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Can Zion's population reach 250.000 in 100 years? Yes.
The One chooses 23 people FROM the Matrix, when the matrix is Reloaded, and these 24 people rebuild Zion. And over the course of the next 100 years they are able to reach a population of 250.000.
Since these 23 are unaware of the past, and are freed by a man, who could “change he Matrix as he sees fit” this gives rise to the prophecy of the One who’ll return.
Let’s assume that each woman gives birth to two children, half are boys, half are girls. And that it takes 6 people 3 months to free a mind from the Matrix. That means in a year 6 people can free 4 minds.
Let’s furthermore assume that half the freed minds are men, the other half are women. Lets also assume that half of the freed minds chooses to stay in Zion to help rebuild. The other half continues to free minds.
Zion year 20: Population 432 (24 originals+32 children+376 free minds). Of these half the children and a quarter of the freed minds will, within the next 20 years, have two children. Half the freed minds and the original 24 can continue freeing minds.
Zion year 40: Population 3159 (432 originals+220 children+2507 freed minds). Lets now in the subsequent calculations exclude half the original population from the previous 20 years, to account for death either premature or of natural causes. There are also not that many hovercrafts. The exact number is unknown, but let’s assume that now 75% of the freed minds stay in Zion instead of half.
Zion year 60: Population 12,045 (1580 survivors + 2100 children+8365 freed minds).
Zion year 80: Population 42,278 (6022 survivors + 8373 children+27,883 freed minds).
The same cycle, except the rate of freeing goes up during the last six months. In this last 20 years the rate of freeing goes up (probably due to the availability of computers. So let’s say it now only takes 3 people 3 months to free a mind – should also account for the extra freeing in the last 6 months).
Zion year 100: Population (Time of the One) 236,309 (21,139 survivors +29,284 children+185,886 freed minds).
This has been based on 2 children calculation, and that half the population from the previous 20 years died. If some women had more than one child the number 250,000 could easily be reached.
(Math, not from the actual movies)