Let’s say that we have an infinitely large, flat world, and that this world is divided into different Ranks, with each taking up 25% of the area of the previous, such that Rank 2 takes up 1/4th of the land, Rank 3 takes up 1/16th, Rank 4 takes up 1/64th, etc, with Rank 1 being an exception and taking up two thirds because it’s what’s left over after the infinite sum is finished. By eating food grown in a high Rank area, a person’s Rank will increase to match it, providing an exponential scaling to all their traits, with a scale factor between 1.1 and 1.9, depending on the trait, their personal talents, etc.
Now, let’s say we have a character in this world who has a special power. They can increase the Rank of an area around themselves, with a radius of 1 mile at Rank 1. This power works in accordance with the exponential area scaling that Rank normally requires- if they wanted to increase an area to Rank 3, they’d need to have made 4 times that area into Rank 2. This means, for instance, that at Rank 1, they’d be able to turn an area of about a square meter (enough for a single bush or fruit tree) into Rank 11, and that they’d then be able to turn it into Rank 12 by walking along the circumference of the circle of the area affected by their power, since doubling the radius would quadruple the area.
However, because the radius affected by their power scales at less than a ratio of 2 per Rank, there should eventually come a point where they’re no longer able to increase their rank with their special power alone. My intuition is saying that this is probably also true even with the use of the circumference walking trick, since the radius needed to increase Rank will increase beyond the range of their power, but I’m not entirely certain.
My question is simply this: for each of the decimal scaling factors between 1.1 and 1.9, what is the maximum Rank that this character can achieve using their power alone?
So, since nobody answered it, I eventually gave up on waiting and wrote an Excel spreadsheet to calculate it for me, by repeatedly following the steps of calculating the maximum range for a given Rank with a given scaling factor, calculating the area of that in square meters, quadrupling that, then taking the floor of the log of that value divided by the log of 4, and then repeating that process on each row of the spreadsheet until it stopped increasing.
These are the values I have calculated, for each scaling factor:
I'm not certain why these particular values were the ones I arrived at, or how you would go about calculating them with pure mathematics rather than a spreadsheet, but hopefully someone else will be able to write an Answer doing so!