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Yes, that is a very good question and it has been discussed profitably on other threads here, and I will be DARNED if I can figure out what search terms to use to turn 'em up, or to reconstruct (via google) the information that was cited, even the stuff I myself remember posting
My best recollection, though, is that recommendations were cited, both from actual (very experienced) chicken breeders and from academic types, that you probably ought to have at least 20-50 hens and 3-8 cocks, managed in some sort of intelligent nonrandom system. Also I have seen it suggested numerous times by (very experienced) chicken breeders that you should probably at most be keeping 10% of your chicks as further breeders. (Of course, the more chicks you produce the more selective you can be). If you are keeping no more than 10% at the most, that gives you a clue about how many you need to be hatching...
(Note that all three of these things -- how many breeders you have, how many chicks you produce, and how stringently you cull the chicks -- are independantly important. The first and third relate mainly to the 'health' of your gene pool, the second relates mainly to your ability to make progress selecting for or against particular traits).
Aha, wait, I have found one bit of it:
https://www.backyardchickens.com/forum/viewtopic.php?id=255023&p=2 post #33 -- "According to R. D. Crawford, poultry scientist at the University of Saskatchewan, the ideal number to minimize genetic losses and thereby indefinitely sustain a flock is 25 hens (to 50) and 20 (or more) cocks"
Those interested in properly, sustainably, effectively working with rare breeds and small population sizes should check with various rare-breed organizations such as ALBS, SPPA, RBC, etc etc etc because many of them offer useful pages, pamphlets or books on managing the gene pool of small flocks/herds.
Or is it possible that with even a dozen birds, lets say 3 roos and 9 hens, that you may end up with offspring that is no good to pass on??
Well of course it depends what you start with
But assuming the initial material is of reasonable quality and reasonably-stable breeding, I don't see any problem starting that way -- as long as you are producing
large numbers of chicks each year, culling strictly down to only the best, and ideally carrying over a few more breeders than that for future years.
FWIW I can tell you that, starting with like 8 (culled down from 12) sussexes that were basically what I'm personally looking for but not of any 'special' lineage, the unfortunately-very-few chicks I managed to produce last year were ALL a considerable improvement over the parent generation; and my big project for this year is to A Whole Big Lot Of Chicks so I can do more-serious culling, in addition to adding a few from the same original bloodlines to broaden the male-side genetic base. I am not aiming for show birds, I want them to basically fit the breed description but mainly be good meat producers and respectably-adequate layers. But the same principle applies if you are breeding for fancy points.
Pat