Bob Mackinnon

Mystical Math

Some feel there is mystic power in certain numbers: 7 is lucky, 13 is unlucky. Scholars in the 10th Century calculated the world would end in the year 1000 AD, a nice round number, albeit a miscalculation. Some fans are attached to the number worn by their sports hero. Is it significant that Alex Rodriguez chose to wear number 13 which in hindsight appears to be an ominous tempting of fate?

I suppose everyone has a favourite number (mine is 5), but what do you think is the most beautiful number? Philosophers from Greek times have considered the most beautiful number to be one-half of 1 plus the square root of 5 (1.61803….). This is known as the Golden Ratio. Not beautiful in itself, perhaps, but Phidias designed the Parthenon using it. Euclid wrote of it. It appears in the design of the pyramids and great mosques of the middle ages and is a feature of the Fibonacci sequence introduced to Europeans in the 13th century along with the decimal system.  Mathematician Luca Pacioli called it The Divine Proportion (1509) and passed on his ideas to Leonardo da Vinci – an Italian bastard, whose mother should be praised for her judgment in choosing to produce such brilliant fellow (even though it is akin to praising the execution of an unintentional double squeeze.) Leonardo kept the concept in mind as he painted The Mona Lisa. The modern architect Le Corbusier thought of the number as an underlying feature of the mathematical order of the universe. Today industrial engineers use it in the design of consumer products. When I learned of this from a Japanese TV program I measured the attractive face of my TV cabinet finding it to conform closely to the Golden Ratio.

The number 13 figures prominently in the game of bridge. Consider 13 cards divided in the ratio of 5 to 8, a ratio of 1.6, a close approximation. The opening bid of a 5-card major reflects this split. The ratio of 8 to 13 is 1.61, an even closer approximation. If partners hold 8 of the 13 cards in a suit, the Law of Total Tricks indicates they are safe to contract for 8 tricks. Also, a 4-4 fit (8 cards) is often the best trump combination. So maybe there is a bit of magic in the division of 13 into the ratio of 5 and 8. If we add the prior number 3 from the Fibonacci sequence 3-5-8-13, we have the a priori expectation of the number of cards in a given suit having been dealt to a given hand.

Pythagoras thought the number 10 was divine. His faith is shared by bridge players who adhere to the Work Point Count. HCPs are used to draw boundaries in bidding systems which are considered sacrilegious to cross, but as with the Ten Commandments, temptation often overrides the inconvenience. It is a matter of the remote possibility of bitter punishment measured against the sweet promise of immediate satisfaction. Here is a hand where I had to break the rules to get to a slam everyone missed.

W
Bob
K95
AQ2
AQ853
A8
 
E
John
AQ84
KJ1065
107
72
Bob
John
2NT
3♦*
4
4NT
5
6
Pass
 

A bidding system is merely a language by which one conveys the tricking taking potential of one’s holding. The HCP content isn’t always the best indicator. My hand had 7 controls and 5 losers. It is the equivalent to a 2NT opening bid with 23 HCP, although, of course, it contains only 19 HCP. When partner transferred to 3, my hand was worth a jump to 4, counting AQx as the equivalent to 4 trumps with potential ruffing capacity in clubs.

Despite the fact that 13 tricks were available, my less than optimal achievement of 12 tricks was enough to score a clear top as 3 pairs played in 4, 3 in 3NT (Garrr) and 1 in 6, off 2. So, on this combination if you follow the standard bidding rules that depend on HCP evaluation you probably won’t reach the right contract. Responder will not be able to act if you don’t super-accept on 3-card support with ruffing potential. To me this goes to demonstrate that the consensus is often wrong. Have you noticed that groups make bad decisions? If my crowd debates which restaurant to go to, I feel lucky to get to dine at the second-best place and drink the third-best wine. Then there is Congress…..

Here is a second example where for the majority slam was hard to reach on just 30 HCP. Again, opener held the dreaded 19 HCP.

W
Bob
A10
AKJ1042
6
AK108
 
E
John
Q9732
7
AQ
J5432
Bob
John
1♣*
1
2
3
4
5
6
Pass

Two other pairs reached 6, 4 were in 3NT and 2 in 5. We began with a Precision 1, but the rest of the auction was natural and game forcing. Over 4 I thought briefly as possible about bidding 4 to show the A, but chose a conservative 5 influenced the quality of my 2 long suits. John had 8 controls and did not hesitate to bid the slam, perhaps expecting to see the K opposite. A spade was led to the J and A. The AK dropped the Q. Declarer played his LHO for the Q and made 13 tricks for a clear top without the need for the diamond finesse.

Leonardo noted that truth comes from observation, but if you are too concerned with appearances and forget to count you make mistakes, as I did on the following hand.

W
Bob
A1097654
A10
A105
K
 
E
John
K
Q74
KQ93
Q10987
W
West
N
North
E
East
S
South
1♣*
Pass
2
2
4
All Pass
 
 

My hand was worth a strong 1 bid and partner replied a natural, game forcing, 2. This gave my RHO the opportunity to enter the auction with a lead directing 2 without great fear of being doubled for penalty. My first mistake was to jump to game on what appeared to be, and nearly was, a misfit. The lead was the 3 to the4, J and my A. I entered dummy with the K, returned with the A and led the A, dropping the QJ from my RHO’s hand.

