1.3 Conditionals and Loops
We use the term flow of control to refer to the sequence of statements that are executed in a program. All of the programs that we have examined to this point have a simple flow of control: the statements are executed one after the other in the order given. Most programs have a more complicated structure where statements may or may not be executed depending on certain conditions (conditionals), or where groups of statements are executed multiple times (loops).
If statements.
Most computations require different actions for different inputs. Program Flip.java uses an ifelse statement to print out the results of a coin flip. The table below summarizes some typical situations where you might need to use an if or ifelse statement.
While loops.
Many computations are inherently repetitive. The while loop enables us to perform a group of statements many times. This enables us to express lengthy computations without writing lots of code. TenHellos.java prints out "Hello World" 10 times.
 PowersOfTwo.java takes a commandline argument N and prints out all of the powers of 2 less than or equal to N.
For loops.
Many loops follow the same basic scheme: initialize an index variable to some value and then use a while loop to test an exit condition involving the index variable, using the last statement in the while loop to modify the index variable. Java's for loop is a direct way to express such loops. For example, the following two lines of code are equivalent to the corresponding lines of code in TenHellos.
for (int i = 4; i <= 10; i = i + 1) System.out.println(i + "th Hello");
 Idioms for singleoperand operations. The idiom i++ is a shorthand notation for i = i + 1.
 Scope. The scope of a variable is the part of the program where it is defined. Generally the scope of a variable is the statements following the declaration in the same block as the declaration. For this purpose, the code in the for loop header is considered to be in the same block as the for loop body.
Loop examples.
Give table containing loop examples. [Loops.java]Applications.
The ability to program with loops and conditionals immediately opens up the world of computation to us. Ruler subdivisions. RulerN.java reads in a command line parameter N and prints out the string of ruler subdivision lengths. This program illustrates one of the essential characteristics of loops  the program could hardly be simpler, but it can produce a huge amount of output.
 Finite sums.
The computational paradigm used in
PowersOfTwo.java is one that you
will use frequently. It uses two variables  one as an index that controls
a loop, and the other to accumulate a computational result.
Program Harmonic.java uses the same
paradigm to evaluate the sum
H_{N} = 1/1 + 1/2 + 1/3 + 1/4 + ... + 1/N
These numbers, which are known as the Harmonic numbers, arise frequently in the analysis of algorithms. They are the discrete analog of the logarithm: they approximate the area under the curve y = 1/x from x = 1 to x = N. This approximation is a special example of quadrature or numerical integration.
 Newton's method. How are functions in Java's Math library such as Math.sqrt() implemented? Sqrt.java uses a classic iterative technique that dates back to Heron of Alexandria in the first century. It is a special case of the more general NewtonRaphson method. To compute the square root of a positive number x: Start with an estimate t. If t is equal to x/t (up to machine precision), then t is equal to a square root of x, so the computation is complete. If not, refine the estimate by replacing t with the average of t and x/t. Each time we perform this update, we get closer to the desired answer.
 Number conversion. Binary.java prints the binary (base 2) representation of the decimal number typed as the command line argument. It is based on decomposing the number into a sum of powers of two. For example, the binary representation of 106 is 1101010, which is the same as saying that 106 = 64 + 32 + 8 + 2, or in binary, 1101010 = 1000000 + 100000 + 1000 + 10. To compute the binary representation of N, we consider the powers of 2 less than or equal to N in decreasing order to determine which belong in the binary decomposition (and therefore correspond to a 1 bit in the binary representation).
 Gambler's ruin. Our next example is representative of a widely used class of programs, where we use computers to simulate what might happen in the real world, so that we can make informed decisions in all kinds of complicated situations. Suppose a gambler makes a series of fair $1 bets, starting with $50, and continue to play until she either goes broke or has $250. What are the chances that she will go home with $250, and how many bets might she expect to make before winning or losing? Program Gambler.java is a simulation that can help answer these questions. It reads in three commandline arguments, the initial stake ($50), the goal amount ($250), and the number of times we want to simulate the game. See the textbook for details. Factors.java takes a commandline argument N and prints its prime factorization. In contrast to many of the other programs that we have seen (which we could do in a few minutes with a calculator or pencil and paper), this computation would not be feasible without a computer. See the textbook for details.
