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2. Problems
Function prototypes:
An additional point about function prototypes that we did not emphasize in lecture
is that you do not need a separate function prototype if you define the entire
function above the point where it is called. If you define square before main, for
instance, you do not need a prototype for square.
A note on global variables:
We discussed in lecture how variables can be declared at file scope – if a variable
is declared outside of any function, it can be used anywhere in the file. For
anything besides global constants, this is usually a bad idea – if you need to access
the same variable from multiple functions, most often you simply want to pass the
variable around as an argument between the functions. Avoid global variables
when you can.
Part 1 – Fix the Function:
1. Identify the errors in the following programs, and explain how you would
correct them to make them do what they were apparently meant to do.
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3. a. #include <iostream>
int main() {
std::cout << square(35);
return 0;
}
int square(int number) { return number * number; }
b. #include <iostream>
int square();
int main() {
int number = 35;
std::cout << square(number);
return 0;
}
int square() { return number * number; }
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4. d. #include <iostream>
int sumOfPositives(int a, int b);
int main() {
std::cout << sumOfPositives(35, 25, 3);
return 0;
}
int sumOfPositives(int a, int b) {
int sum = 0;
if(a > 0)
sum += a; if(b > 0)
sum += b;
}
2. What do the following program fragments do? (Try to answer without
compiling and running them.) Briefly explain your results.
a. char *hello = "Hello world!";
void printHello(char *hello) {
std::cout << hello;
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5. }
int main() {
...
printHello("This is not the hello message!");
...
}
b. int mystery(int &number) { std::cout << number; return ++number; }
int main() {
for(int i = 0; i < 10; i++) {
cout << mystery(i);
}
...
}
c. int mystery(int number) { std::cout << number; return ++number; }
int main() {
for(int i = 0; i < 10; i++) {
cout << mystery(i);
}
...
}
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6. Part 2 – Random numbers:
Random number generation:
C++ provides the rand() function, which returns a pseudorandom integer
between 0 and RAND_MAX (a constant integer defined by the compiler). The
integer returned is not truly random because the random number generator
actually uses an initial “seed” value to determine a fixed sequence of almost
random numbers; the sequence is the same each time the same seed is used.
To initialize the random number generator, we call the srand function with the
desired seed as an argument. This only needs to be called once in the entire
program (having one-time-per-program initializer functions is not uncommon).
The seed passed to the function is often the current time, which changes on each
execution and is obtained via the time(0) function call. Once the initialization has
been performed, we can just call rand() each time we want another random
number.
Putting it all together, generating a random number looks something like this:
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7. #include <iostream>
#include <cstdlib> //C standard library – defines rand(), srand(),
RAND_MAX
#include <ctime> //C time functions – defines time()
int main() {
srand( time(0) ); // Initialize random number generator
int random1 = rand(), random2 = rand();
cout << "random1: " << random1 << "nrandom2: " << random2 <<
endl;
}
If we want to restrict the random numbers to a particular range, we can
use the modulus (%) operator as follows:
int dieRollValue = rand() % 6 + 1;
The %6 ensures that the number is between 0 and 5, and adding one
shifts this to be between 1 and 6 (the appropriate range for a die roll).
(Recall that a % b returns the remainder of a / b.)
We can also convert the random integers to decimal values in the range
[0, 1] simply by dividing by RAND_MAX:
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8. double randomDecimal = rand() / static_cast<double>(RAND_MAX);
The static_cast keyword creates a temporary copy of the variable in
parentheses, where the copy is of the type indicated in angle brackets.
This ensures that our division is done as floating-point, and not integer
division. This is called “casting” RAND_MAX to a double.
Potshots at Pi:
Let’s calculate pi! We’re not going to do it by measuring circumferences,
though; we’re going to do it with Monte Carlo methods – using random
numbers to run a simulation.
