133
votes

### Mathematical games interesting to both you and a 5+-year-old child

One evening at the dinner table, when my oldest daughter was 3 or 4, I was in a teasing mood, and I called her a goose. She didn't want to be a goose, so she refuted the claim, "I am not a goose!" ...

Community wiki

98
votes

### Mathematical games interesting to both you and a 5+-year-old child

The game "Set" seems to fit the bill. It's a card came where there are cards that show images which have four different features, each of which comes in three possibilities:
number (1, 2, or 3 ...

Community wiki

73
votes

### Mathematical games interesting to both you and a 5+-year-old child

Another topological game: Sprouts.
Rules:
The game is played by two players, starting with a few spots drawn on a sheet of paper. Players take turns, where each turn consists of drawing a line ...

Community wiki

68
votes

Accepted

### Ideas for introducing Galois theory to advanced high school students

I have now twice taught Galois theory to advanced high school students at PROMYS. This is a six week course, meeting four times a week, for students who already are comfortable with proofs and, in ...

Community wiki

53
votes

Accepted

### Zorn's lemma: old friend or historical relic?

I agree with almost everything in your post. But still, I believe I know why people use Zorn's lemma.
My answer. Zorn's lemma encapsulates succinctly many of the consequences of AC via transfinite ...

- 205k

51
votes

Accepted

### History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$

I now believe that my question (and suggestion that proof $(1)$ should have become standard before Lacroix) relied on the misconception that tangent was easier to differentiate than arctangent. In ...

- 28.8k

49
votes

### Mathematical games interesting to both you and a 5+-year-old child

Knot or not?
The topological game involves a projection of a knot, drawn onto paper, such as:
Player 1 picks an intersection and assigns a crossing (which segment of the curve is "above" and which "...

Community wiki

46
votes

Accepted

### Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?

No, it’s not consistent.
Let $V=k^{(\omega)}$ be the vector space of finite sequences of elements of $k$. Then $V^*$ can be identified with the vector space $k^\omega$ of all sequences, and elements ...

- 30.9k

42
votes

### Not especially famous, long-open problems which anyone can understand

Here is another easy to state problem which is 140 years old but not very famous.
Consider the potential of finitely many positive charges:
$$u(x)=\sum_{j=1}^n\frac{a_j}{|x-x_j|},\quad x,x_j\in R^3,\...

Community wiki

40
votes

### Short papers for undergraduate course on reading scholarly math

I think this is a delightful paper:
Hull, Thomas C. "Solving cubics with creases: The work of Beloch and Lill." The American Mathematical Monthly 118, no. 4 (2011): 307-315.
(PDF download.)
It ...

Community wiki

40
votes

### Zorn's lemma: old friend or historical relic?

I agree with the existing answers, but I personally like Zorn's lemma both pedagogically and mathematically for an additional reason: the "poset of partial solutions" that it introduces is a ...

- 20.2k

39
votes

### Why do we need random variables?

An honest answer should start with the fact that probabilists usually care more about the distributions of random variables than the underlying probability spaces. Terry Tao has a blog post in which ...

- 11.5k

38
votes

### How do you mentor undergraduate research?

As a former faculty member in several elite college and university departments, I have worked with and mentored a number of talented undergraduate and graduate students including several mathematics ...

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37
votes

Accepted

### What is the standard 2-generating set of the symmetric group good for?

Let $f\in\mathbb{Q}[x]$ be an irreducible polynomial of prime degree $p$, with exactly $2$ non-real roots.
You can view the Galois group of $f$ (i.e., the Galois group of the splitting $f$) as a ...

- 2,394

37
votes

### What is so special about Chern's way of teaching?

Louis Auslander has described his experiences on S.S. Chern as a teacher:
Somehow Chern conveyed the philosophy that making mistakes was normal
and that passing from mistake to mistake to truth was ...

- 158k

34
votes

### Not especially famous, long-open problems which anyone can understand

Does there exist a point in the unit square whose distance to each of the four corners is rational?
This is sometimes called the rational distance problem, although that name often refers to a more ...