Other conditional and loop constructs.
To be complete, we consider four more Java constructs related to conditionals and loops. They are used much less frequently than the if, while, and for statements that we've been working with, but it is worthwhile to be aware of them. Dowhile statements.
A dowhile loop is almost the same as a while
loop except that the loop continuation condition is omitted the
first time through the loop. If the conditional initially holds,
there is no difference. For example,
the following code sets x and y such that (x, y) is randomly
distributed inside the circle centered at (0, 0) with radius 1.
double x, y, r; do { x = 2.0 * Math.random()  1.0; y = 2.0 * Math.random()  1.0; r = x*x + y*y; } while (r > 1);
With Math.random() we get points that are randomly distributed in the 2by2 square center at (0, 0). We just generate points in this region until we find one that lies inside the unit disk. We always want to generate at least one point so a dowhile loop is most appropriate. Note that we declare x and y outside the loop since we will want to access their values after the loop terminates.
 Break statements. In some situations we want to immediate exit a loop, without letting it run to completion. Java provides the break statement for this purpose. Program Prime.java takes a commandline argument N and prints out true if N is prime, and false otherwise. It illustrates that we can break out of a loop using the "break" statement (as soon as we find an some integer that divides i, we know that i is not prime and can stop looking). Determining whether a number is prime is easy to do if the number is small, but is a challenge if the number is huge. We will look at some better methods later on. Huge prime numbers have important applications in cryptography, so researchers still seek better methods for testing primality.
 Continue statement. Java also provides a way to skip to the next iteration of a loop: the continue statement. When a continue is executed within a loop body, the flow of control immediately transfers to the next iteration of the loop. Usually, it is easy to achieve the same effect with an if statement inside the loop.
 Conditional operator.
The conditional operator ?: is a ternary operator (three operands)
that enables you to embed a conditional within an expression.
The three operands are separated by the ? and :
symbols. If the first operand (a boolean expression) is true,
the result has the value of the second expression; otherwise it
has the value of the third expression.
int min = (x < y) ? x : y;
 Labeled break and continue statements. The break and continue statements apply to the innermost for or while loop. Sometimes we want to jump out of several levels of nested loops. Java provides the labeled break and labeled continue statements to accomplish this. Here is an example.
Q + A
Q. My program is stuck in an infinite loop. How do I stop it?
A. For DrJava, click the menu option Tools > Reset Interactions. For OS X Terminal and Linux shell, type <ctrlc>. That is, hold down the key labeled ctrl or control and press the c key. For Windows Command Prompt type <ctrlz>.
Q. How can I check whether two strings are equal? Using == doesn't work.
A. This is one of the difference between primitive types (int, double, boolean) and reference types (String). We'll learn about testing strings for equality in Section 3.1.
Q. Why doesn't the statement if (a <= b <= c) work?
A. The <= operator cannot be chained. Instead, use if (a <= b && b <= c).
Q. Is there an example where you cannot remove the curly braces from a onestatement block?
A. The first code fragment is legal (but pointless), while the second is a compiletime error. Technically, the second line in the second fragment is a declaration and not a statement.
// legal for (int i = 0; i <= N; i++) { int x = 5; } // illegal for (int i = 0; i <= N; i++) int x = 5;
Q. Is there an example for when the following for and while loops are not equivalent?
for (<init stmnt> <boolean expr>; <incr stmnt>) { <body statements> } <init stmnt>; while (<boolean expr>) { <body statements> <incr stmnt> }
A. Yes. Use a continue statement.
Exercises
 Write a program that takes three integer commandline arguments and prints equal if all three are equal, and not equal otherwise.
 Write a more general and robust version of Quadratic.java that prints the roots of the polynomial ax^{2} + bx + c, prints an appropriate error message if the discriminant is negative, and behaves appropriately (avoiding division by zero) if a is zero.

