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\section*{Core Part 1 (Scala, 3 Marks)}

\mbox{}\hfill\textit{``The most effective debugging tool is still careful thought,}\\

\mbox{}\hfill\textit{coupled with judiciously placed print statements.''}\smallskip\\

\mbox{}\hfill\textit{ --- Brian W. Kernighan, in Unix for Beginners (1979)}\bigskip

\IMPORTANTNONE{}

\noindent

Also note that the running time of each part will be restricted to a

maximum of 30 seconds on my laptop.

\DISCLAIMER{}

\subsection*{Reference Implementation}

Like the C++ assignments, the Scala assignments will work like this: you

push your files to GitHub and receive (after sometimes a long delay) some

automated feedback. In the end we take a snapshot of the submitted files and

apply an automated marking script to them.\medskip

\noindent

In addition, the Scala coursework comes with a reference implementation

in form of \texttt{jar}-files. This allows you to run any test cases on

your own computer. For example you can call Scala on the command line

with the option \texttt{-cp collatz.jar} and then query any function

from the template file. Say you want to find out what the functions

\texttt{collatz} and \texttt{collatz\_max} produce: for this you just

need to prefix them with the object name \texttt{C1}. If you want to

find out what these functions produce for the argument \texttt{6}, you

would type something like:

\begin{lstlisting}[language={},numbers=none,basicstyle=\ttfamily\small]

$ scala -cp collatz.jar

scala> C1.collatz(6)

...

scala> C1.collatz_max(6)

...

\end{lstlisting}%$

\subsection*{Hints}

\noindent

\textbf{For the Core Part 1:} useful math operators: \texttt{\%} for modulo, \texttt{\&} for bit-wise and; useful

functions: \mbox{\texttt{(1\,to\,10)}} for ranges, \texttt{.toInt},

\texttt{.toList} for conversions, you can use \texttt{List(...).max} for the

maximum of a list, \texttt{List(...).indexOf(...)} for the first index of

a value in a list.\bigskip

\newpage

\subsection*{Core Part 1 (3 Marks, file collatz.scala)}

This part is about function definitions and recursion. You are asked

to implement a Scala program that tests examples of the

\emph{$3n + 1$-conjecture}, also called \emph{Collatz

  conjecture}.\video{https://www.youtube.com./watch?v=LqKpkdRRLZw}

This conjecture can be described as follows: Start with any positive

number $n$ greater than $0$:

\begin{itemize}

\item If $n$ is even, divide it by $2$ to obtain $n / 2$.

\item If $n$ is odd, multiply it by $3$ and add $1$ to obtain $3n +

  1$.

\item Repeat this process and you will always end up with $1$.

\end{itemize}

\noindent

For example if you start with, say, $6$ and $9$, you obtain the

two \emph{Collatz series}

%

\[

\begin{array}{@{}l@{\hspace{5mm}}l@{}}

6, 3, 10, 5, 16, 8, 4, 2, 1 & \text{(= 8 steps)}\\

9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1  & \text{(= 19 steps)}\\

\end{array}

\]

\noindent

As you can see, the numbers go up and down like a roller-coaster, but

curiously they seem to always terminate in $1$. Nobody knows why. The

conjecture is that this will \emph{always} happen for every number

greater than 0.\footnote{While it is relatively easy to test this

conjecture with particular numbers, it is an interesting open problem to

\emph{prove} that the conjecture is true for \emph{all} numbers ($> 0$).

Paul Erd\"o{}s, a famous mathematician you might have heard about, said

about this conjecture: ``Mathematics may not [yet] be ready for such

problems.'' and also offered a \$500 cash prize for its solution.

Jeffrey Lagarias, another mathematician, claimed that based only on

known information about this problem, ``this is an extraordinarily

difficult problem, completely out of reach of present-day mathematics.''

There is also a \href{https://xkcd.com/710/}{xkcd} cartoon about this

conjecture\here{https://xkcd.com/710/}). If you are able to solve this

conjecture, you will definitely get famous.}\bigskip

\noindent

\textbf{Tasks}

\begin{itemize}

\item[(1)] You are asked to implement a recursive function that

  calculates the number of steps needed until a series ends

  with $1$. In case of starting with $6$, it takes $8$ steps and in

  case of starting with $9$, it takes $19$ (see above). We assume it 

  takes $0$ steps, if we start with $1$. In order to

  try out this function with large numbers, you should use

  \texttt{Long} as argument type, instead of \texttt{Int}.  You can

  assume this function will be called with numbers between $1$ and

  $1$ Million. \hfill[1 Mark]

\item[(2)] Write a second function that takes an upper bound as

  an argument and calculates the steps for all numbers in the range from

  1 up to this bound (the bound including). It returns the maximum number of

  steps and the corresponding number that needs that many steps.  More

  precisely it returns a pair where the first component is the number

  of steps and the second is the corresponding number. \hfill\mbox{[1

    Mark]}

\item[(3)] Write a function that calculates \emph{hard

    numbers} \here{https://medium.com/cantors-paradise/the-collatz-conjecture-some-shocking-results-from-180-000-iterations-7fea130d0377}

  in the Collatz series---these are the last odd numbers just before a

  power of two is reached.  For this, implement an

  \textit{is-power-of-two} function which tests whether a number is a

  power of two. The easiest way to implement this is by using the

  bit-operator $\&$ of Scala. For a power of two, say $n$ with $n > 0$, it

  holds that $n \;\&\; (n - 1)$ is equal to zero. I let you think why

  this is the case.

