Question 33
Let $y_{1}, y_{2}, y_{3}, \ldots$ be a sequence such that $y_{1}=1$ and, for $k\gt 0$, is defined by the relationship:
$$
\begin{gathered}
y_{2k}= \begin{cases}2 y_{k} & \text{ if } k \text{ is even } \\
2 y_{k}+1 & \text{ if } k \text{ is odd }\end{cases} \\
y_{2k+1}= \begin{cases}2 y_{k} & \text{ if } k \text{ is odd } \\
2 y_{k}+1 & \text{ if } k \text{ is even }\end{cases}
\end{gathered}
$$
Show that the sequence $y_{1}, y_{2}, y_{3}, \ldots$ takes on every positive integer value exactly once.
$$
\begin{gathered}
y_{2k}= \begin{cases}2 y_{k} & \text{ if } k \text{ is even } \\
2 y_{k}+1 & \text{ if } k \text{ is odd }\end{cases} \\
y_{2k+1}= \begin{cases}2 y_{k} & \text{ if } k \text{ is odd } \\
2 y_{k}+1 & \text{ if } k \text{ is even }\end{cases}
\end{gathered}
$$
Show that the sequence $y_{1}, y_{2}, y_{3}, \ldots$ takes on every positive integer value exactly once.