You need a population of Planet size * Planet size * 2500, as that page states. That means you have to calculate what number of flats would lead to that population figure.

The Planetary Economy rules tell us that the Base Population (BP) is calculated as BP = (Ln (Flats) x (Flats)^2) + (Flats).

So for a sample planet of size 10x10, you'd need 10*10*2500=250,000 population for control.

250,000=(Ln (X) * (X)^2) + (X)
X = 215.604

So to control a planet of size 10x10 you'd need to place 216 flats to ensure sufficient population.

That is almost the whole truth, but not quite Mikel. What you have calculated is base population. It's worth pointing out that total population is affected by the tax level of a planet. For a newly found planet, the tax level is 0% so the population modifier is 1.15 times the base population. So the minimum number of flats to secure a 10 x 10 planet is actually 203.

(Note that credit for this knowledge goes to Ellias Aubec and his fantastic super calculator which tells you all these things.)

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"Civilized men sleep soundly because rough men stand ready to do violence on their behalf..."

Because if you have 203 flats, the BP = (Ln (203) * (203)^2) + (203) = 219,155
Thus at 0% tax, the total population = 1.15 * 219,155 = 252,028 which is just greater than 250k, the required population to control a 10 x 10 planet.

If you take Mikel's figure of 216 flats, the BP = (Ln (216) * (216)^2) + (216) = 251,005
The total population is actually 1.15 * 251,005 = 288,656 which will of course give control but is much more than required.

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"Civilized men sleep soundly because rough men stand ready to do violence on their behalf..."