2 Questions

Adonnus said:

1. How do I integrate/annex satellite states?

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Unfortunately, there is no way of doing this in game, aside from cheating. I'n the spoiler tags below is the way to do this;

Adonnus said:

2. How do I get cores on my provinces?

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Again, unfortunately you cannot gain cores in the game.

Adonnus said:

I am playing as Prussia and trying to unite Germany, but I don't see any annex option for Saxony which just appears as a blob in my otherwise perfect Germany, and also my country only has cores on Prussia which hurts my manpower.

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While in other Paradox titles you can generally gain cores after a certain period of time has passed, MotE has such a short time frame that it would not really work. Boosting manpower as Prussia is possible however. I'd advise grabbing Bavarian land as soon as possible, there are some very populous cities there which will have a positive impact, and you should build depots in every province which generates manpower. Use the ledger to view all your cities, and sort by manpower generated. Build the depots in the larger cities first as it adds a small percentage to the manpower.