Community wiki

34
votes

Accepted

### Historical (personal) examples of teaching-based research

The first time I taught forcing, I wanted to mention, as motivation, the fact that the independence of the continuum hypothesis (CH) or even of the axiom of constructibility (V=L) cannot be proved by ...

Community wiki

33
votes

### What kid-friendly math riddles are too often spoiled for mathematicians?

Here's a few, two I got to solve myself as a kid and one (a trickier one, in my opinion) that was spoiled for me.
There are $1000$ lights all in a line and turned on. At time $n$, person $n$ comes by ...

Community wiki

32
votes

### Advice for PhD Supervisors

There is a previous MO thread about this: Resources for mathematics advising.
Here is a list of resources:
Section 2B of Indiana University's How to be a Good Grad Student is Advice for Advisors.
...

Community wiki

31
votes

### Not especially famous, long-open problems which anyone can understand

Erdos's problem on the length of lemniscates (it is somewhat famous in certain narrow circles).
Let $P$ be a polynomial, and consider the set $E=\{ z:|P(z)|=1\}$ in the complex plane.
What is the ...

Community wiki

30
votes

### Mathematical games interesting to both you and a 5+-year-old child

There is an infinite indexed family of family-friendly, $\geq2$-player, perfect-information, draw-free-if-finite, cheap-to-construct, two-player combinatorial1, solved, sequential games. They are ...

Community wiki

30
votes

### Can one deduce the fundamental theorem of algebra from real calculus and linear algebra?

I don't think so. Indeed the result that every symmetric matrix is diagonalizable is true for some orderable non-real-closed field $K$ (see this answer by Will Sawin to Over which fields are symmetric ...

- 54.8k

27
votes

### What kid-friendly math riddles are too often spoiled for mathematicians?

To make this suitable for MO rather than math.SE, perhaps we can define a "too often spoiled" puzzle to be one that can be recognized instantly by a mathematician even with what looks like ...

Community wiki

26
votes

### Mathematical games interesting to both you and a 5+-year-old child

Dots and Boxes
Is a pencil-and-paper game for two players. It's quite simple to explain but quite hard to play. Five year olds should be able to learn it and with some training maybe also being good ...

Community wiki

25
votes

### History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$

The Madhava–Gregory series, by R. C. Gupta, attributes (3) to Indian mathematician-astronomer Madhava of Sangamagrama (circa 1350–1425). He also writes that a geometric derivation which is basically ...

- 15.6k

25
votes

Accepted

### Languages beyond enumerable

Yes, for starters there is the arithmetical hierarchy, where enumerable = $\Sigma^0_1$ and it continues $\Pi^0_1$, $\Delta^0_2$, $\Sigma^0_2$ etc.
See also the Computability Menagerie.

- 24.1k

25
votes

### Mathematical games interesting to both you and a 5+-year-old child

What about Spot It! (US), also known as Dobble (Europe)?
We are given a deck of 55 cards. Each card contains 8 different symbols, such that any pair of cards in the deck has precisely one symbol in ...

Community wiki

25
votes

Accepted

### A conjecture in which both "if" and "only if" are near misses

False claim: A Hausdorff topological space is compact if and only if it is sequentially compact.
It's believable if your intuition of Hausdorff spaces comes entirely from metric spaces (where the ...

- 11.9k

24
votes

### What are the most misleading alternate definitions in taught mathematics?

One often sees the cumulants of a probability distribution defined by saying the cumulant-generating function is the logarithm of the moment-generating function:
$$
\sum_{n=1}^\infty \kappa_n \frac {t^...

Community wiki

23
votes

### Short papers for undergraduate course on reading scholarly math

I think Calkin and Wilf’s lovely short paper on what is now known as the Calkin-Wilf tree would be suitable.
Neil Calkin and Herbert S. Wilf, Recounting the Rationals. The American Mathematical ...

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