Sounds a bit ambitious, so … let’s get
right down to the proof.

Applying the Dirichlet'sPrinciple from
mathematics, we see that THE WORST-CASE scenario when
talking about fit and misfit is that you either have
at least two 7-card fits (the so called “Italian”
fits) or one 8+ card fit.

We have 13 + 13 = 26 Dirichlet’s Balls (the cards
in both hands) and 4 Dirichlet’s Drawers (the
suits). You can easily see that (13 + 13) – 4
x 6 = 2(the
6 comes from filling-in all 4 “boxes” to
6 each) and these 2 “loose” cards will have
to fall in the one or two of the Dirichlet'sPrinciple Drawers (suits in our case),
making the “fit” of at least 6+1 = 7.

This means that you virtually always have a fit or fits somewhere.

The best-case scenario is of course a board with two
13-card fits:

A Q x x x x

K J x x x x x

___

___

K J x x x x x

A Q x x x x

___

___

Do you like this board? I don’t – it would
be a wash on any tournament …Almost everybody
will bid a GRAND. Unless someone decides to fish for
a top and bids 7 NT, hoping for a favorable lead :-)

The Theorem has deep
implications on the bidding process simply because
you do know that your goal is to findthe pre-existing fit(s)
rather than approaching the bidding trying to find out whether or not you have fits. Think about it!

The best example is the balancing. You have noticed that part of the “aggressiveness”
of the experts is that these guys will almost never
let you play at low levels, provided that you wish to stop there. They will try to push you up or to get the contract
in “their” suit.

Do they know they “have” a suit? At least
they “hope”, that’s for sure :-) The
Theorem gives you the confidence to shoot for finding
your best spot, simply because you know that it exists.

If you get a little greedy and ask the question “How
often do I get into the worst-case scenario of 2 7-card
fits”, I have good news for you. For that to happen,
you have to have special cases of only four possible combinations: 4333 vs. 4333, 4333 vs. 4432, 5332
vs. 4432, and 5332 vs. 5332 distributions. Which special
cases? The following ones: 4333 vs. 3433, 4333 vs. 2344,
4432 vs. 3244, 5332 vs. 2335, and 5332 vs. 2434 (and
slight variations of these - the unbalanced combinations
like 6160 vs. 1606, 7060 vs. 0706 etc. have a negligible
probability ~0%).

So when you run the probabilities, you reach the following
form of The Theorem:

In bridge you always have a fit:

–about 85% of the time at least
one 8-card fit

–about 15% of the time at least
two 7-card fits

If you are a careful reader, you probably have noticed
that there is a chance for you to have 3 7-card fits
(the 4432 vs. 3343 case). This chance is included in
the 15% chance for having only 7-card fits. These results
were also re-checked with several generations of deals
via DealMaker, DealPump, and Deals programs (chunks
of 1,000,000 boards each time).

One last word on this subject, stemming from the fact
that if you have only 7-card fits (neglecting the cases
with ~0% probability like 7060 vs. 0706), both hands are balanced.In this case the only thing that matters is brute
HCP power. If you have it - generally play in NT.
If they have it - let them suffer, because if you show
your head ‘above the water’, they’ll
make a salad out of you.

Think twice before balancing with 4333 despite The
Theorem. Better yet – think twice and then
pass :-)

The only exception to that rule would be when in Matchpoints
you have to push your opponents out of a non-vulnerable
1 NT - arguably the point in which you have to be most
aggressive. But think once before doing it :-)

Let’s now get back to the other bridge theory
addressing the issue of levels of play – The Law,
and how it can be extended in made more explicit. This
is the subject of the following “Superfits Theorem”.