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\title{Assignment 3 \\ FN204 \\ Algorithms in Ancient India}

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\maketitle

Instructions:\\
\begin{itemize}
\item Due date for submission: $4^{th}$ May 2018.
\item Numbers to the right of each question indicate the marks.
\end{itemize}
\begin{enumerate}
\item Show that $x = \frac{2a}{N-a^2}$ is a possible solution for the equation $Nx^2+1=y^2$.\\ \hspace{10cm} \textbf{(5)}
\item Solve the equation $32 x^2 + 1 = y^2$ by Cakrav{\=a}la method. \hspace{1cm} \textbf{(8)}
\item There are 8 books (A,B,C,D,E,F,G,H) that are interdisciplinary in nature.	 Various disciplines to which these books belong to are as shown below.
\begin{itemize}
\item Discipline 1: D,E
\item Discipline 2: B,C,D,F,G,H,I
\item Discipline 3: A,B
\item Discipline 4: F,I
\item Discipline 5: C,D,E,F,G,H,I
\item Discipline 6: G,H
\end{itemize}

\begin{enumerate}
\item Is it possible to arrange these books in such a way that all the books belonging to a certain discipline are together?
\item If yes, in which order should they be arranged?\\
If there are multiple ways of arranging them, indicate all of them.\flushright	\textbf{(10)}
\end{enumerate}
\item Below are given a set of s{\=u}tras from P{\=a}\d{n}ini's A\d{s}\d{t}{\=a}dhy{\=a}y{\=\i}, in their expanded form (by borrowing the words from the previous s{\=u}tras in order to make each s{\=u}tra self sufficient in itself). The borrowed words are shown in parenthesis. Show that if we follow the factorisation as in Mathematics, we arrive at the s{\=u}tras with words excluding those in parenthesis.	\hspace{8cm}\textbf{(10)}
\begin{itemize}
\item 6-1-87 \={a}d gu\d{n}a\d{h} (sa\d{m}hit\={a}y\={a}m aci eka\d{h} p\={u}rvaparayo\d{h})
\item 6-1-88 v\d{r}ddhi\d{h} eci (sa\d{m}hit\={a}y\={a}m  eka\d{h} p\={u}rvaparayo\d{h} \={a}d)
\item 6-1-89 etyedhaty\={u}\d{t}hsu (sa\d{m}hit\={a}y\={a}m eka\d{h} p\={u}rvaparayo\d{h} \={a}d v\d{r}ddhi\d{h} eci)
\item 6-1-90 \={a}\d{t}a\d{h} ca (sa\d{m}hit\={a}y\={a}m aci eka\d{h} p\={u}rvaparayo\d{h}  v\d{r}ddhi\d{h})
\item 6-1-91 upasarg\={a}t \d{r}ti dh\={a}tau (sa\d{m}hit\={a}y\={a}m  eka\d{h} p\={u}rvaparayo\d{h} \={a}d v\d{r}ddhi\d{h})
\item 6-1-92 v\={a} supi \={a}pi\'{s}ale\d{h} (sa\d{m}hit\={a}y\={a}m eka\d{h} p\={u}rvaparayo\d{h} \={a}d v\d{r}ddhi\d{h} upasarg\={a}t \d{r}ti dh\={a}tau)
\item 6-1-93 \={a} ot am\'{s}aso\d{h} (sa\d{m}hit\={a}y\={a}m aci eka\d{h} p\={u}rvaparayo\d{h})
\item 6-1-94 e\.{n}i parar\={u}pam (sa\d{m}hit\={a}y\={a}m  eka\d{h} p\={u}rvaparayo\d{h} \={a}d  upasarg\={a}t dh\={a}tau)
\end{itemize}
Explain in what way anuv\d{r}tti is different from the factorisation in Mathematics.
\item What is a praty{\=a}h{\=a}ra? How many praty{\=a}h{\=a}ras are possible with the following \'{s}ivas{\=u}tras?	\hspace{6cm} \textbf{(7)}\\
\begin{center}
{\it a i u (\d{N})}\\
{\it \d{r} \d{l}(K)}\\
{\it e o (\d{N}) }\\
{\it ai au (C) }\\
{\it ha ya va ra (\d{T}) }\\
{\it l (\d{N})}\\
\end{center}
 Is there any ambiguity in the praty{\=a}h{\=a}ras? If yes, which praty{\=a}h{\=a}ras are ambiguous?
\end{enumerate}
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\end{enumerate}
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