Let’s take a circle of radius 1 around the origin, circumscribed by a square
with side length 2, like so:
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9. Imagine that this square is a dartboard, and that you are tossing many,
many darts at it at random locations on the board. With enough darts, the
ratio of darts in the circle to total darts thrown should be the ratio between
the area of the circle and the area of the square. Since you know the area
of the square,
you can use this ratio to calculate the area of the circle, from which you
can calculate pi using
1. We can make the math a little easier. All we really need to calculate
is the area of one quadrant and multiply by 4 to get the full area. This
allows us to restrict the domain of our x and y coordinates to [0,1].
a. Write two variable declarations representing the x-coordinate
and y-coordinate of a particular dart throw. Each one should be
named appropriately, and should be initialized to a random
double in the range [0,1].
b. Place your variable declarations within a loop that increments a
variable named totalDartsInCircle if the dart’s location is within
the circle’s radius. You’ll need to use the Euclidean distance
formula ( d 2 = x2 + y2 ). (Assume for the moment that
totalDartsInCircle has already been declared.)
c. Now use your loop to build a function that takes one argument,
specifying the number of “dart throws” to run, and returns the
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10. decimal value of pi, using the technique outlined above. Be sure to
name your function appropriately. You should get pretty good
results for around 5,000,000 dart throws.
Function overloading:
C++ supports a very powerful feature called function overloading. This means
that you can define two functions by the same name, but with different argument
lists. The argument lists can differ by number of arguments, argument type, or
both. When you call the function, the compiler will examine the arguments that
you pass and automatically select the function whose prototype matches the
arguments specified. If there is no function whose prototype matches the call, or
there is more than one (i.e. the call is ambiguous), the compiler will spit out a
syntax error.
2. Wouldn’t it be nice if we didn’t have to do all this modulus and subtraction
stuff each time we wanted a random number? Write two functions, both called
randomInRange, to simplify random number generation. One function should
take an upper and lower bound on the range of allowed integers and return a
random integer in that range. The other should do the same thing for doubles.
Test your functions to make sure they produce correct results. Note: The actual
random number range for both functions should be [lower, upper]. Your code
will be slightly different for the int and double versions.
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11. 3. Why, given your definitions for randomInRange, is randomInRange(1, 10.0) a
syntax error?
Hint: Think about promotion and demotion – the conversion of arguments between
types in a function call.
Part 3 – Maxing out:
1. a. Write a single maximum function that returns the maximum of two integers.
(The ternary operator, though not necessary, is a very elegant way to do this in
one line.)
b. Write a set of 3 functions capable, collectively, of finding the maximum of
anywhere between 2 and 4 numbers. Each one should take integer arguments and
return an integer. The function call required to find any such maximum should
look something like maximum(40, 24, 83). Hint: Use overloading, and think
about how you can build larger functions out of smaller ones.
2. Write a single maximum function capable of handling an arbitrary number of
integers to compare. It should take two arguments, and return an integer. Hint:
What data types can you use to represent an arbitrarily large set of integers in two
arguments?
3. Use default arguments to define a single maximum function that can be called
with anywhere from 2 to 4 explicit arguments. In defining your default values,
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12. you may want to take advantage of the constant INT_MIN, defined in standard
include file climits, which is a compiler-specific constant holding the lowest
possible integer value.
Templates:
Function overloading allows us to define different but similar operations to
perform on different argument types. Sometimes, though, you may find that you
want to program for multiple data types in exactly the same way. For instance,
you may want to define a square function that works for both ints and doubles.
The operation is identical – square(x) should return x * x – but any given function
must take and return either ints or doubles. To allow reuse of the same code, C++
provides templates, which allow you to define a function generically to work with
a broad spectrum of data types. The C++ compiler, a very clever beast,
automatically infers from each function call what data type is being passed in, and
internally generates a new compiled version of the function for each argument
type.
The syntax for a template function declaration is as follows:
template< class T > // or template< typename T >
T square(T number) { return number * number; }
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13. T is called a formal type parameter – a generic, unspecified type representing
whatever type the function happens to be called with. In the copy of the function
generated from the function call square(10), all references to T would be replaced
by int. In the call square(10.0), T would translate to double.
