suit, the distribution of greater expectation is the 4-4-3-2 (45%).

• If the player has five cards, suppose hands 5-3-3-2 or 5-4-3-1 (31% and 26% respectively).

• When the player has six cards, assume 6-3-2-2 or 6-4-2-1 (34% and 28% respectively).

• For a player with a seventh card suit, is likely to find it with the 7-3-2-1 (53%). And,

• If the player has eight cards in a suit, assume the distribution 8-2-2-1 (41%).

Would be idle to comment, because is more than obvious according with these numbers, that for playing percentages will have whenever to assume, first, that the pattern of a hidden hand follows the trend even or odd of the known suit, and second, the more likely distribution is the one balanced in the other three suits, two of which are of the same type even or odd than the number of cards of the known suit.

The second part of pattern´s play by percentages is easy and is related with the allocation of cards in each of the three unknown suits. For making this there is a notion that is repeated many times in the game of bridge, constituting a procedure for use of both the experts or the common players: assign the bigger addendum to the suit with less number of cards in the two seen hands, and the lower addendum to the suit that is observed with greater number of cards. In the examples that follow this procedure will be perfectly clarified.

After studying dozens of boards played in major tournaments, in which mistakes in play of just one board had consequences on the results, I have come to the conclusion that, apart from the errors motivated by fatigue or the losing of player´s concentration, many of the poor results could have been reversed if the players had been making decisions that keep the percentage play lined with the numbers from the table above and the distributional trend of a known suit, rather than imagining plans with different distributions when there are no clear evidence in such sense, facilitating with this the intellectual work to avoid the tiredness. If the experts begin to use as an additional tool to their remarkable skills, these simple percentage play guides, I am sure that they would get even better results than those achieved in international competitions.

Let's see a few examples of recent tournaments, always from the point of view of declarer, as we will see later, in defending the hands the distributional theory is widely favorable to the interests of the defenders. In the first example, E/W vulnerable and dealer South (hands turned for easier appreciation):

North

♠

QJ864

♥

7

♦

AJ763

♣

A2

South

♠

A952

♥

A6

♦

K1095

♣

QJ4

W

N

E

S

1NT

P

2♥

P

3♠

P

4♣

P

4♦

P

4♥

P

4NT

P

5♠

P

6♠

P

P

P

Lead: ♣10

1st. Trick: West ♣ 10, 2, K, 4

2nd. Trick: East ♥ 4, A, 10, 7

3rd. Trick: South ♥ 6, 9, ♠4, 2

4th. Trick: North ♠ Q, K, A, 7

5th. Trick: South ♠ 2, 10, J, 3

6th. Trick: North ♦ 3, 2, K, 4

7th. Trick: South ♦ 10, 8, 6, Q One down.

This board is extremely interesting from the point of view of patterns and percentage play since without information from the auction result is only defined by the fall of the diamond suit. Guided by the concept of "empty spaces" declarer decided at the table that West has great chances of having 3 cards in diamonds, if as seems by West spots played at second and third tricks (♥10/9) East looks as having six cards in hearts, but that assumption is wrong if analyzing the hole hand in respect to probabilities.

Beginning with pattern’s deductions, the sums of the seen cards give same type of count in the four suits, uneven (9-3-9-5), and declarer can assign EVEN patterns to BOTH defenders at the same time after the 5th trick. There are four distributions with even patterns as the more frequent when we know by fact that both hands have two cards in spades: 2-4-4-3 (26%), 2-5-4-2 (26%), 2-6-3-2 (14%) and 2-6-4-1 (8%) with a total of 40% of probability for the two kind of permutations related with falls 2-2 in spades and diamonds (second and third), rather than the two ones related with the fall 3/1 in diamonds (first and fourth) with a total of 34%.

Since the odds give to the fall 2-2 in both suits a greater likelihood of presentation, declarer who wants to play with the percentages in favor will be obligated to also expect same favorable distribution in diamonds and win the contract in consequence. The East-West hands at the table were: WEST: ♠107, ♥KJ109, ♦84, ♣109863 and EAST: ♠K3, ♥Q85432, ♦Q2, ♣K75.

The second example is typical in the discovering of defender´s distributions just applying the concepts of following percentages and pattern´s play. Dealer East and both vulnerable (hands turned for easier appreciation):

North

♠

K98

♥

A8

♦

J1082

♣

A854

South

♠

Q105

♥

KQ10764

♦

A

♣

1072

W

N

E

S

P

1♥

X

XX

P

2♥

P

4♥

P

P

P

Lead: ♣K

Playing normally declarer holdup the lead, take in second with the ♣A in dummy and plays the ♥A finding the bad news when observing the void in the West hand. Time to stop and assign the patterns: West with the void will have an even pattern of the type 0-5-4-4 (if having the 0-6-4-3 may have made an overcall instead of doubling), and East with five cards in trumps an odd one of the type 5-3-3-2. As the sums of the seen cards give two even results in the majors and two odd in the minors, declarer can choose either defender to apply the procedure, but not both. Suppose that West is the chosen one to assign the count: 5 cards would go with the diamond suit since declarer sees only five cards in it, so the 4 cards will be in spades and clubs where are seen 6 and 7 cards respectively. As a result, the most likely distribution of that player will be 4-0-5-4, in which the 4th suit is diamonds. The correspondent East hand will be 3-5-3-2 in which the 4th suit is clubs with two cards, the related suit with diamonds in declarer´s ED26.

