\section*{5. For ALL APPLICANTS.}
Given an $n \times n$ grid of squares, where $n > 1$, a tour is a path drawn within the grid such that:

\begin{itemize}
  \item along its way the path moves, horizontally or vertically, from the centre of one square to the centre of an adjacent square;
  \item the path starts and finishes in the same square;
  \item the path visits the centre of every other square just once.
\end{itemize}

For example, below is a tour drawn in a $6 \times 6$ grid of squares which starts and finishes in the top-left square.\\
\includegraphics[max width=\textwidth, alt={}, center]{373b6bde-f147-4e16-aaa9-78f86989fc96-14_522_525_909_762}

For parts (i)-(iv) it is assumed that $n$ is even.\\
(i) With the aid of a diagram, show how a tour, which starts and finishes in the top-left square, can be drawn in any $n \times n$ grid.\\
(ii) Is a tour still possible if the start/finish point is changed to the centre of a different square? Justify your answer.

Suppose now that a robot is programmed to move along a tour of an $n \times n$ grid. The robot understands two commands:

\begin{itemize}
  \item command $R$ which turns the robot clockwise through a right angle;
  \item command $F$ which moves the robot forward to the centre of the next square.
\end{itemize}

The robot has a program, a list of commands, which it performs in the given order to complete a tour; say that, in total, command $R$ appears $r$ times in the program and command $F$ appears $f$ times.\\
(iii) Initially the robot is in the top-left square pointing to the right. Assuming the first command is an $F$, what is the value of $f$ ? Explain also why $r + 1$ is a multiple of 4 .\\
(iv) Must the results of part (iii) still hold if the robot starts and finishes at the centre of a different square? Explain your reasoning.\\
(v) Show that a tour of an $n \times n$ grid is not possible when $n$ is odd.