Proof Writing & Problem Solving · Challenge problem

For a positive integer $q$, let $d(q)$ be the distance from $q\sqrt2$ to the nearest integer. Placing the 11 fractional parts of $0, \sqrt2, 2\sqrt2, \ldots, 10\sqrt2$ into 10 intervals of length $\tfrac{1}{10}$ shows that $d(q) < \tfrac{1}{10}$ for some $q \le 10$. What is the SMALLEST positive integer $q$ with $d(q) < \tfrac{1}{10}$?