OK, first of all, don't laugh, this is related to a social iOS game, but I promise there is some real juicy math here...
I am attempting to derive a general formula or algorithm that can predict the next 9 numbers in a series based on a single starting number. Every series has exactly 10 numbers, no more, no less. Here are four samples that use the correct (unknown) algorithm:
Starting numbers: ("Initial cost") 7,500 4,000 2,500 1,000 Next 9 numbers: ("Upgrades") 12,500 6,680 4,175 1,670 21,000 11,000 7,000 2,790 35,000 18,500 11,500 4,660 58,000 31,000 19,000 7,775 97,500 52,000 32,500 13,000 163,000 87,000 54,000 21,700 270,000 144,000 90,000 36,250 450,000 240,000 150,000 60,500 750,000 400,000 250,000 101,000
More examples can be found at the following forum page under "money buildings", ignoring "gold buildings". (Remember, I said don't laugh.) The data here is good; I have confirmed both the validity of the sample data provided, and that these series are indeed based on only the "initial cost" by inspecting the manifest files that are sent by the game manufacturer (they only contain the first number in the series, implying that the game's code figures out the rest).
Summary of my findings so far:
- Each number is approximately 5/3 of the previous number, but not exactly.
- There is no obvious pattern surrounding the rounding up or down of values.
- Extrapolating a simple exponential trend (in Excel) does not produce an exact or "close enough" solution.
- I have solved other algorithms based on other series in this game, and some involved rules, such as "the 10th item in the list shall be multiplied by constant X", so it is possible that similar rules exist.
- There is a small chance that the numbers in this series is dependent on other series not listed here but is listed on the forum page linked above. If someone makes a good case for it, I can include that data here as well.
- My best guess for the actual solution probably involves some quirky rounding method.
It is worth mentioning that I have already attempted using the OEIS suggested by the excellent question, "Predict next number from a series". Along with my own trials and errors, I have hit a dead end and am hoping that someone in the community here can figure this out.
I will accept any of the three following answers:
- Proof that an algorithm cannot exist or cannot be found without psychic powers.
- An exact answer.
- A general approach that will eventually lead to an answer.
(If the solution turns out to be interesting, it would be a good candidate to submit to the OEIS.) And for those wondering why, there is no need for a "why", I am just trying to learn new techniques for solving these types of problems, if they exist.