Please have mercy on a simple, non-math person like me. I'm trying to find a formula (or even be told it's impossible) for a unique and difficult compound interest scenario pertaining to the Golden Goose. At least it's unique and difficult to me.

**Setup:** Suppose I have a "Golden Goose" which is actually just a regular goose except it lays a solid gold egg every day. Also if I wait and accumulate 100 golden eggs, I can exchange them for *another* goose.

**Scenario:** So I've got a golden goose and I've got a friend, Jimmy, that also has a golden goose! We agree to pool our eggs so we can collectively buy more geese more quickly.

- So instead of each of each us waiting 100 days to get another goose, we combine our eggs and reach 100 eggs in 50 days. Jimmy then buys a goose bringing us to 3 geese total. We wait 34 more days, combine our 100 eggs again and then I get a goose.
- So at the end of this 84 day egg-pooling-cycle we each finish with 1 more goose than we started with. We each got a 2nd goose 16 days earlier than if we waited on our own!

Now let's try it again adding a 3rd friend with more geese than either of us.

- We have another friend, Sally, with 4 golden geese. So I have 2, Jimmy has 2, and Sally has 4, which brings us to 8 total. We decide to pool our eggs again to double our geese.
- Since Sally has double the geese we each do and is therefore contributing double the eggs, it's only fair that she gets double the geese during this next egg pooling cycle.
- So in 13 days (we're rounding up from 12.xxx days) we accumulate 100 eggs and Sally gets a goose, bring the new total to 9 geese.
- In 12 more days, I get a goose. New total = 10 geese
- In 10 more days, Sally gets a goose. New total = 11 geese
- In 10 more days, Jimmy gets a goose. New total = 12 geese
- In 9 more days, Sally gets a goose. New total = 13 geese
- In 8 more days, I gets a goose. New total = 14 geese
- In 8 more days, Sally gets a goose. New total = 15 geese
- In 7 more days, Jimmy gets a goose. Final total = 16 geese

- So at the end of this cycle, which lasted a total of about 77 days, Sally finishes with 8 geese, I have 4, and Jimmy has 4.

- Since Sally has double the geese we each do and is therefore contributing double the eggs, it's only fair that she gets double the geese during this next egg pooling cycle.

So we all figure, wow! This is great! The more people with geese we pool together the more everyone gets geese more quickly. Right?

Right?

**WRONG!!**

If Jimmy and I would have simply pooled our 4 geese without Sally (or if Sally would have accumulated on her own without us), we/she would have reached 8 total geese in 77 days still. We didn't save any time by adding Sally.

**So now we're left asking ourselves**

- Was our math right and fair? Did everyone get the number of geese they were supposed to?
- Is there a scenario where pooling eggs is inefficient for some or all participants?
- What if it was 10 people each with 1 goose?
- 20 people each with 1 goose?
- 3 people each with 1 goose and 2 people with 4 geese?

- Is there a ratio for pooling eggs that is MOST efficient?

# And now the real question:

- What is a formula to find the best ratio and/or length of time for pooling eggs and gaining geese based on the number of participants AND the number of geese each participant begins with?

*Note*: Assume the geese all lay one egg at the same time, in unison, in the morning. So if you acquired 100 geese in the morning after they had already laid their eggs for their previous owner, you would have to wait 1 full day for them to lay again next morning.

I was surprised to know that there was already a question about the Golden Goose, but it's not super related.