I wanted to comment on HDE's answer. It's an excellent answer (meaning I agree with it), but I wanted to add the following observations.
First, if you want your character to be a student, you'd best make him a graduate student. As HDE mentions, at the undergraduate level you might have a basic understanding of linear algebra and PDE's, but not enough ...
For this to work in a good story, you need something that is "bad" mathematics that is used by physicists and easily understood by readers (in its simplest form). The option that comes to mind for me is zeta function regularization, in particular Ramanujan summation. The most famous example of this was "proven" by my favorite mathematician and is below.
For basic physics, you need (multivariable) calculus and linear algebra. This is basic literacy. You won't get anywhere in physics without them. There are some differential equations too, but one tends to learn that on a case-by-case basis as one studies examples.
For general relativity, you need Riemannian geometry. Talking about curved spacetime only ...
Let's assume that this student wants to begin by understanding the twin pillars of modern physics: quantum mechanics and general relativity. There are several major tools in the toolkit of anyone studying both of these theories at a basic level:
Calculus (single-variable and multivariable)
Operators such as divergence, gradient, ...
Math does this all the time, you don't even have to go to great lengths
The set of rational numbers "Q" is contained within the set of real numbers "R". Both sets are infinite, but "Q" is countably infinite an "R" is uncountably infinite (therefore "larger").
Some of the neat properties of this relationship are:
"Q" doesn't need the real numbers to do ...
Anything noteworthy that E discovers and knows P is not working on.
This happened in our history - in 1535, Antonio Maria Fiore challenged Niccolò Tartaglia to a public contest. He gave problems that lead to solving depressed cubic equations, something which was considered difficult if not unsolvable and he (Fiore) found a solution to the problem.
Digits of pi
There are many estimates for pi, starting from 22/7 and working from there. Your Mr E may even have memorised it to some number of digits.
But Mr P has access to the Newton-Raphson iterative method. (Newton is very much on the border between medieval and Renaissance; indeed he is one of the people who created the Renaissance. So we can justify ...
A lot of good solutions have been given already, I think there are a couple that could be really nice because they can be shown to the public immediately and can be computed very accurately with relative little ease:
the period of a pendulum
the failure load of a truss structure
the biggest fish the fishermen in the next town over have caught based on a ...
Length of a Coastline
This is a bit of an underhanded cheat, but it's one way you can guarantee a win for Mr. P. Ask for the length of a coastline around an island or lake, or from one port to another on a seashore, etc. Thing about this is that due to the Coastline Paradox, the length can be almost anything you want it to be depending on how closely you ...
Anything involving mathematical chaos or general sensitivity to initial conditions will be difficult for the estimator. The trick is to find something which will be reasonably easy for the calculator, which means something where the initial conditions can be reproduced.
My favorite would be showing some rows of Rule 30 and asking for a row a few rows on.
Something concerning Benford's law.
Benford's law is an observation about the frequency distribution of
leading digits in many real-life sets of numerical data.
In short, smaller digits appear to more frequently lead a number.
For example, if asked for an estimation about how many times 1 would be the leading digit in the whole 90000 tax records of ...
The Birthday Paradox (simplified).
Have six jars, each with twenty numbered balls (the ten jars can hold differently coloured balls to make sorting simpler), 1 to 20.
Say we extract one ball from each jar, thus obtaining a collection of six balls.
How often will (at least) two of those balls share a number?
Calculate the probability of this ...
What if instead of having to calculate something very precisely, you proposed a MCQ (Multiple Choice Questions) instead?
A MCQ is a list of questions with (most of the time) 4 proposed answers given for each.
The sky is :
D. Above our head
To make it even trickier, sometimes MCQs have malus points if you answer something wrong. ...
I, the King, wish to share the Kingdom's wealth with the People. If the Kingdom's population keeps growing, how long before they collectively are richer than the Royal Family?
An estimate would say 'Probably 100 years'. An exact formula says never.
Stick with me here.
Let's say this is a verrrryy nice king. What goes around comes around- he shares his ...
Decrypt an RSA-style Message
Although the specifics of RSA encryption and decryption - or even more generally public key cryptography didn't come around until the 1900s, the ideas of factorization go back to the Greeks.
Furthermore, RSA itself works on rather simple mathematics: multiplication and modulus operations. Wikipedia even has a simple example ...
A simple, trivial, one would even say childish. Calculation of how much of a square foots the kingdom is. While knowing it's two dimensions.
Your Mr. P is John Napier. Who wrote a book called
Description of the Wonderful Rule of Logarithms
Using word Wonderful was just to smear in the face of Mr. E (of whom history forgot) that John found a way to ...
The Archimedes eureka problem.
Have a number of strangely shaped objects made out of different materials.
The challenge is to work out which of them is made of the densest material.
You can't estimate that as they are strangely shaped. You can't just weigh them as they are different volumes.
The solution is to weigh them, then sink them in water and see ...
Date of the next lunar/solar eclipse. See “Antikythera mechanism”:
Eclipses follow such a complex pattern, they cannot be estimated from previous events, but as the Greek mechanism shows, they could be calculated.
Estimate exponential growth using grains of rice and a chessboard.
The story goes:
The ruler or India was so pleased with one of his palace wise men, who
had invented the game of chess, that he offered this wise man a reward
of his own choosing and he said to the man: “Name your reward!”
The man responded: “Oh emperor, my wishes are simple. I ...
The Problem of Proving
Any test they would have to do needs a proof. Its nice that MR. P would calculate the exact right value and Mr. E guesses exactly the same value, both might be wrong and nobody would know it. So you need something that is not know, hard to estimate but possible to measure or reassure. Problem is that some calculations might require ...
The Sand Reckoner of Archimedes
Archimedes of Syracuse was a Greek mathematician who lived in the 3rd century before the common era. He was probably the greatest mathematician of the antiquity
Among many other things, he was interested in devising a notation for very large numbers. In order to present his suggestion for a system to represent very large ...
Anything that can be calculated can be estimated. However something with a sinusoidal function like the tides requires significant precision even on an estimate, if you're betting on ebb and you get flow, or you're betting on a high tide and you get low, you're in a reasonable amount of trouble. There's no slack for adding a margin for error, you're ...
Where a cannon ball will land. A problem of extreme importance in the Middle Ages, why Newton and Galileo were studying gravity, and (because of the scale involved) nearly impossible to estimate with the precision desired by commanders.
Because of the great mass of a cannon ball, the effect of wind and complicating factors of air resistance are ...