What (if anything) is wrong with each of the following statements?
 if (a > b) then c = 0;
 if a > b { c = 0; }
 if (a > b) c = 0;
 if (a > b) c = 0 else b = 0;
 Write a code fragment that prints true if the double variables x and y are both strictly between 0 and 1 and false otherwise.
 Improve your solution to Exercise 1.2.25 by adding code to check that the values of the commandline arguments fall within the ranges of validity of the formula, and also adding code to print out an error message if that is not the case.

Suppose that i and j are both of type int. What is
the value of j after each of the following statements is executed?
 for (i = 0, j = 0; i < 10; i++) j += i;
 for (i = 0, j = 1; i < 10; i++) j += j;
 for (j = 0; j < 10; j++) j += j;
 for (i = 0, j = 0; i < 10; i++) j += j++;
 Rewrite TenHellos.java to make a program Hellos.java that takes the number of lines to print as a commandline argument. You may assume that the argument is less than 1000. Hint: consider using i % 10 and i % 100 to determine whether to use "st", "nd", "rd", or "th" for printing the ith Hello.
 Write a program FivePerLine.java that, using one for loop and one if statement, prints the integers from 1000 to 2000 with five integers per line. Hint: use the % operator.
 Write a program that takes an integer N as a commandline argument and uses Math.random() to print N uniform random variables between 0 and 1, and then prints their average value. (see Exercise 1.2.30).
 Describe what happens when you invoke RulerN.java with an argument that is too large, such as java RulerN 100.
 Write a program FunctionGrowth.java that prints a table of the values of log N, N, N log N, N^{2}, N^{3}, and 2^{N} for N = 16, 32, 64, ..., 2048. Use tabs ('\t' characters) to line up columns.

What is the value of m and n after executing
the following code?
int n = 123456789; int m = 0; while (n != 0) { m = (10 * m) + (n % 10); n = n / 10; }

What does the following code print
out?
int f = 0, g = 1; for (int i = 0; i <= 15; i++) { System.out.println(f); f = f + g; g = f  g; }
 Expand your solution to CarLoan.java from Exercise 1.2.24 to print a table giving the total amount paid and the remaining principal after each monthly payment.

Unlike the harmonic numbers, the
sum 1/1 + 1/4 + 1/9 + 1/16 + ... + 1/N^{2}
does converge to a constant as N grows to
infinity. (Indeed, the constant is π^{2} / 6, so
this formula can be used to estimate the value of π.)
Which of the following
for loops computes
this sum?
Assume that N is an int initialized to 1000000
and sum is a double initialized to 0.
(a) for (int i = 1; i <= N; i++) sum = sum + 1 / (i * i); (b) for (int i = 1; i <= N; i++) sum = sum + 1.0 / i * i; (c) for (int i = 1; i <= N; i++) sum = sum + 1.0 / (i * i); (d) for (int i = 1; i <= N; i++) sum = sum + 1 / (1.0 * i * i);
 Using the fact that the slope of the tangent to a (differentiable) function f(x) at x = t is f'(t), find the equation of the tangent line and use that equation to find the point where the tangent line intersects the x axis to show that you can use Newton's method to find a root of any function f(x) as follows: at each iteration, replace the estimate t by t  f(t) / f'(t). Then use the fact that f'(t) = 2t for f(x) = x^{2}  c to show that Sqrt.java implements Newton's method for finding the square root of c.
 Use the general formula for Newton's method in the previous exercise to develop a program Root.java that takes two command line arguments c and k, and prints the kth root of c. Assume k is a positive integer. You may use Math.pow(), but only with an exponent that is a nonnegative integer.
 Suppose that x and t are variables of type double and N is a variable of type int. Write a code fragment to set t to x^{N} / N!.
 Modify Binary.java to get a program Modify Kary.java that takes a second command line argument K and converts the first argument to base K. Assume the base is between 2 and 16. For bases greater than 10, use the letters A through F to represent the 11th through 16th digits, respectively.
 Write a code fragment that puts the binary representation of an int N into a String s.
 Write a version of Gambler.java that uses a for loop instead of a while loop.
 Write a program GamblerPlot.java that traces a gambler's ruin simulation by printing a line after each bet that has one asterisk corresponding to each dollar held by the gambler.
 Modify Gambler.java to take an extra command line parameter that specifies the (fixed) probability that the gambler wins each bet. Use your program to try to learn how this probability affects the chance of winning and the expected number of bets.
 Modify Gambler.java to take an extra command line parameter that specifies the number of bets the gambler is willing to make, so that there are three possible ways for the game to end: the gambler wins, loses, or runs out of time. Add to the output to give the expected amount of money the gambler will have when the game ends.
 Modify Factors.java to print just one copy of each of the prime divisors.
 Run quick experiments to determine the impact of using the loop continuation condition (i <= N/i) instead of (i*i <= N) in Factors.java. For each loop continuation condition, find the largest integer M such that when you type in an M digit number, the program is sure to finish within 10 seconds.

