Question 79
Let $x_{11}, x_{21}, \ldots, x_{n1}$, $n\gt 2$, be a sequence of integers. We assume that all of the numbers $x_{i1}$ are not equal. Assuming that the numbers $x_{1k}, x_{2k}, \ldots, x_{nk}$ have been defined, we set
$$
\begin{aligned}
& x_{i, k+1}=\frac{1}{2}\left(x_{ik}+x_{i+1, k}\right), \quad i=1,2, \ldots, n-1 \\
& x_{n, k+1}=\frac{1}{2}\left(x_{nk}+x_{1k}\right)
\end{aligned}
$$
Show that for $n$ odd, $x_{jk}$ is not an integer for some $j, k$. Does the same conclusion hold for $n$ even?
$$
\begin{aligned}
& x_{i, k+1}=\frac{1}{2}\left(x_{ik}+x_{i+1, k}\right), \quad i=1,2, \ldots, n-1 \\
& x_{n, k+1}=\frac{1}{2}\left(x_{nk}+x_{1k}\right)
\end{aligned}
$$
Show that for $n$ odd, $x_{jk}$ is not an integer for some $j, k$. Does the same conclusion hold for $n$ even?