## Exercises for Week 2: Recursion and Datatypes

Here are some exercises designed to help you practice writing and reasoning about recursive definitions and datatypes.

If you do not have time to do all these exercises, don't worry.
The exercises are intended to provide enough work to keep the *most*
experienced students busy. If you do all exercises marked with an (*) you have
probably understood this week's material.

Good luck!

### 1. (*) The Maximum Function

Complete the following function definition:

-- maxi x y returns the maximum of x and y

(We call this function maxi, because there is a standard function max which does the same thing. Of course, you should not use it.)

Begin by writing a type signature for maxi, and a left hand side "equal to undefined". Before you write the definition, think of at least one property that you will use for testing your code.

You will need to consider two cases: what are they? Write the left hand sides, and make sure GHCi accepts them.

Complete your definition and test it with QuickCheck.

### 2. Sum of squares.

Define a function which computes the sum of the squares of the numbers from 1 to n.

-- sumsq n returns 1*1 + 2*2 + ... + n*n

*Hint:* use recursion to compute sumsq n from sumsq (n-1); **do not**
use a formula (such as the one below) to compute this without recursion.

Some say that sumsq n is equal to n (n+1) (2n + 1) / 6. Use QuickCheck to investigate whether this holds.

### 3. (*) The Towers of Hanoi

The Towers of Hanoi is an ancient puzzle, consisting of a collection of rings of different sizes, and three posts mounted on a base. At the beginning all the rings are on the left-most post as shown, and the goal is to move them all to the rightmost post, by moving one ring at a time from one post to another. But, at no time may a larger ring be placed on top of a smaller one!

Can you find a strategy for solving the puzzle based on recursion? That is,
if you already know how to move `n`-1 rings from one post to another,
can you find a way to move `n` rings?

If you try out your strategy, you will quickly discover that quite a large number of moves are needed to solve the puzzle with, say, five rings. Can you define a Haskell function

which computes the number of moves needed to move n rings from one post to another using your strategy? How many moves would be needed to solve the puzzle with ten rings?hanoin

Legend has it that the original version of the puzzle has 32 rings, and is being solved at this very moment by Bhuddist monks in a monastery. When the puzzle is complete, the world will end.

**More difficult** (only do this if you want a challenge): Now
suppose we add a fourth post to the puzzle. Can you think of a strategy
which makes use of the fourth post? Define a Haskell function to compute the
number of moves your new strategy needs to solve the puzzle. How many moves are
now required to solve the puzzle with ten rings? How much closer would the end
of the world be if the monks added a fourth post to their puzzle?

Can you explain this behaviour? *Hint*: what kind of recursion do your
two strategies use?

### 4. Fibonacci Numbers

The *Fibonacci numbers* are defined by

F _{0}= 1 F _{1}= 1 F _{n+2}= F _{n+1}+ F_{n}

so the sequence of values begins 1, 1, 2, 3, 5, 8...

Write a recursive function

based on the mathematical definition above. Use it to compute the 10th, 15th, 20th, 25th, and 30th Fibonacci number. What do you notice? Can you explain this behaviour?-- fib n computes the nth Fibonacci number

**More Difficult:** There is a faster way to compute Fibonacci numbers.
Suppose we define a function fibAux which satisfies the property

fibAuxi(fibn) (fib(n+1))==fib(n+i)

Notice that this is *not* a definition, it is a property!

*From this property* can you see how we might *redefine* fib by
using fibAux? (Hint: try setting n to 0 in the property).

It is possible to *derive* a recursive definition of fibAux from this
property just using algebra: the two cases we want are

fibAux0ab=...fibAuxiab|i>0=...

See if you can use the property to figure out what the right hand sides should be.

Use QuickCheck to test whether the new definition of fib (in terms of fibAux) satisfies the property above.

Use the new version of fib to compute the 10th, 15th, 20th, 25th, and 30th Fibonacci number. What do you notice?

Calculate `fibAux 4 1 1`

by hand. If you have programmed before,
then observe the way the values of i, a, and b change in successive
recursive calls. Does it remind you of anything?

### 5. Factors.

A prime number `p` has only two factors, 1 and `p` itself.
A composite number has more than two factors. Define a function

smallestFactorn

which returns the smallest factor of `n` larger than one.
For example,

smallestFactor14==2

smallestFactor15==3

Before you program `smallestFactor`

, write at least two QuickCheck
properties that it should satisfy. You will need functions for integer
division and remainder: investigate the standard functions
`div`

and `mod`

for this purpose.

*Hint:* if you want to practise recursion, write
`smallestFactor`

by using an
auxiliary function `nextFactor k n`

which returns the
smallest factor of n larger than k. You can define `smallestFactor`

using `nextFactor`

, and `nextFactor`

by recursion.
Write QuickCheck properties of `nextFactor`

before defining it also.

Now define

numFactorsn

which computes the number of factors of n in the range 1..n.

### 6. (*) Multiplying list elements

Define a function

which multiplies together all the elements of a list. (Think: what should its value be for the empty list?). For examplemultiply::Numa=>[a]->a

`multiply [1,2,3,4,5]`

120

(This is actually a standard function, called `product`

).

### 7. Avoiding Duplicates

In many situations, lists should not contain *duplicate* elements. For
example, a pack of cards should not contain the same card twice. Define a
function

duplicates::Eqa=>[a]->Bool

which returns `True`

if its argument contains duplicate elements.

`duplicates [1,2,3,4,5]`

> False`duplicates [1,2,3,2]`

True

*Hint:* the standard function `elem`

,
which tests whether an element occurs in a list, is helpful here.

One way to *ensure* a list contains no duplicates is to start with a
list that might contain duplicate elements, and remove them. Define a function

which returns a list containing the same elements as its argument, but without duplicates. Test it using the following property:removeDuplicates::Eqa=>[a]->[a]

prop_duplicatesRemoved::[Integer]->Boolprop_duplicatesRemovedxs=not(duplicates(removeDuplicatesxs))

Does this property guarantee that `removeDuplicates`

behaves correctly? If not,
what is missing?

(`removeDuplicates`

is actually a standard function, called `nub`

).

### 8. Testing

Take the definition of `rankBeats`

from the lecture,
and sabotage it by
changing one of the `False`

results to `True`

,
or vice versa. Thorough testing of
`rankBeats`

ought to reveal the error. Your task is to define one or more
properties of `rankBeats`

, which are True for the correct definition, but which
enable you to reliably detect the error in the sabotaged definition using
QuickCheck.

(*Note:* Because QuickCheck chooses test data at random, it is always
possible that no error is found among the first 100 test cases. If you expect an
error, and none is found, just keep testing the same property. If it really is
False, then QuickCheck should eventually find a test case that fails).

### 9. (*) Defining Types

Define a data type `Month`

to represent months, and a function

which computes the number of days in a month, given also the year. (You can ignore leap centuries and the like: just assume that every fourth year is a leap year).daysInMonth::Month->Integer->Integer

Define a data type `Date`

, containing a year, month, and day, and
a function

that returnsvalidDate::Date->Bool

`True`

if the day in the date lies between 1 and the
number of days in the month.
Also define a function

that computes tomorrow's date for a given date.tomorrow::Date->Date