Some of my more interesing answers, not necessary most upvoted:
What is the distribution of R2 in linear regression under the null hypothesis? My answer includes a comprehensive graphic of the distribution of $R^2$ for multiple regression on a small sample.
Ink to data ratio and plot backgrounds - a fascinating question about the recent trend for plots to have grey backgrounds.
Uniform random variable as sum of two random variables - if $U$ is a uniform variable, why can't we find i.i.d. variables $X$ and $Y$ such that $U=X+Y$? A nice solution by thinking about limits on kurtosis.
Random walk on the edges of a cube - if a spider walks at random along the edges, what is the expected number of steps to reach an ant trapped on the opposite vertex? Parity considerations give an elementary pen-and-paper method.
Correlation between a nominal (IV) and a continuous (DV) variable - sometimes we have to explain that something can't be done, but suggest alternatives or analogues.
Finding an unbiased estimator with the smallest variance - answered with Lagrange multipliers.
Prove the equivalence of the following two formulas for Spearman correlation - using only high school methods. This is a bit clunky, but the topic sometimes comes up at high school level (at least in my country), so it's nice not to have to rely on higher level maths.
Expectation of Quotient of Sums of IID Random Variables (Cambridge University Worksheet) - an interesting question that can be answered by (a)symmetry considerations.
Transformation Chi-squared to Normal distribution - including a trick using a Rademacher variable, and some very pretty graphs to show deciles (and R code).
Addressing some elementary misconceptions about summary statistics, with nice counterexamples: Does mean = median imply that a unimodal distribution is symmetric? and Will two distributions with identical 5-number summaries always have the same shape?