4. Use templates to generalize your array-based maximum function to allow
anything the compiler can treat as a number – double, int, char, etc. Give sample
function calls for two different data types, and indicate what the value of your
formal type parameter would be for each.
Part 4 – Toying with Triangles:
Out parameters:
C++ does not allow returning multiple values. One common method of
circumventing this language limitation is to pass in reference arguments. Instead
of setting a variable equal to the return value of the function, the caller declares
several variables and passes them by reference into the function. By changing the
values of these variables (called out parameters), the callee can simulate returning
multiple values.
1. a. Give a function prototype for a function that takes as arguments the lengths of
two sides of a triangle, the angle in between them in radians, and three out
parameters. The out parameters should be set up to allow the function to simulate
returning three values:
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14. 1. a. Give a function prototype for a function that takes as arguments the lengths
of two sides of a triangle, the angle in between them in radians, and three out
parameters. The out parameters should be set up to allow the function to
simulate returning three values: the length of the remaining side and the size of
the two remaining angles in radians.
b. Now write the body of the function. You
will probably want to make use of the law of
cosines c2 = a2 +b2 − 2ab cos (γ ) , with a, b, c,
and γ as defined in the diagram above. You
may also want to use the cos, acos, pow, and
sqrt functions from the standard cmath include
file. It will probably help to first write out the
math for each one of the out parameters on a
piece of pape
Recursion:
One common design pattern in creating functions is to set up a function to call
itself – to make it a recursive function. One classic example is calculating the
nth term of the Fibonacci sequence: the nth term is defined as the sum of term n
− 1 and term n − 2 , except for the first two terms, which are both defined as 1.
We can thus define the function fibonacci as follows:
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15. int fibonacci(int n) {
if(n == 0 || n == 1)
return 1;
else }
return fibonacci(n - 1) + fibonacci(n - 2);
}
n == 0 || n == 1 is called the base case – the situation in which
fibonacci does not call itself (i.e. it does not recurse). Every recursive
function must have a base case, and every time the function calls itself
the new calls must be one step closer to reaching the base case;
otherwise it will keep calling itself forever! (Or at least until the program
dies from an overdose of function calls.)
Recursion is a difficult technique to master, and we will not be covering
it in-depth for this course. However, it is still worth plowing through one
example of defining a recursive function.
2. a. One convenient way to define n! is n ( n − 1)! . (The factorial of 0 is
1.) Using this mathematical definition, define a recursive function
factorial that takes one integer argument and returns an integer. You
should first make sure to define a base case – the situation in which
you do not need to call factorial again to calculate the factorial
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16. . b. Pascal’s triangle is a triangular arrangement of numbers in which each
number in the triangle is the sum of the two numbers above it:
and so on. As it happens, the ith element of row n in this triangle can be
expressed as
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, for the mathematically inclined). For the first row, n= 0 ,
and
similarly i = 0 for the first column. i. Using your function from part (a),
indicate in just a few lines of code how you would print out the i th
element of row n. Assume that i and n are defined as integer variables.
ii. Using the code you wrote for part (i), write a loop that outputs the
entire row n. iii. Now turn your loop into a function that prints out the
nth row of Pascal’s triangle. It should take n as an argument, and
return nothing.

17. 2. Write a program to input an array of 10 integers and output the
average of all the elements of that array.
Passing arrays as arguments:
It is often useful to pass an array as an argument to a function. The
syntax for declaring a function that takes such an argument is as follows:
int myFunc(int arr[], const int arrayLen, …) { … }
The arr variable does not need to be declared with a particular length;
that is only necessary when creating an array. However, the function has
no way of knowing how long the array being passed in is, and since all
integer arrays are of the same data type, you cannot simply specify that
this function takes, say, an array of length 5. The solution is to pass the
length of the array as another argument.
Arrays are always passed by reference.
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Part 5 – Arrays:
1. Enter an array of 20 characters and check if each of them is a non-
alphanumeric character. Display the character and its position if it is.
