Question 18
Find a sequence $a_{1}$, $a_{2}$, ..., $a_{25}$ of 0s and 1s such that the following sums are all odd:
$$
\begin{array}{rl}
& a_{1}a_{1} + a_{2}a_{2} + \ldots + a_{25}a_{25} \\
& a_{1}a_{2} + a_{2}a_{3} + \ldots + a_{24}a_{25} \\
& a_{1}a_{3} + a_{2}a_{4} + \ldots + a_{23}a_{25} \\
& \ldots \\
& a_{1}a_{24} + a_{2}a_{25}
\end{array}
$$
Show that we can find a similar sequence of $n$ terms for some $n \gt 1000$.
$$
\begin{array}{rl}
& a_{1}a_{1} + a_{2}a_{2} + \ldots + a_{25}a_{25} \\
& a_{1}a_{2} + a_{2}a_{3} + \ldots + a_{24}a_{25} \\
& a_{1}a_{3} + a_{2}a_{4} + \ldots + a_{23}a_{25} \\
& \ldots \\
& a_{1}a_{24} + a_{2}a_{25}
\end{array}
$$
Show that we can find a similar sequence of $n$ terms for some $n \gt 1000$.