Has anyone ever conquered the entire map?

[Landwalker]

Precisely. We need a new formula to take into account the culminative effect of expansion speed.

Happily.

As the Romani, you begin play with 5 provinces. There are a total of 199, and you have 1144 turns in which to achieve this. The solution is to find out what your rate of expansion relative to your existing size you must average per turn in order to conquer the entire map in the allotted time. This accounts for the "snowball effect" of expansion, where greater territory means greater resources means greater ability to prosecute wars. It only considers rate of expansion, however, not the many variables (like financing and order) necessary to maintain that rate.

Warning: The following contains math.

We begin with the equation s*(r^t) = F, where s is your starting number of provinces, r is your expansion rate relative to existing size, t is the number of turns of play, and F is your final number of provinces. Note that the ^ symbol indicates "to the nth power" (so that 2^3 would be "two cubed", or "two to the third power", or 2*2*2 = 8.

For example, if we began play with three provinces, and wished to achieve a total of 40 after 25 turns, we would have the equation of 3*(r^25) = 40.

Since the Romani begin with 5 provinces and we are allowing all 1144 turns to conquer 199 provinces, the equation becomes: 5*(r^1144) = 199. We want to solve for r, so that we know the necessary rate of expansion. So, here we go:

'  5 * r^1144 = 199
       r^1144 = 199/5 = 39.8
         1144 = log[base r](39.8)
         1144 = log(39.8) / log(r)
1144 * log(r) = log (39.8)
       log(r) = log(39.8) / 1144
       log(r) = 0.0013984992
    10^log(r) = 10^0.0013984992
            r = 1.003225354 

This means that each turn, your empire will need to be 1.003225354 times larger than it was on the turn before in order to reach 199 provinces in 1144 turns from a starting point of five provinces. This translates into a roughly 0.323% growth per turn.

If you would like to extrapolate this process to other conditions--shorter times, different number of starting provinces, or a different target number of ending provinces (particularly handy if you want to get it right down to the victory conditions themselves), here is the "end" formula:

r = 10^{[log(F/s)]/t}

So, going back to the original example of 3 starting provinces, 25 turns, and 40 end provinces, you're looking at an r value of about 1.1092, or a brique 11% growth per turn.

Got all that?

Cheers.

Editted what to make with the pretty alignment of equal signs.

Edit 2: It's worth pointing out that once you have "r", you can easily determine both A) How many provinces you need to conquer this turn, C*r, where C is your current number of provinces, and, if C*r is significantly less than 1, B) how many turns it should take you to conquer your next province, t = log[(C+1)/C] / log(r).

Edit 3: Mixed up a division and a multiplication in the "root formula" for r. This has been corrected, as well as its impact on the determination of how many turns it should take you to conquer your next province.

[Moosemanmoo]

Happily.

Warning: The following contains math

*screams and runs for the door*!

[Tellos Athenaios]

It's not really a realistic model given that it assume exponential growth. Something which is quite doable with smallish empires in which one settlement more or less matters a great deal in %; but downright impossible to achieve with medium-large-hughe empires. \

For reliable growth models you'd be looking at some kind of 'natural' growth i.e.: something like: dy/dt=0.4y-0.1y^2. Where y is the amount of provinces you have.

[Landwalker]

It's not really a realistic model given that it assume exponential growth. Something which is quite doable with smallish empires in which one settlement more or less matters a great deal in %; but downright impossible to achieve with medium-large-hughe empires. \

For reliable growth models you'd be looking at some kind of 'natural' growth i.e.: something like: dy/dt=0.4y-0.1y^2. Where y is the amount of provinces you have.

On the scale of EB, it isn't radically impractical to use an exponential growth model, because the rate of growth is so low (0.32% for the Romani, and even single-province factions have it as low as 0.46%). At no point during the 1144 turns do you need to be capturing more than one settlement per turn (even when you're up in the 190 province area) in order to meet the goal, and given that by the time you get up that high a character probably has at least half a dozen major armies and might well capture three settlements in a single turn, the expectation, while perhaps not a realistic model, is nevertheless not unreasonable.

A better model of growth would probably be a logistic one, where growth starts slow, then booms, then slows again as the hassles of maintaining a large empire start to occupy more of your resources. But even I'm not so bored as to try to derive a logistical model of growth for EB. ;)

Cheers.

[Gebeleisis]

gAH ,i come back home from maths at high to relax on the forums and i see more maths













SOME ONE PLS RESCUE MEEEE!!
*screams and runs trough the closed door*

[Landwalker]

Cheers.

[Hax]

Meh.

I liked Pintsize better.

"Gravity. I hate gravity."

[zooeyglass]

Cheers.

ah, qc pops up everywhere

[Tellos Athenaios]

A better model of growth would probably be a logistic one, where growth starts slow, then booms, then slows again as the hassles of maintaining a large empire start to occupy more of your resources. But even I'm not so bored as to try to derive a logistical model of growth for EB. ;)

Natural growth. Logistic growth. Same thing.
And while the Math behind that one isn't too complicated... I completely agree with you: don't do it unless you are really bored.

[duncan.gill]

How's this??




I didn't cheat. Chose not to take out the Casse as EB v1.0 came out and I was playing the game on v0.8