# What formula could I use to find out how many paperclips my factories could produce?

I'm playing Universal Paperclips and I'm near the end of the game. I currently have 2.3 quadrillion factories. Each factory un-upgraded can produce 100 billion clips per second. The first upgrade increases each factory's performance by 100x. The second increases performance by 1000x. The last upgrade says "Each factory added to the network increases every factory's output 1,000x".

I'm trying to figure out what formula I can use to determine the theoretical maximum number of paperclips my factories can produce per second with all three upgrades.

If I'm understanding this right after the first two upgrades the "base" value of 100 billion per second becomes (100 billion * 100 * 1000 =) 1 quintillion per second. But how do I express the part where each factory added increases every factory's output 1000x?

• Apologies if this is tagged incorrectly, I'm fairly new to this site and wasn't sure with most of the tags I saw. Jul 17, 2020 at 18:15
• Do the upgrades stack or do higher level ones replace earlier ones? Jul 17, 2020 at 19:13
• The game doesn't clearly explain it but I'm reasonably certain that they stack from the wording. Jul 17, 2020 at 19:22

$$\text{total clips/sec from factories} = (n \text{ factories}) \times \left(\left(10^{11} \frac{\text{clips}/\text{sec}}{\text{factory}}\right) \times 100 \times 1000 \times \left(\frac{1000}{\text{factory}} \times n\text{ factories}\right)\right)$$
Increasing your number of factories from $$n$$ to $$n + 1$$ would lead to an increase of:
$$10^{19} \times (2n+1) \text{ clips/sec}$$
This can be calculated by taking your rate at $$n+1$$ factories and subtracting your rate at $$n$$ factories.
You should confirm with your findings in-game to determine if the upgrades indeed stack in the above way. Otherwise, you'll be making as much as $$10^5$$ times less.