Write a program Checkerboard.java that
takes one commandline argument N and prints out a two dimensional
NbyN checkerboard pattern with
alternating spaces and asterisks, like the following 4by4 pattern.
* * * * * * * * * * * * * * * *
 Write a program GCD.java that find the greatest common divisor (gcd) of two integers x and y using Euclid's algorithm, which is an iterative computation based on the following observation: If x > y, then if y divides x, the gcd of x and y is y; otherwise the gcd of x and y is the same as the gcd of x % y and y.
 Write a program RelativelyPrime that takes one commandline argument N and prints out an NbyN table such that there is an * in row i and column j if the gcd of i and j is 1 (i and j are relatively prime) and a space in that position otherwise.
 Write a program PowersOfK that takes an integer k as commandline argument and prints all the positive powers of k in the Java long data type. Note: The constant Long.MAX_VALUE is the value of the largest integer in long.
 Generate a random point (x, y, z) on the surface of a sphere using Marsaglia’s method: Pick a random point (a, b) in the unit disk using the method described at the end of this section.Then, set x = 2 a sqrt(1  a^2  b^2) and y = 2 b sqrt(1  a^2  b^2) and z = 1  2 (a^2 + b^2).
Creative Exercises

Ramanujan's taxi.
S. Ramanujan was an Indian mathematician who became famous for his
intuition for numbers. When the English mathematician G. H. Hardy came to
visit him in the hospital one day, Hardy remarked that the number of his
taxi was 1729, a rather dull number. To which Ramanujan
replied, "No, Hardy! No, Hardy! It is a very interesting number.
It is the smallest number
expressible as the sum of two cubes in two different ways."
Verify this claim by writing a program
Ramanujan.java that takes a
command line argument N and prints out all integers less than
or equal to N that can be expressed as the sum of two cubes in
two different ways  find distinct positive integers a,
b, c, and d
such that a^{3} + b^{3} = c^{3} + d^{3}.
Use four nested for loops.
Now, the license plate 87539319 seems like a rather dull number. Determine why it's not.
 Checksums. The International Standard Book Number (ISBN) is a 10 digit code that uniquely specifies a book. The rightmost digit is a checksum digit which can be uniquely determined from the other 9 digits from the condition that d_{1} + 2d_{2} + 3d_{3} + ... + 10d_{10} must be a multiple of 11 (here d_{i} denotes the ith digit from the right). The checksum digit d_{1} can be any value from 0 to 10: the ISBN convention is to use the value X to denote 10. Example: the checksum digit corresponding to 020131452 is 5 since is the only value of d_{1} between 0 and and 10 for which d_{1} + 2*2 + 3*5 + 4*4 + 5*1 + 6*3 + 7*1 + 8*0 + 9*2 + 10*0 is a multiple of 11. Write a program ISBN.java that takes a 9digit integer as a command line argument, computes the checksum, and prints out the 10digit ISBN number. It's ok if you don't print out any leading 0's.

Calendar.
Write a program Calendar that takes two command
line arguments m and y and prints out the monthly
calendar for the mth month of year y. For example,
your output for Calendar 2 2009 should be
February 2009 S M Tu W Th F S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Hint: see programs LeapYear.java and DayOfWeek.java.
 Counting primes. Write a program PrimeCounter.java that takes one commandline argument N and prints out the number of primes less than N. Use it to print out the number of primes less than 10 million. Note: if you are not careful to make your program efficient, it may not finish in a reasonable amount of time. Later in Section 1.5, we will learn about a more efficient way to perform this computation called the Sieve of Eratosthenes.
 Dragon curves. Write a program Dragon.java that takes a commandline argument N and prints out the instructions for drawing a dragon curve of order N. See exercise 1.2.18.
 What happens when you try to compile the following code fragment?
double x; if (a >= 0) x = 3.14; if (a < 0) x = 2.71; System.out.println(x);
Answer: it complains that the variable x might not have been initialized (even though we can clearly see that x will be initialized by one of the two if statements). You can avoid this problem here by using ifelse.

