Intereting Posts

Rounding to nearest
Distribute n identical objects into r distinct groups
Iterations with matrices over simple fields approximating solutions for more complicated fields.
Is the space of $G$-maps $G/H \to X$ naturally homeomorphic to $X^H$?
Bibliography for Singular Functions
What's wrong with my understanding of the Freyd-Mitchell Embedding Theorem?
Integral of $\frac{\sqrt{x^2+1}}{x}$
Do we gain anything interesting if the stabilizer subgroup of a point is normal?
Radical/Prime/Maximal ideals under quotient maps
Integrate: $ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $ without using complex analysis methods
Graph with 10 nodes and 26 edges must have at least 5 triangles
Expected maximum absolute value of $n$ iid standard Gaussians?
Solving $2x – \sin 2x = \pi/2$ for $0 < x < \pi/2$
Is $\frac{1}{\exp(z)} – \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ entire?
If $n^2+4n+10$ is a perfect square, then find the possible integer values of $n$.

This game will be familiar to many mathematicians, and it is always good fun to play.

I am looking to find a list of good questions with short, when-you-see-it solutions. The kind of question one could solve pretty much anywhere: over dinner, while taking a walk – essentially, without pen & paper, and where the solution lies in spotting a clever trick or fact (rather than via some monstrously power theorem).

To get an idea of what I mean, here is an example:

- List of matrix properties which are preserved after a change of basis
- Examples of bi-implications ($\Leftrightarrow$) where the $\Rightarrow$ direction is used in the proof of the $\Leftarrow$ direction.
- Counterintuitive examples in probability
- Examples of results failing in higher dimensions
- Nice examples of finite things which are not obviously finite
- Do there exist groups whose elements of finite order do not form a subgroup?

Q.A real number is calledrepetitiveif its decimal expansion contains arbitrarily long blocks which are the same. Prove that the square of a repetitive number is repetitive.

(please do not post a solution, since I am sure anyone can figure it out given enough time)

Does anyone have similar chestnuts to offer?

- Differentiable functions satisfying $f'(f(x))=f(f'(x))$
- Examples of Infinite Simple Groups
- Applications of the Isomorphism theorems
- What are some examples of mathematics that had unintended useful applications much later?
- Your favourite application of the Baire Category Theorem
- What is the deepest / most interesting known connection between Trigonometry and Statistics?
- Proofs of the Cauchy-Schwarz Inequality?
- What Do Mathematicians Do?
- If there are obvious things, why should we prove them?
- For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .

Evaluate:

$$\int_0^1\!\!\int_0^1\! \dfrac{1}{1-xy} \, \mathrm{d}x\mathrm{d}y$$

Consider a 8×8 chessboard and 2×1 dominos. Clearly one can easily cover the whole chessboard with 32 dominos.

Now omit the squares on the bottom-left and top-right. There are 62 squares left on the chessboard. Can you cover them using 31 dominos?

$1$, $z$, $z^2$, and $z^3$ (all distinct) lie on the same circle on the complex plane. What is the center of that circle?

A rectangular room in the museum has a floor tiled with tiles of two shapes: $1 \times 4$ and $2 \times 2$. The tiles completely cover the floor of the room, and no tile has been damaged, cut in half, or otherwise had its integrity interfered with. One day, a heavy object is dropped on the floor and one of the tiles is cracked. The museum handyman removes the damaged tile and goes to the storage closet to get a replacement. But he finds that there is only one spare tile, and it is of the other shape. Can he rearrange the remaining tiles in the room in such a way that the spare tile can be used to fill the hole?

This is an old putnam problem, that I think is an excellent example of this.

Let $a_n = 1111…1 $($n$ ones) for instance $a_2 = 11$. Find all polynomials $f$ such that for all $a_n$, there is some $a_k$ where $f(a_n) = a_k$.

Hint: Reduce the problem to finding polynomials $g$ where $g(10^n)=10^k$.

If the probability of rain on Saturday is 50% and the probability of rain on Sunday is 50%, which is the probability of rain on the weekend?

It can be very fun during dinner with non-mathematicians friends…

Someone plays chess at least once a day and no more than 10 times in 7 consecutive days. Prove that there is a period of n consecutive days where the man or woman played 23 times.

- General Information about Eigenvalues for an 3×3 symmetric matrix
- One root of an irreducible polynomial in an extension field, so is the other.
- “Direct” proof for a bound on the difference between endpoints of a path?
- Extending to a disc means fundamental group is trivial
- If $G$ is non-nilpotent and $M$ is non-normal subgroup of $G$, then $|G: M|=p^{\alpha}$?
- A criterion for independence based on Characteristic function
- Infinite sum involving ascending powers
- Lesser-known integration tricks
- How does one calculate genus of an algebraic curve?
- Why does $(A/I)\otimes_R (B/J)\cong(A\otimes_R B)/(I\otimes_R 1+1\otimes_R J)$?
- A improper integral with complex parameter
- Find $\lim_{n\rightarrow\infty}\frac{n}{(n!)^{1/n}}$
- How can we determine the closed form for $\int_{0}^{\infty}{\ln(e^x-1)\over e^x+1}\mathrm dx?$
- $S=\{(n,{1\over n}):n\in\mathbb{N}\}$ is closed in $X$?
- Difference between proof and plausible argument.