Recent content by LulThyme

Problem 2 is ill-defined. The answer depends on the distribution you choose for the points. Even "natural" choices may lead to different answers. See https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

And, because this surely will come up, you don't need infinitesimals to define limits. The common modern definition of limits involves only real numbers, by clever use of quantifiers.

It's not true that these are the same. Here is one important difference: In fact, in 0.99999..., before any given "9", there are only FINITELY many "9"'s before and thus every "9" has an actual, FINITE position. In your proposed example, the "8" does not have a defined finite position, so...

Strictly speaking, science is not in the proof business. You only accrue more and more evidence. If you want to argue on that level, then the Earth being roughly spherical (rather than flat) or the existence of Venus is also not a fact (and neither is anything really, with the possible exception...

No, Aleph_1 is the next cardinal after Aleph_0. (I think you need the axiom of choice for this to be true strictly, but some version of it is true in ZF.) What IS undecidable is whether the continuum (which is also 2^(Aleph_0)) is equal to Aleph_1, in other words, if there is a cardinal between...

What exactly is confusing about the Wikipedia formulas? To be really honest, I do think everybody's time would be spent more productively if you tried to learn how to parse such formulas. Especially in the long run, you wouldn't have to ask again for every new formula (teach a man to fish and...

Yes, it seems airdrops are not enough, even if you forget about the unit limit per turn (which I'm not sure exists). Just the delay to build them means that you can't expand the outer border of your empire faster than by foot. On the other hand, railroads work! You can produce a number of...

Same here. I've played thousand of hours of Civ 2 but it's been a long time. I thought of railroads already but then convinced myself it wouldn't work, because you still need go get to the new square. But now that I think about it, as the number of engineers grows, you should be able to get...

You seem to be mixing up very different things. The fact that the ratio of consecutive Fibonnaci number converges is well-known (and yes, this implies that the sequence grows exponentially). What this has to do with our problem at hand, I don't know. I think too many of you were focused on the...

It's been a while since I've played either of these games... For Civ 2, I think the growth of the number of cities would be limited for the same reasons (movement speed). If you can do exponential population growth, it would seem it would have to be through very fast city growth. I seemed to...

This was the problem with your argument. Exponential growth means that the growth is proportional with the value. As the empire gets bigger, the ratio that is in the outermost x rings goes to 0, hence if only that contributes, it certainly can't be exponential growth of number of cities...

In conclusion : assuming linear population growth in cities not building settlers, the best total population growth that can be achieved is cubic.

Now that we have an upper bound, let us try to achieve it. Assuming movement as in civ and that cities build settlers in at most constant time, we can demonstrate quadratic growth in the number of cities, as follows. 1. Each city produces 2 settlers. (at most constant time) 2. For each city...

I'm surprised everybody assumed or predicted exponential growth when it is so clearly wrong. You have only shown that there is an exponential upper bound, you havn't really show that exponential growth can be achieved. You are very easily satisfied ;) In fact, while trying to explain my...

The following lemma suffices: Lemma : In any sequence of 10 consecutive positive integers, there is one that is divisible by neither 2, 3, 5 or 7. Once we have the lemma, it suffices to pick an integer x with the property guaranteed by lemma. If x is not relatively prime to the product of all...