Interesting mathematical problems for 1st year university students

Question

Can you explain some mathematical problems that you find the most interesting (NB: the problem must be accessible to a 1st year university student: that is, a problem for which there is an elegant solution that a student can find). Also, why do they interest you?

Answer 1

I think if you take the three-course calculus sequence and differential equations and an intro physics course, then you might find the calculus of variations very interesting. Consider the following question: What is the brachistochrone curve, i.e., the curve of fastest descent? In other words, what is the path that will carry a particle from one place to another in the least amount of time? A related question is what is the tautochrone, namely, the curve for which the time taken by a particle sliding without friction, under the influence of gravity, to its lowest point, is independent of its starting point? Another related curve is known as the catenary. This curve appears everywhere in nature. The calculus of variations allows one to answer such questions. Personally, I found this problems very interesting when I was an undergrad. Richard Feynman also found this intriguing. He used the principles of the calculus of variations in quantum mechanics to develop something known as quantum electrodynamics (QED).

Another memorable problem I found interesting is known as the Basel problem: Find the value of $$\sum_{n = 1}^\infty \frac{1}{n^2} = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots$$

It is hard to find a mathematician who did not find this problem and the answer fascinating as an undergrad.

It is hard to come up with the top three. It varies from person to person. Some people like applied math, while others like pure math.

Answer 2

I like this question because it is easy to understand yet incredibly hard to solve. Students may come up with a short proof but that often has a logical flaw.

We allow for equality in the definition of increasing and decreasing and call a function monotonic if it is increasing or decreasing. If $f:\mathbb R\to \mathbb R$ is not monotonic, are there three points $x<y<z$ such that $f(y)<f(x),f(z)$ or $f(y)>f(x),f(z)$?

Answer 3

If you find number patterns interesting you may like the following problem. It amazed me when I first learned about it because I did not expect odd numbers and square numbers to be so closely connected.

What is the sum of the first $n$ odd numbers? Prove it.

I'll concede that this problem is really simple -- the answer is "square numbers" -- but it generalizes quite nicely. Here are a few directions you can generalize it in:

1. Prove that your solution is correct in as many different ways as you can. (There are seven proofs in Knuth's book Concrete Math, for instance. This could be a good segue into any number of topics in discrete mathematics.)
2. Which sequence has the property that the sum of its first $n$ terms is $n^k$, for $k\geq1$ an integer?
3. What is the sum of the first $n$ numbers of the form $k(k+1)\dots(k+l-1)$?

These problems are trivial to most mathematicians but are probably not trivial to many first-years. They would be particularly accessible to first-years because they rely on little mathematical background and are easy to tackle with experimentation.

Answer 4

the ~¾ century old Collatz conjecture is a great problem to experiment/study on esp with CS based approaches (writing code, visualizing results); its possible/ultimate/eventual(?) solution is regarded as very hard by experts. however there are many basic variants of exercises on it that have "correct solutions" that are accessible to undergraduates. (example: create a Finite state transducer to calculate iterates.) it has remarkable/amazing aspects of/connections to many areas of active mathematical & CS research:

• fractals
• chaotic behavior
• dynamical systems
• number theory
• (un)decidability
• automata theory
• random walks
• pseudorandom number generators

see eg this Collatz conjecture experiments page for some basic starting leads & many links to standard/recent references