We can easily determine what our minimum fit is based on what we learn from the auction and the opponents.
Hypothesis: When they have a fit, we have a fit.
Rule #1: When the opponents have an eight card fit our minimum fit is 7.
This rule is proven in the following manner:
The opponents have the following auction:
1
P 2 ?
The opponents have a minimum fit of 8 Spades. That leaves no more than 5 Spades for us. Since each of our hands has 13 cards for a total of 26 cards for the partnership, and we – as a partnership – hold no more than 5 Spades, 21 of our cards must be Hearts, Diamonds and Clubs, the 3 remaining suits. When we divide 21 by 3, the product is 7 – the number of cards that we could have as a partnership in each suit.
If you held Ax 
Axxx 
Kxxxx 
xx, the only instance in which your partnership would not have an 8 card fit is when your partner held 3 3 2 5.
This is represented in the following equation: 26 – (13 – their minimum fit)/3 = our minimum fit.
26: the total cards held by the partnership.
(13 – their minimum fit): the minimum number of cards held by us in their suit.
3: the number of the suits remaining.
Rule #2: If the opponents have a nine card fit, we must have at least one eight card or longer fit.
1 P 3 ?
In this auction the opponents typically have 9 or more Spades.
Using the fit formula we can determine the following: 26 – (13 – 9)/3 = 7.333
26: the total cards held by the partnership
(13 – 9): their minimum fit subtracted from 13 (the total number of Spades)
3: the total number of suits remaining.
Since cards are always whole numbers (you cannot have 1/3 of a heart for example) we must have at least one 8 card fit.
An alternative way of expressing this rule follows:
Rule #2A: Whenever the partnership has fewer than 5 cards in any given suit, we must have at least one 8 card fit.
By the use of this formula we can also determine the following minimum fits:
When the opponents have a 10 card fit:
26 – (13-10)/3 = 7.666. One 9+ fit, or two 8 card fits.
When the opponents have an 11 card fit:
26 – (13-11)/3 = 8. One 10+ fit, exactly one 9 AND one 8 card fit, or three 8 card fits.
When the opponents have a 12 card fit:
26 – (13-12)/1 = 8.333. One 11+ fit or exactly one 10 AND one 8+ card fit, or two 9 card fits.
When the opponents have a 13 card fit:
26 – (13-13)/3 = 8.666 one 12+ fit or exactly one 11 AND one 8+ card fit, or exactly one 10 AND one 9 card fit, or one 10 AND two 8 card fits, or exactly two 9 AND one 8 card fits.
Intellectual Property of GS Jade Barrett not to be distributed without permission.
Copyright 20 June 2001