My second mistake to assume too hastily that I had 3 remaining losers, the J, the K and the A. One of these might be discarded on the 4th diamond if the J fell in 3 rounds. I tried this after drawing the 3rd trump, but my luck was out as I held myself to 11 tricks. This tied for bottom as 4 pairs made 6 and one pair bid and made slam.

First, I should have bid 3, not 4, giving my partner the chance of bidding 3NT, but once in the common contract, it was too easy to assume that our Precision auction had worked to our disadvantage in allowing a disruptive overcall, as it is so often claimed to do by those who don’t play the system. I even went so far as to congratulate my RHO on her brave action. (2* would have been down 4, vulnerable.)

Leonardo wrote in his notebook that clear thinking is best achieved in a small room by oneself. Later at home without distraction it became obvious I should have run the spades to this 5-card ending:

 
N
Dummy
Q7
KQ9
 
W
Bob
4
10
105
K
 
E
RHO
K
J87
A
 
S
 
 

When the 7 is discarded on the 4 it is natural for my RHO to discard a diamond. Now I play on the diamonds and collect my 12th trick. If I had really thought ahead, I would have played the Q from dummy at the first trick, leaving the T hidden in my hand in the squeeze position. Now she might discard the J fatally allowing for a progressive squeeze to 13 tricks. With no cost involved, it pays to try even if the odds are against it.

So, here is my advice. When playing the hand once dummy has appeared, put aside ideas of what could have or should have been and concentrate one’s mind on doing what’s best under the current circumstances. Books provide useful preparation, but the cards are what’s real, and the lie of the cards is what determines the outcome here and now. Imagine yourself alone at a desk in a quite room at midnight, when time is suspended between yesterday and tomorrow. There is only the timeless present, without thirst, without hunger, without regret, and (be careful) without triumph. Count the cards, and keep counting as those are the numbers wherein you’ll discover the magic.


22 Comments

Keith GeorgeFebruary 13th, 2014 at 3:54 pm

A strong 2NT makes effective slam bidding hard, so as advocated by David Bird in the EBU Magazine, I now complete the 3D/H transfers over 2NT to show 3 card support, and break the transfer to the 4 level to show four card support. Playing at my club last night I held:
♠ K 10 9 6 3
♥ A 6
♦ J 10 5 3
♣ 10 7
and over my 2H response to my partner’s 2D (Multi) he rebids rebids 2NT(22-23). I bid 3H, transfer and partner completes. Obviously we have 30-31 points and I did a brief check on my estimate of ZAR points and came up with the magic 62. (Since I heard about ZAR in your blog I’ve been tracking how good it is as a predictor of slams both at the table and in the post-mortem. AS an example your first hand has ZAR=62 and the second ZAR=67). As completing the transfer with 3 cards is an innovation I decided not to risk any cue bidding and bid 4NT RKCB, partner showed 2 with the Q of trumps and I bid 6S. Admittedly this is not a huge advance compared to my bidding 3NT and then partner bidding at the 4 level to show a spade fit and me bidding on, but the missing agreements can be filled in in due course. Partner had:
♠ A Q 7
♥ K Q 10
♦ A K
♣ K J 9 8 4
The CA was right, but spades split 4-1 over the K so 6 was one off, the S10 was not quite enough!

bob mFebruary 14th, 2014 at 4:02 pm

Nice try, Keith! Many use HCPs as an excuse to underbid with distribution which is given high priority in ZAR evaluation. Holding back is a protection against the bad split. Rather sad, actually.

bob mFebruary 14th, 2014 at 4:48 pm

Ok, I just checked my team results from yesterday in a local club Swiss. Playing 2/1 my partner and I 4 times ended in contracts of 5 of a minor, making an overtirck each time. So 4 legitimate slams missed. For these pathetic efforts we gained 12 IMPs. Something definitely wrong there.

Rainer HerrmannMarch 3rd, 2014 at 1:17 pm

“Despite the fact that 13 tricks were available, my less than optimal achievement of 12 tricks was enough to score a clear top as 3 pairs played in 4♥, 3 in 3NT (Garrr) and 1 in 6♦, off 2. So, on this combination if you follow the standard bidding rules that depend on HCP evaluation you probably won’t reach the right contract.”

Appearances are deceptive.
The slam requires the diamond finesse (certainly on a club lead) and could be down even with the diamond finesse working, for example if the red suits do not break well.
So what the right contract is can be argued.

RafalMarch 9th, 2014 at 7:46 pm

Hi,

I wanted to appreciate your book on bridge probabilities which I have recently read. Nicely done 🙂

I would like to pose a practical question using as an example a deal played in a match. Lets suppose you’ re in 3H contract with Axx, Qxxx, Ax, xxxx vs xx, AKxx, xxxx, AJx (all the spots are low ones). Information from the bidding is that your LHO had shown six spades (spades 6-2). Basically you want to choose between two lines of play : A/ basing on assumption of H(3/2) and C(3/3), B/ 2H and 3D on your left. You might add some possibilities of KQ in C onside. My question is how should one tackle this at the table in the most efficient manner (due to time/pressure constraints). I would be grateful if you could demonstrate it in sort of a step-by-step approach.

Best of regards

Rafal

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