(You may want to use the isalnum function from ).

18. As an aside, it is often useful to use named constants to indicate the length of an
array – that way, if you change the length of the array, you only need to change
the constant in one place. For example:
const int NUM_STUDENTS = 47;
double studentGrades[NUM_STUDENTS];
3. Adapt your program from Problem 2 to contain a main function that inputs the
array, which it then passes to an average function. This function should return a
decimal value representing the average of the values in the array.
Default initialization of arrays:
You do not need to explicitly initialize all elements of an array of numerical
values. If you specify an explicit size for the array, but initialize it with a shorter
list, the compiler assumes you want to initialize any unspecified elements to 0.
For instance,
int numbers[10] = {0};
initializes all 10 elements of numbers to 0.
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19. 4. In the Fibonacci sequence of numbers, each number is the sum of the previous
two numbers. Thus, the sequence begins 0, 1, 1, 2, 3, 5, 8, 13, etc. Write a
program which creates and displays the first 20 terms of Fibonacci sequence. Use
arrays to compute the elements of this series (you may initialize only the first two
elements of the array explicitly).
5. Write a program that enters an array of characters from the user and reverses
the sequence without creating a new array. (Print the reversed sequence at the
end.) For example, INPUT: a r k a n s a s OUTPUT: s a s n a k r a
6. Enter an array of characters from the user and remove all the duplicate
characters. (You may use a second array for this.) For example, INPUT: a e s e d f
k d m n v s o j n OUTPUT: a f k m v o j
7. Enter an array of characters from the user and move all the lower case
characters to the left side and the upper case characters to the right side of the
array. You may want to use the islower and/or isupper functions from the standard
cctype include file.
8. Enter an array of 10 integers and sort the first half in an ascending order and
the second half in a descending order. For example,
INPUT: 1, 38, 7, 13, 28, 19, 64, 18, 22, 11
OUTPUT: 1, 7, 13, 28, 38, 64, 22, 19, 18, 11
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20. Solutions
Part 1:
1. a. A function prototype for square should be added before main (or
the whole function should be moved earlier).
b. Either number should be added as a parameter to square, or
number should be made a global variable and removed from the
function call to square. The former method is preferable, since
the latter entails creating global variables unnecessarily.
c. Change the square prototype and definition to take an int&.
sumOfPositives should return
d. sum. Also, the call to it has too many arguments, which can be
corrected either by adding an argument to the prototype and
definition or by deleting one of the arguments.
2. a. It prints "This is not the hello message!".
b. It prints the numbers from 0 to 9 (the print statements that
actually print these numbers alternate between main and the
function).
c. It prints the numbers from 0 to 10, repeating each of the
numbers in between twice. There are two print statements: the
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21. gets executed for a given number, and then, since the parameter was
passed by reference, the original counter is not updated by mystery, but
only by the for loop, so the same number is printed again by the cout
statement in main.
Part 2:
1. double calculatePi( int numDarts ) {
int dartsInCircle;
for( int i = 0; i < numDarts; i++ ) {
double xVal = rand() / static_cast<double>(RAND_MAX),
yVal = rand() / static_cast<double>(RAND_MAX);
if( sqrt(xVal * xVal + yVal * yVal) <= 1 )
dartsInCircle++;
}
// pi = circle area / r^2, but r == 1
return static_cast<double>(dartsInCircle) / numDarts;
}
2. int randomInRange( int lowerBound, int upperBound ) {
int range = upperBound – lowerBound + 1;
return rand() % range + lowerBound;
}
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22. double randomInRange( double lowerBound, double upperBound ) {
double range = upperBound - lowerBound;
double randomDecimal = rand() / static_cast(RAND_MAX);
return randomDecimal * range + lowerBound;
}
3. The call is ambiguous – the compiler knows both how to convert a double to
an int and vice versa, so it doesn’t know which of the parameters it should
convert and which it should leave alone.
