## Top new questions this week:

### Problems that become easier in a more general form.

When solving a problem, we often look at some special cases first, and then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, …

(soft-question) (proof-strategy) (examples-counterexamples) (problem-solving)

### How far can one see over the ocean?

Since Earth is a sphere, one has only a limited visibility radius. How far is that, actually? This Q&A was inspired by this question, about whether or not Legolas can see the 24km distant Riders …

(geometry)

### Geometry problem involving infinite number of circles

What is the sum of the areas of the grey circles? I have not made any progress so far.

(sequences-and-series) (geometry)

### Mathematical literature to lose yourself in

H.M. Edwards in the preface to his book on the Riemann Zeta Function, summarises his philosophy on learning Mathematics: ...I have tried to say to students of mathematics that they should read the …

(reference-request) (soft-question) (recreational-mathematics) (big-list)

### Simulate a 7-sided die with a 6-sided die

What is the most efficient way to simulate a 7-sided die with a 6-sided die? I've put some thought into it but I'm not sure I get somewhere specifically. To create a 7-sided die we can use a …

(probability) (dice)

### Which is greater, $20 \uparrow\uparrow\uparrow\uparrow 20$ or $4 \uparrow\uparrow\uparrow\uparrow\uparrow 4$?

This past Wednesday's What-If had this image at the bottom: In particular, I am interested in $20 \uparrow\uparrow\uparrow\uparrow 20$. I immediately thought of Graham's Number, but clearly that …

(number-theory) (recreational-mathematics)

### Where did the negative answer come from?

The question is to evaluate $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$ $$x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$ $$x^2=2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$ $$x^2=2+x$$ …

## Greatest hits from previous weeks:

### How to Determine if a Function is One-to-One

I am looking for the "best" way to determine whether a function is one-to-one, either algebraically or with calculus. I know a common, yet arguably unreliable method for determining this answer would …

(calculus) (algebra-precalculus) (functions)

### Mathematicians ahead of their time?

In every field there's always that person who's just years ahead of their time. For instance, Paul Morphy (born 1837) is said to have retired from chess because he found no one to match his technique …

(soft-question) (math-history) (big-list) (mathematicians)

### Explicit description of small open set containing the rationals

We know that the set $\mathbb{Q}$ of rational numbers has measure zero because it is countable. In fact, if $(q_n)_{n=1,2,\ldots}$ is an enumeration of $\mathbb{Q}$, then …

(measure-theory)

### Sum the series $\sum_{n = 0}^{\infty} (-1)^{n}\{(2n + 1)^{7}\cosh((2n + 1)\pi\sqrt{3}/2)\}^{-1}$

In one of his letters to G. H. Hardy, Ramanujan gave the following sum \dfrac{1}{1^{7}\cosh\left(\dfrac{\pi\sqrt{3}}{2}\right)} - \dfrac{1}{3^{7}\cosh\left(\dfrac{3\pi\sqrt{3}}{2}\right)} + …

(sequences-and-series)
Let $k$ be an integer such that $1 \leq k \leq \left(\dfrac{p-1}{2}\right)$ for some odd prime $p$. Let $a$ be another integer such that $1 \leq a \leq (p-1)$. Then find the number of integral values …