In one of the stories I'm designing, in Middle Age time, the main character, Mr. P, is a good mathematician serving the King and other lords in different works (army numbers and suppliers counts, ballistic calculations, civil and militar buildings design, etc.).
He is very good making precise calculations.
The problem is that another character, Mr. E, not evil but deceiver after all, tries to get over him.
Beyond his traps, in order to get choosen for the contracts, Mr. E offer always his results earlier than Mr. P.
To get these response times, Mr. E always makes estimations and he adds/substract a safe margin, depending the situation.
Mr. P knows that most of the time Mr. E gets good enough results. For that, he wants to challenge Mr. E in a public meeting. His idea is to link some number related questions, for example:
- How many people there is in the meeting/party?
- How many houses can be found in all the kingdom?
- ... even how many stars can be observed in the sky.
Mr. E will find a good answer for everything. At that point, I need Mr. P to ask Mr. E something (known in this age) that is extremely difficult/not possible to estimate but could be calculated by Mr. P with his calculations.
The problem is that I cannot imagine something calculable and not estimable, that could be demostrated/validated by audience.
If this is not possible, I could settle for a calculation which estimation would be a long way from the real result, and how to demonstrate quickly to any audience.
TL;DR: I need something extremely difficult/not possible to estimate which can be well calculated using equations and able to be checked (in a Middle age scenario, so advanced physics and that stuff is not useful).