  The function \textit{is-hard} calculates whether

  $3n + 1$ is a power of two.  Finally the \textit{last-odd} function

  calculates the last odd number before a power of 2 in the Collatz

  series. This means for example when starting with 9,

  we receive 5 as the last odd number.  Surprisingly a lot of numbers

  have 5 as last-odd number. But for example for 113 we obtain 85,

  because of the series

  %

  \[113, 340, 170, \,\fbox{85}\,, 256, 128, 64, 32, 16, 8, 4, 2, 1\]

  The \textit{last-odd} function will only be called with numbers that are not

  powers of 2 themselves.\hfill\mbox{[1 Mark]}

\end{itemize}

\noindent

\textbf{Test Data:} Some test cases are:

\begin{itemize}

\item 1 to 10 where $9$ takes 19 steps 

\item 1 to 100 where $97$ takes 118 steps,

\item 1 to 1,000 where $871$ takes 178 steps,

\item 1 to 10,000 where $6,171$ takes 261 steps,

\item 1 to 100,000 where $77,031$ takes 350 steps, 

\item 1 to 1 Million where $837,799$ takes 524 steps

  %% runs out of stack space

  %% \item[$\bullet$] $1 - 10$ million where $8,400,511$ takes 685 steps

\item 21 is the last odd number for 84

\item 341 is the last odd number for 201, 604, 605 and 8600

\end{itemize}

\end{document}

%%%%%%% Historical Stuff

\newpage

This part is about web-scraping and list-processing in Scala. It uses

online data about the per-capita alcohol consumption for each country

(per year?), and a file containing the data about the population size of

each country.  From this data you are supposed to estimate how many

litres of pure alcohol are consumed worldwide.\bigskip

\noindent

\textbf{Tasks (file alcohol.scala):}

\begin{itemize}

\item[(1)] Write a function that given an URL requests a

  comma-separated value (CSV) list.  We are interested in the list

  from the following URL

\begin{center}

  \url{https://raw.githubusercontent.com/fivethirtyeight/data/master/alcohol-consumption/drinks.csv}

\end{center}

\noindent Your function should take a string (the URL) as input, and

produce a list of strings as output, where each string is one line in

the corresponding CSV-list.  This list from the URL above should

contain 194 lines.\medskip

\noindent

Write another function that can read the file \texttt{population.csv}

from disk (the file is distributed with the assignment). This

function should take a string as argument, the file name, and again

return a list of strings corresponding to each entry in the

CSV-list. For \texttt{population.csv}, this list should contain 216

lines.\hfill[1 Mark]

\item[(2)] Unfortunately, the CSV-lists contain a lot of ``junk'' and we

  need to extract the data that interests us.  From the header of the

  alcohol list, you can see there are 5 columns

  \begin{center}

    \begin{tabular}{l}

      \texttt{country (name),}\\

      \texttt{beer\_servings,}\\

      \texttt{spirit\_servings,}\\

      \texttt{wine\_servings,}\\

      \texttt{total\_litres\_of\_pure\_alcohol}

    \end{tabular}  

  \end{center}

  \noindent

  Write a function that extracts the data from the first column,

  the country name, and the data from the fifth column (converted into

  a \texttt{Double}). For this go through each line of the CSV-list

  (except the first line), use the \texttt{split(",")} function to

  divide each line into an array of 5 elements. Keep the data from the

  first and fifth element in these arrays.\medskip

  \noindent

  Write another function that processes the population size list. This

  is already of the form country name and population size.\footnote{Your

    friendly lecturer already did the messy processing for you from the

  Worldbank database, see \url{https://github.com/datasets/population/tree/master/data} for the original.} Again, split the

  strings according to the commas. However, this time generate a

  \texttt{Map} from country names to population sizes.\hfill[1 Mark]

\item[(3)] In (2) you generated the data about the alcohol consumption

  per-capita for each country, and also the population size for each

  country. From this generate next a sorted(!) list of the overall

  alcohol consumption for each country. The list should be sorted from

  highest alcohol consumption to lowest. The difficulty is that the

  data is scraped off from ``random'' sources on the Internet and

  annoyingly the spelling of some country names does not always agree in both

  lists. For example the alcohol list contains

  \texttt{Bosnia-Herzegovina}, while the population writes this country as

  \texttt{Bosnia and Herzegovina}. In your sorted

  overall list include only countries from the alcohol list, whose

  exact country name is also in the population size list. This means

  you can ignore countries like Bosnia-Herzegovina from the overall

  alcohol consumption. There are 177 countries where the names

  agree. The UK is ranked 10th on this list by

  consuming 671,976,864 Litres of pure alcohol each year.\medskip

  \noindent

  Finally, write another function that takes an integer, say

  \texttt{n}, as argument. You can assume this integer is between 0

  and 177 (the number of countries in the sorted list above).  The

  function should return a triple, where the first component is the

  sum of the alcohol consumption in all countries (on the list); the

  second component is the sum of the \texttt{n}-highest alcohol

  consumers on the list; and the third component is the percentage the

  \texttt{n}-highest alcohol consumers drink with respect to the

  the world consumption. You will see that according to our data, 164

  countries (out of 177) gobble up 100\% of the World alcohol

  consumption.\hfill\mbox{[1 Mark]}

\end{itemize}

\noindent

\textbf{Hints:} useful list functions: \texttt{.drop(n)},

\texttt{.take(n)} for dropping or taking some elements in a list,

\texttt{.getLines} for separating lines in a string;

\texttt{.sortBy(\_.\_2)} sorts a list of pairs according to the second

elements in the pairs---the sorting is done from smallest to highest;

useful \texttt{Map} functions: \texttt{.toMap} converts a list of

pairs into a \texttt{Map}, \texttt{.isDefinedAt(k)} tests whether the

map is defined at that key, that is would produce a result when

called with this key; useful data functions: \texttt{Source.fromURL},

\texttt{Source.fromFile} for obtaining a webpage and reading a file.

\newpage

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