Good problems that uncover difficult points in a theory

Question

There is a great quote of Yitz Herstein:

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver."

A number of such problems can be found in Herstein's classic book "Topics in Algebra".

Given appropriate background, certain mathematical problems lead us to discover things we otherwise might not, often with elegance and surprise. These problems make us feel more clever than we actually are by our discovery of important and deep ideas. They have the delight and surprise of a competition problem, but reveal an important idea for a theory...and the discovery of the idea is accompanied by a lovely sense of ownership since the "would-be solver" has found it. (I'm thinking about a particular problem from Herstein's book that basically forces the idea of a quotient group on the investigator. I don't reveal which problem this is since I don't want to ruin it.)

Anyhow, the purpose of this question is to collect examples of such problems.

Question: What are some good problems that require the discovery of central and difficult concepts of a theory?

Since such problems usually require background, please include the requisite background with your answer.

Please note: I'm not so sure it is a good idea to post excellent problems along with the concepts they lead to...if a problem does what I want it to, one should be able to discover the idea that is needed by working on the problem. A good answer to this question might omit explicit mention of what idea the problem leads to.

Answer 1

Edit (Dec 2016): Encouraged by a few comments on SE, and a few direct emails about this post, I wrote up the ideas below for a journal of math education. The citation, and linked pre-print, are:

Dickman, B. (2017). Enriching Divisibility: Multiple Proofs and Generalizations. Mathematics Teacher, 110(6), 416-423. Pre-Print (no pay-wall).

---

(Adapted from a lecture described here.)

Requisite background: Secondary school mathematics for the problem, but moving forward is a nice way to introduce concepts from Number Theory and Abstract Algebra.

Problem: Observe that $(n-1)n = n^2 - n$ is always even. How would you generalize this result?

In particular, consider the following equivalent formulations:

Statement A: The product of any $2$ consecutive natural numbers is divisible by $2$.

Statement B: If $n \in \mathbb{N}$, then $2$ divides $n^2 - n$.

For me, generalization is a cornerstone of mathematical thinking and an important learning outcome for mathematics students. It is discussed in Polya's (1945) How to solve it, and there are some nice related comments in Wertheimer's (1953) Productive Thinking, but I shan't delve deeper into this point here.

For the question at hand: I think the most natural generalizations come from replacing $2$ with $k$.

Generalization A: The product of any $k$ consecutive natural numbers is divisible by $k$.

Generalization B: If $n \in \mathbb{N}$, then $k$ divides $n^k - n$.

(Side-note: I once distributed forms to fifty prospective mathematics teachers in a course on Problem Solving - not taught by me - where twenty-five received Statement A and the other twenty-five received Statement B, and all were asked to state a generalization without needing to provide a proof. The result was that almost no student/prospective teacher was able to complete the task properly, suggesting there is some deep misunderstanding insofar as what constitutes a generalization in mathematics. I won't include sample responses here, since I find them very disheartening. However, you might consider replicating such a task with your own students to see how they fare.)

The two generalizations above now "require the discovery of central and difficult concepts of a theory." More precisely, proving the first statement requires the very essence of modular arithmetic, which students might understand without having formally studied Number Theory, whereas the latter generalization is false as stated. (For example, take $k = 4$.) The natural follow-up is to classify $k$ for which Generalization B holds, which is the content of Fermat's Little Theorem, proved most easily (in my opinion) as a corollary of Lagrange's Theorem in a first course on Group Theory.

Answer 2

I think there are several good problems that can be explored (e.g., using the Moore method) by beginning with a word or term and trying to axiomatize it. I happen to think that assembling several of these words/terms and axiomatizing them would make for a nice textbook, but I digress...

Back to the question: Some examples.

What should "bigger than" mean?

(Similarly: What should "at least as big as" mean?)

Alternatively, what should "equivalent" mean?

For this answer, I will explore briefly what happens when you ask what "closeness" should mean.

Required Background: Some experience with proof-writing; naive set theory.

Problem: (to be explored with the Moore method or otherwise scaffolded) Let $X$ be a non-empty set and let $\mathbf{K}: P(X) \rightarrow P(X)$ be a function whose input and output are subsets of $X$. Given a subset $A \subset X$, we wish to define our function so that $\mathbf{K}(A)$ captures the set of elements "close" to $A$. What sort of properties should $\mathbf{K}$ have?

Next, one might proceed by considering $A, B \subset X$ and the various ways in which they can be related.

What could (or should) we say about $\mathbf{K}(A \cup B), \mathbf{K}(A \cap B), \mathbf{K}(A - B)$ or $\mathbf{K}(A)$ and $\mathbf{K}(B)$ when $A \subset B$? What about $\mathbf{K}(\emptyset)$ or $\mathbf{K}(X)$? What if we repeatedly apply $\mathbf{K}$ to some subset? Etc.

These questions can also be thought of (or translated into) sentences about closeness.

For example, the statement "nothing is close to nothing" corresponds to $\mathbf{K}(\emptyset) = \emptyset$. (1)

Similarly, it seems reasonable to assert something like "everything in a subset is also close to that subset." This can be written as: For all $A \subset X$, we have $A \subset \mathbf{K}(A)$. (2)

In addition to (1) and (2) above, one can similarly motivate two more statements. Namely,

$\mathbf{K}(\mathbf{K}(A)) \subset \mathbf{K}(A)$ (3) and $\mathbf{K}(A \cup B) = \mathbf{K}(A) \cup \mathbf{K}(B)$ (4). (Describe (3) and (4) in words!)

Given the four items as above, we now have the Kuratowski closure axioms. In particular, our exploration of "closeness" has led us to an axiomatization equivalent to the standard one in Topology!

We might even make up some terminology to go along with our definitions. For example, suppose for some $A \subset X$ we find that $A = \mathbf{K}(A)$. In such a case, we could call $A$ closed (with respect to $\mathbf{K}$).

Indeed, this notion of being closed is equivalent to the standard topological one, as well.

Answer 3

Here are a couple problems to shatter misconceptions about homomorphisms, while introducing the student to constructive thinking in group theory.

Are the following statements true? Prove them or provide a counterexample.

• If $K_1$ and $K_2$ are isomorphic subgroups of $G$, then $G/K_1$ is isomorphic to $G/K_2$.

• If $\varphi:G\rightarrow H$ is a group homomorphism, a subgroup $A$ of $G$ exists such that $G/\operatorname{ker}(\varphi)\cong A$.

To play fair, it's best to have introduced direct products and cyclic groups for the first one, and the quaternion group $Q_8$ for the second one.