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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.

10 votes

What are your favorite puzzles/toys for introducing new mathematical concepts to students?

Slightly off the mark, however you can build a toy model to accomplish the same objective. I took my section out of the classroom to a spot not far away when I was a TA for calculus. There was a dome …
Gerhard Paseman's user avatar
6 votes

Excellent mathematical explanations

I think explanatory is a group concept: a proof is explanatory when it affects a group of readers in such a way that they can explain the proof to others after they read it. There may be a more phil …
2 votes

Positive results coming from paradoxes

More in the spirit than the letter of the question, some of Alan Turing's work can be seen as applications of contradiction. The development of the Turing Machine and using it to resolve Hilbert's E …
15 votes

Examples of seemingly elementary problems that are hard to solve?

One problem that my advisor infected me with was a conjecture of Peter Frankl: Let C be a finite collection of finite sets closed under union. The easy version asks: is there an element x in union C …
1 vote

Graphical representation of mathematical structures (in the spirit of unified modeling langu...

There are areas where a graphical presentation may poorly communicate the idea. In my work on hyperassociativity, I spent many hours writing equation after equation after equation (given that hyperas …
Gerhard Paseman's user avatar
2 votes

Believing the Conjectures

Rather than give an example with characterization, I will give an example and invite characterization. Harry Altman gives a nice description of $c(n)$, what I call the one-complexity of an integer $n …
9 votes

What's a magical theorem in logic?

Birkhoff's HSP Theorem I liked the preservation theorems personally. Birkhoff's HSP theorem which identifies model classes of certain equational theories as being those classes (known in universal a …
1 vote

What's a magical theorem in logic?

This example is for me sheer wizardry: Given a class of relational structures of the same type, all with finite underlying universes and including one with a one-element universe, such a class has un …