Part 3: 1. (10 for working valid solution, 5 for using smaller functions in larger)
a. int maximum( int num1, int num2 ) {
return ( num1 > num2 ? num1 : num2 );
}
b. int maximum( int num1, int num2 ) { return (
num1 > num2 ? num1 : num2 ); }
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23. int maximum( int num1, int num2, int num3 ) {
int preliminaryMax = maximum(num1, num2);
return ( preliminaryMax > num3 ? preliminaryMax : num3 );
}
int maximum( int num1, int num2, int num3, int num4 ) {
int preliminaryMax = maximum(num1, num2, num3);
return ( preliminaryMax > num4 ? preliminaryMax : num4 );
}
2.
int maximum( const int numbers[ ], const int arrayLength ) {
int max = numbers[0];
for( int i = 1; i < arrayLength; i++) {
if( numbers[i] > max ) {
max = numbers[i];
}
}
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24. return max;
}
3.
template < class T>
T maximum( const T numbers[], const int arrayLength ) {
T max = numbers[0];
for( int i = 1; i < arrayLength; i++) {
if( numbers[i] > max ) {
max = numbers[i];
}
}
return max;
}
Sample calls:
char chars[ ] = {'c', 'a', ':', '0'};
maximum(chars, 4); // T = char
char doubles[ ] = {234.2, 12.324, 938.23, 282.23423};
maximum(doubles, 4); // T = double
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25. 4. int maximum( const int num1, const int num2,
const int num3 = INT_MIN, const int num4 = INT_MIN ) {
int max = ( num1 > num2 ? num1 : num2 );
if( num3 > max )
max = num3;
if( num4 > max )
max = num4;
return max;
}
Part 4:
1.
a. void calculateTriangleValues( const double sideA,
const double sideB, const double angleGamma,
double &sideC, double &angleAlpha, double &angleBeta );
b. void calculateTriangleValues( const double a,
const double b, const double angleGamma,
double &c, double &angleAlpha, double &angleBeta ) {
// Use law of cosines to find c
c = sqrt( a*a + b*b – 2*a*b * cos(angleGamma) );
// Next use proportions of angles and sides to find others
angleBeta = asin(b * sin(angleGamma) / c );
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26. angleAlpha = 180 – angleGamma -
angleBeta;
/* Alternatively,
angleBeta = acos( (a*a + c*c – b*b)/(2*a*c)
); angleAlpha = 180 – angleGamma -
angleBeta;
*/
}
2. a. int factorial( int n ) {
if( n < 0 ) // Error check – good to include it,
return 0; // but we didn't require it
else if( n == 0 )
return 1;
else
return n * factorial( n - 1 );
}
b. void printPascalRowN( int n ) {
for( int i = 0; i < n; i++ ) {
cout << factorial(n) /
( factorial(n – i) * factorial(i) );
}
}
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27. Part 5:
1. #include <iostream>
#include <cctype>
using namespace std;
int main()
{
char arr[20];
cout << "Enter 20 characters: ";
for(int i = 0; i < 20; i++)
cin >> a[i];
for (int j = 0; j < 20; j++) {
if (!(isalnum(arr[j])))
cout << "The non-alphanumeric character at
position " << i + 1 << " is " << arr[j] << endl;
}
return 0;
}
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28. 2. #include <iostream>
using namespace std; i
nt main( )
{
int arr[10];
int sum=0;
cout << "Enter 10 integers: ";
for(int i=0; i<10; i++)
cin>>arr[i];
for(int j = 0; j < 10; j++)
sum+=arr[j];
cout << "The average is: "<< sum/10.0 << endl;
return 0;
}
3. #include <iostream>
using namespace std;
double average(int arr2[]) {
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29. int sum=0;
for (int i=0; i<10; i++)
sum+=arr2[i];
return sum / 10.0;
}
// Optional:
int main() {
int arr[10];
cout << "Enter 10 integers: ";
for(int i = 0; i < 10; i++)
cin>>arr[i];
cout << "The average is: " << average(arr); return 0;
}
4.
