Here is a reminder of Anirban DasGupta’s puzzles, Student Puzzle 59. Anirban says, “You can send a solution to one, two, or ideally all three! We hope you will enjoy thinking about these problems.”

Deadline: February 1, 2026

Puzzle 59.1
Suppose $X, Y, Z$ are iid standard normal. Find, without doing any calculations, the distribution of $\frac{X+YZ}{\sqrt{1+Z^2}}$.

Puzzle 59.2
Suppose $X_1, X_2, \cdots $ are iid $U[0,1]$. For $n \geq 2$, let $X_{n-1:n}$ be the second largest observation among $X_1, \cdots , X_n$. Find sequences $a_n, b_n$ and a nondegenerate distribution $G$ such that $\frac{X_{n-1:n} – a_n}{b_n}$ converges in distribution to a random variable with distribution $G$.

Puzzle 59.3
Consider the standard linear model $\bf{Y} = \bf{X}\,\bf{\beta} + \bf{\epsilon}$. Find an element $u$ in the column space of $X$ such that $||u-\bf{Y}||^2 < ||v – \bf{Y}||^2$ for all $v$ different from $u$ in the column space of $X$.

Student members of IMS are invited to submit solutions to bulletin@imstat.org (subject “Student Puzzle Corner”). If correct, we’ll publish your name (and photo, if there’s space) with the solution in the next issue. The Puzzle Editor is Anirban DasGupta. His decision is final.