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Successful settlements need four critical things: beds, food, water, and defense. In the same respect, power is required to meet these needs beyond the most basic sense, especially for defense. There is an easy and quick way to understand how much of each you need:
Beds: One per settler (max of 10 settlers plus 1 settler per Charisma point).
Food: One food per settler.
Water: One water per settler.
Defense: One point per settler, although more is more helpful.
Power: Add all of the power consumption things you need running, that’s the required power.
As noted, you’ll probably have a max of 11 to 20 settlers, depending on your build.
So make sure you have enough water, food and defense with at least 1 charisma to get up to 11 and more.
So the max should be 21 as it is possible to get a SPECIAL over 10, with charisma at 11, that would make possible to have 21 settlers with the same amount of beds, food, water and defense.
Charisma boosting gear also increases the max, the calculation is 10+CHA stat(maximum of 11 with bobblehead)+CHA bosting gear/drugs=max pop, though there are some conflicting reports about the longevity of that solution
also a settlements defence needs is calculated as such:
Defence rating > local food production + local water production
So a settlement with 25 food, and 25 water needs atleast 50 defence to satisfy it's defence need
+ the SPECIAL book if you choose to use the point it gives towards CHA.
This brings up a good question: Do connected settlements take the extra from other connected settlements when it comes produce or water that they lack?
I have two industrial water purifiers as well with their own dedicated large generator. The first day I saw 60(ish) purified water (which was in the mid 20s the day before) in my workshop inventory. But then I connected a couple more settlements and I haven't seen the number go up at all.
I have 7 charisma, nothing to boost that. The Castle had 16 settlers. It got attacked and everyone wiped out. I know if you go there the settlers respawn but they didn't. I waited a long time. Still all zeroes.
As usual I had overpopulation problems. As the Castle was still empty I sent them there. Next thing I know there are 29 settlers there. I gave them all beds and atm there are even 31 settlers.
Does the reincarnation of the first bunch not count towards the population limit?