Quick answer: If you want the pond to stay the same size each year, restock exactly the number you sold — that is, restock half the pond. If you only want to make sure there are some fish next season, even restocking 1 fish will keep at least one fish, but to keep size steady you must restock half the pond.
Step-by-step explanation:
1) Start with a number for the fish. Call it P (for example, P = 100 fish).
2) You sell half the pond this year. That means you sell P/2 fish and you have P/2 left.
3) Now you restock with R fish. So the number of fish next season will be:
P_next = P/2 + R
4) If you want the pond to be the same size next season as this season (so you can keep repeating the plan), set P_next = P and solve for R:
P = P/2 + R ⇒ R = P/2.
So you must restock exactly P/2 fish — the same number you sold.
Example with numbers:
- Suppose P = 100 fish.
- You sell half → sell 50, so 50 left.
- Restock R = 50. Next season you have 50 + 50 = 100 again.
If you restock less than half: the pond will shrink over time. There is a neat long-term fact: if every year you restock R fish after selling half, the pond will settle (in the long run) to about 2R fish. That means if you want at least S fish in the long run, pick R ≥ S/2.
One-sentence link to Clause 65 (Capitulary de Villis): Medieval rules like Clause 65 warned managers not to sell everything at once so resources would last — mathematically that is exactly the idea: leave half (or restock what you sold) so the pond keeps producing fish next year.
If you want, tell me how many fish you start with (a number for P) and I will show the exact numbers year by year.