A spaceship is on a very long voyage. It starts with a crew of 4 women and 4 men, none of whom are related by blood. How many descendants at most can this 8-person crew produce without inbreeding? Assume each woman (whether original crew, or a descendant) produces at most 2 children, and it's all done naturally. There is no outside DNA, no cloning, no gene-editing, no surrogate baby-growing vats, etc.
I'll take "at most" to mean the absolute theoretical maximum, and the "no inbreeding" to mean the parents share no ancestors whatsoever, no matter how distant.
Since it is never better to add two ancestors where one would do, the best result can be achieved when the only men to reproduce are the ones in the original crew. At the theoretical maximum, then, all the offspring (in the generations that are still able to reproduce) will be female.
To keep track of the lineage, let's label the men A-D and the women 1-4.
Original males: A B C D (4)
Original females 1 2 3 4 (4)
Generation 1 (all female): A1 A1 B2 B2 C3 C3 D4 D4 (8)
Generation 2 (all female): AB1 x4, BC2 x4, CD3 x4, DA4 x4 (16)
Generation 3 (all female): ABC1 x8, BCD2 x8, CDA3 x8, DAB4 x8 (32)
Generation 4: ABCD1 x16, BCDA2 x16, CDAB3 x16, DABC4 x16 (64)
Since we were able to achieve maximum population increase in every generation, and at the fourth generation everyone is by necessity related to all the original males, we know we have found the theoretical maximum population, which is
and since we were asked to find the maximum number of descendants, we should deduct the original crew, for the final result of
I think they can, in theory,
produce 24 descendants (three further generations) before inbreeding becomes strictly necessary.
In the following way.
Assuming monogamous pairs is the best way to produce a second generation with as few blood relations as possible (no half-siblings). So we divide the 8 astronauts into 4 heterosexual pairs.
We're given that each woman produces at most 2 children, so let's assume it's exactly 2 for each woman. Let's further assume that two women produce only girls and the other two produce only boys.
Now we have a second generation consisting of
4 boys and 4 girls, such that none of the boys are related to any of the girls. This means that, in the second generation, blood relations impose no limitations for heterosexual pairings.
Again, 4 monogamous pairs. For each pair of sisters, their children will be blood-related anyway, so let's matchmake them with a pair of brothers to minimise limitations.
Again, assume each woman produces exactly 2 children. Since we have two brother-sister double couples, let's assume one of those produces 4 boys and the other one produces 4 girls.
Now we have a third generation consisting of
4 boys and 4 girls, such that none of the boys are related to any of the girls, although all of the boys are now blood-related to each other, and ditto all of the girls.
This time we can form 4 monogamous pairs arbitrarily. No matter how we do it, all of the next generation will be blood-related, since their fathers are all related and their mothers are all related.
Again, each woman producing exactly 2 children will lead to a total of 8 children in the next generation.
Now we have a fourth generation consisting of
8 children who are all cousins (to some degree) of each other.
- I've assumed no intergenerational breeding: e.g. we can't have one of the original male astronauts making a second-generation woman pregnant. But I'm not sure if this makes a difference, since every woman is producing her maximum allowed number of 2 children anyway.