Let $n$ be a positive integer. An $n$-brick is a rectangle of height 1 and width $n$. A 1 -tower is defined as a 1 -brick. An $n$-tower, for $n \geqslant 2$, is defined as an $n$-brick on top of which exactly two other towers are stacked: a $k _ { 1 }$-tower and a $k _ { 2 }$-tower such that $1 \leqslant k _ { 1 } \leqslant n - 1$ and $k _ { 1 } + k _ { 2 } = n$. The $k _ { 1 }$-tower is placed to the left of the $k _ { 2 }$-tower so that side-by-side they fit exactly on top of the $n$-brick. For example, here is a 4 -tower: [Figure] (i) Draw the four other 4 -towers. (ii) What is the maximum height of an $n$-tower? Justify your answer. (iii) The area of a tower is defined as the sum of the widths of its bricks. For example, the 4 -tower drawn above has area $4 + 4 + 3 + 2 = 13$. Give an expression for the area of an $n$-tower of maximum height. (iv) Show that there are infinitely many $n$ such that there is an $n$-tower of height exactly $1 + \log _ { 2 } n$. (v) Write $t _ { n }$ for the number of $n$-towers. We have $t _ { 1 } = 1$. For $n \geqslant 2$ give a formula for $t _ { n }$ in terms of $t _ { k }$ for $k < n$. Use your formula to compute $t _ { 6 }$. (vi) Show that $t _ { n }$ is odd if and only if $t _ { 2 n }$ is odd.
\section*{5. For ALL APPLICANTS.}
Let $n$ be a positive integer. An $n$-brick is a rectangle of height 1 and width $n$. A 1 -tower is defined as a 1 -brick. An $n$-tower, for $n \geqslant 2$, is defined as an $n$-brick on top of which exactly two other towers are stacked: a $k _ { 1 }$-tower and a $k _ { 2 }$-tower such that $1 \leqslant k _ { 1 } \leqslant n - 1$ and $k _ { 1 } + k _ { 2 } = n$. The $k _ { 1 }$-tower is placed to the left of the $k _ { 2 }$-tower so that side-by-side they fit exactly on top of the $n$-brick. For example, here is a 4 -tower:\\
\includegraphics[max width=\textwidth, alt={}, center]{df72dfe2-9f61-49a0-a6b9-a4ad637d3c86-16_285_291_769_849}\\
(i) Draw the four other 4 -towers.\\
(ii) What is the maximum height of an $n$-tower? Justify your answer.\\
(iii) The area of a tower is defined as the sum of the widths of its bricks. For example, the 4 -tower drawn above has area $4 + 4 + 3 + 2 = 13$. Give an expression for the area of an $n$-tower of maximum height.\\
(iv) Show that there are infinitely many $n$ such that there is an $n$-tower of height exactly $1 + \log _ { 2 } n$.\\
(v) Write $t _ { n }$ for the number of $n$-towers. We have $t _ { 1 } = 1$. For $n \geqslant 2$ give a formula for $t _ { n }$ in terms of $t _ { k }$ for $k < n$. Use your formula to compute $t _ { 6 }$.\\
(vi) Show that $t _ { n }$ is odd if and only if $t _ { 2 n }$ is odd.