include <iostream>
using namespace std;
int main() {
int arr[20];
arr[0]=0;
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30. arr[1]=1;
cout << arr[0] << ", " << arr[1] << ", ";
for(int i = 2; i < 20; i++) {
arr[i] = arr[i-1] + arr[i-2];
cout << arr[i] << ", ";
}
return 0;
}
5. #include <iostream>
using namespace std;
int main() {
const int ARRAY_SIZE = 20;
char arr[ARRAY_SIZE];
cout << "Enter " << ARRAY_SIZE << " characters: ";
for(int i = 0; i < ARRAY_SIZE; i++)
cin >> arr[i];
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31. for(int i = 0; i < ARRAY_SIZE; i++) {
char temp = arr[i];
arr[i] = arr[ARRAY_SIZE – i - 1];
arr[ARRAY_SIZE – i - 1] = temp;
cout << arr[i];
}
return 0;
}
6. #include <iostream>
using namespace std;
int main() {
const int ARRAY_SIZE = 10;
char arr[ARRAY_SIZE];
char currentChar;
cout << "Enter " << ARRAY_SIZE << " characters: " << endl;
int i = 0;
for ( ; i < ARRAY_SIZE; i++)
cin >> arr[i];
for(i = 0; i < ARRAY_SIZE; i++) {
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32. currentChar = arr[i];
for(int j = 0; j < ARRAY_SIZE; j++) {
if( (arr[j] == currentChar) && (j != i) ) {
arr[i] = '0';
arr[j] = '0';
}
}
}
cout << "Array with duplicate characters removed: " << endl;
for(i = 0; i < unrepeatedChars; i++) {
cout << finalArray[unrepeatedChars];
}
return 0;
}
// Alternate solution:
int main() {
const int ARRAY_SIZE = 10;
char initialArray[ARRAY_SIZE], finalArray[ARRAY_SIZE];
int unrepeatedChars = 0;
bool repeat;
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33. cout << "Enter " << ARRAY_SIZE << " characters: " << endl;
int i = 0;
for ( ; i < ARRAY_SIZE; i++)
cin >> initialArray[i];
for(i = 0; i < ARRAY_SIZE; i++) {
repeat = false;
for(int j = 0; j < ARRAY_SIZE; j++) {
if( (initialArray [j] == initialArray [i]) && (j!=i) ) {
repeat = true;
break;
}
}
if( !repeat ) {
finalArray[unrepeatedChars] = initialArray[i];
unrepeatedChars++;
}
}
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34. cout << "Array with duplicate characters removed: " << endl;
for(i = 0; i < unrepeatedChars; i++) {
cout << finalArray[unrepeatedChars];
}
return 0;
}
7. #include <iostream>
#include <cctype>>
using namespace std;
int main() {
const int ARRAY_SIZE = 10;
char arr[ARRAY_SIZE];
int i = 0,j = 0;
cout << "Enter " << ARRAY_SIZE << " characters:" << endl;
for ( ; i < ARRAY_SIZE; i++)
cin >> arr[i];
for(i = 0; i < ARRAY_SIZE; i++) {
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35. for(j = 0; j < i; j++) {
if( islower(arr[i]) && isupper(arr[j]) ) {
int temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
}
cout << "The array is :" << endl;
for(i = 0; i < 10; i++)
cout << arr[i] << " ";
return 0;
}
8. #include <iostream>
using namespace std;
int main() {
int arr[10];
int i, j;
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36. cout << "Enter 10 integers: ";
for (i = 0; i < 10; ++i)
cin >> arr[i];
for (i = 0; i <= 3; ++i) {
for (j = i+1; j <= 4; ++j) {
if( arr[i] >= arr[j] ) {
int temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
}
for ( i = 5; i <= 8; ++i) {
for (j = i+1; j <= 9; ++j) {
if (arr[i] <= arr[j]) {
int temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
} } }
for (i = 0; i < 10; ++i)
cout << arr[i] << " "; return 0; }
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