If the ♠A rests in the West hand, there is no way to fulfill the contract because he will cash the ♣J while East discards an spade, ending with any chance of an end play in trumps, but if the ♠A is at the other side, the hand is played as a double dummy: ♥8 covered by the ♥9 and ♥10 in the fourth trick, ♦A in the fifth, ♠Q, ♠10 and ♠5 in that order following in the next three tricks, a diamond ruff in the ninth and the play of the last club in the tenth, to receive in a trump coup the last three tricks. Contract made.

The reader can obtain the same solution playing percentages if he had chosen East hand to assign the count: knowing the presence of five cards in trumps, the most likely distribution will be the 5-3-3-2 and the position of the suit with a doubleton will be precisely clubs with 7 cards in sight. The hand against which declarer plays will be consequently the 3-5-3-2 in East and its corresponding 4-0-5-4 in West, getting the same result. The actual hands were: WEST ♠J732, ♦KQ975, ♣KQJ9 and EAST ♠A64, ♥J9532, ♦643, ♣63.

In a final example, West is the dealer and vulnerable North/South:

North

♠

AJ93

♥

K987

♦

AQ

♣

K83

South

♠

K1082

♥

Q10

♦

1087

♣

QJ42

W

N

E

S

P

1NT

P

2♣

P

2♥

P

3NT

P

4♠

P

P

P

Lead: ♣9 (East)

In this board, very important given the situation of the match, both declarers went one down despite having located the position of the trump queen. The lead of the ♣9 was won by declarer with the ♣K (West showing interest in the continuation of the suit), following with ♠A and ♠J to the finesse, receiving bad news when West discards an small diamond. The best declarer´s play was performed in one table eliminating all the trumps, West discarding two additional small diamonds, and continuing through the immediate finesse in hearts running the ♥10 towards the ♥A in East hand, but at the end he also fell from grace when trying a second finesse in diamonds playing against a KJxxx distribution which did not exist in West hand.

Without any information in the auction, the distributional declarer will make a stop at third trick to square the hands: the sums of cards in the four suits offer even results in the majors and odd ones in the minors, which forces to isolate only one defender. Taking East with four cards in trumps as the easier reference in view of a 45% of chances of having the 4-4-3-2 hand as the most probable distribution for playing percentages. The second suit with room for other four cards will be located in diamonds since declarer observes only 5 cards in it, the doubleton’s suit is clubs, with 7 seen cards, and the three cards suit will be hearts, in which he is looking at six cards. The hand imagined for East defender will be accordingly 4-3-4-2 and the corresponding from West 1-4-4-4, proceeding to play against those distributions.

The play will be like the one used at the table: two more trumps, West discarding two additional diamonds; a direct finesse passing the ♥10 up to the ♥A in East hand; West wins the return of the ♣5 with the ♣A to repeat ♣10 which declarer wins with the ♣Q in dummy, East discarding small diamond confirming his original doubleton in clubs. Now declarer cashes ♥Q and, as if playing double dummy, ends his master play with the ♦7 up to the ♦A in hand, falling the ♦J from the West hand; cashes the ♥K and returns the ♦Q to East ♦K in the twelfth trick, winning the last one with the ♦10 in dummy. Of course, if West returns the ♦J instead of the ♣10 at eight trick, declarer takes in hand with the ♦A and returns to dummy with the ♥Q to finish also brilliantly, cashing the ♣Q and giving the lead to West with the fourth club, as a pole towards the finesse in hearts at the end. The actual hands of the defenders were as declarer imagined: WEST: ♠5, ♥J432, ♦J653, ♣A1076 and EAST: ♠Q764, ♥A65, ♦K942, ♣95.

As a final recommendation for the players interested in pattern’s deductions in percentage play, when the time arrives to isolating one defender´s hand, choose the one with the count in the known suit with better odds for assigning distributions. For example if the fall of the cards in the known suit is 4/5, chose the hand with four cards, as the likely distribution 4-4-3-2 has 45% of probabilities. The same with a seven card count as the distribution 7-3-2-1 has even higher odds (53%). Memorizing the better odds for each number of cards in a suit, given above in this article, will be a great aid for everybody.

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