Let's write down the numbers $x_{k}=[k \sqrt{2}](k=1,2, \ldots)$ in a row, and below them, the integers $0<y_{1}<y_{2}<\ldots$ that do not appear among the $x_{k}$ numbers. Determine the difference $y_{k}-x_{k}$ as a function of $k$.
Solution. Since for every positive integer $k$, $1<(k+1) \sqrt{2}-k \sqrt{2}<2$, the value of the difference $x_{k+1}-x_{k}$ can only be 1 or 2. Therefore, the $r$-th missing number, $y_{r}$, for some $k$ is of the form

$$
y_{r}=[k \sqrt{2}]-1
$$

For this $k$, thus $[k \sqrt{2}]-1 \neq[(k-1) \sqrt{2}]$, that is, $[k \sqrt{2}] \neq[k \sqrt{2}-(\sqrt{2}-1)]$, or $0=[\{k \sqrt{2}\}] \neq [\{k \sqrt{2}\}-(\sqrt{2}-1)]$. This means that $\{k \sqrt{2}\}<\sqrt{2}-1$, so $\{k \sqrt{2}\}(\sqrt{2}+1)<(\sqrt{2}-1)(\sqrt{2}+1)=1$, that is, $-1<-\{k \sqrt{2}\}(\sqrt{2}+1)<0$, hence $[-\{k \sqrt{2}\}(\sqrt{2}+1)]=-1$.

Accordingly, up to this point, a total of $r=x_{k}-k=[k \sqrt{2}]-k$ numbers have been left out from the upper row. The $r$-th number in the upper row is

$$
\begin{aligned}
x_{r} & =[r \sqrt{2}]=[([k \sqrt{2}]-k) \sqrt{2}]=[(k \sqrt{2}-\{k \sqrt{2}\}-k) \sqrt{2}]= \\
& =[2 k-k \sqrt{2}-\{k \sqrt{2}\} \sqrt{2}]=2 k-[k \sqrt{2}]+[-\{k \sqrt{2}\}(1+\sqrt{2})]= \\
& =2 k-[k \sqrt{2}]-1
\end{aligned}
$$

Thus,

$$
y_{r}-x_{r}=[k \sqrt{2}]-1-(2 k-[k \sqrt{2}]-1)=2[k \sqrt{2}]-2 k=2 r
$$