Exponential function.
Assume that x and sum are variables of type double.
Write a code fragment to use the Taylor series expansion to set the value of
sum to
e^{x} = 1 + x + x^{2}/2! + x^{3}/3! + x^{4}/4! + . . .

Trigonometric functions.
Write two programs
Sin.java and
Cos.java that compute sin x and cos x
using the Taylor series expansions
sin x = x  x^{3}/3! + x^{5}/5!  x^{7}/7! + . . .
cos x = 1  x^{2}/2! + x^{4}/4!  x^{6}/6! + . . .  Experimental analysis. Run experiments to determine the relative costs of Math.exp() and the following three methods from Exercise 2.3.36 for the problem of computing e^{x}: the direct method with nested for loops, the improved method with a single for loop, and the latter with the termination condition (term > 0). For each method, use trialanderror with a command line argument to determine how many times your computer can perform the computation in 10 seconds.
 2D random walk. A two dimensional random walk simulates the behavior of a particle moving in a grid of points. At each step, the random walker moves north, south, east, or west with probability 1/4, independently of previous moves. Determine how far away (on average) the random walker is from the starting point after N steps. (Theoretical answer: on the order of sqrt(N).)
 Game simulation. In the 1970s game show called "Let's Make a Deal", a contestant is presented with three doors. Behind one door is a valuable prize, behind the other two are gag gifts. After the contestant chooses a door, the host opens up one of the other two doors (never revealing the prize, of course). The contestant is then given the opportunity to switch to the other unopened door. Should the contestant do so? Intuitively, it might seem that the contestant's initial choice door and the other unopened door are equally likely to contain the prize, so there would be no incentive to switch. Write a program MonteHall.java to test this intuition by simulation. Your program should take a command line parameter N, play the game N times using each of the two strategies (switch or don't switch) and print the chance of success for each strategy. Or you can play the game here.
 Chaos. Write a program to study the following simple model for population growth, which might be applied to study fish in a pond, bacteria in a test tube, or any of a host of similar situations. We suppose that the population ranges from 0 (extinct) to 1 (maximum population that can be sustained). If the population at time t is x, then we suppose the population at time t+1 to be rx(1x), where the argument r, sometimes known as the fecundity parameter, controls the rate of growth. Start with a small population—say x = 0.01—and study the result of iterating the model, for various values of r. For which values of r does the population stabilize at x = 1  1/r? Can you say anything about the population when r is 3.5? 3.8? 5?
 Euler's sumofpowers conjecture. In 1769 Leonhard Euler formulated a generalized version of Fermat's Last Theorem, conjecturing that at least n nth powers are needed to obtain a sum that is itself an nth power, for n > 2. Write a program to disprove Euler's conjecture (which stood until 1967), using a quintuply nested loop to find four positive integers whose 5th power sums to the 5th power of another positive integer. That is, find a, b, c, d, and e such that a^5 + b^5 + c^5 + d^5 = e^5. Use the long data type.
Web Exercises
 Write a program RollDie.java that generates the result of rolling a fair sixsided die (an integer between 1 and 6).
 Write a program RollLoadedDie.java that generates the result of rolling a loaded sixsided die (an integer between 1 and 5 with probability 1/8 and the integer 6 with probability 3/8).
 Write a program that takes three integer commandline arguments a, b, and c and print out the number of distinct values (1, 2, or 3) among a, b, and c.
 RGB to HSB converter. Write a program RGBtoHSV.javathat takes an RGB color (three integers between 0 and 255) and transforms it to an HSB color (three different integers between 0 and 255). Write a program HSVtoRGB.java that applies the inverse transformation.
 Write a program that takes five integer commandline arguments and prints out the median (the third largest one).
 (hard) Now, try to compute the median of 5 elements such that when executed, it never makes more than 6 total comparisons.

How can I create in an infinite loop with a for loop?
Answer: for(;;) is the same as while(true).

What's wrong with the following loop?
boolean done = false; while (done = false) { ... }
boolean done = false; while (!done) { ... }

What's wrong with the following loop that is intended to
compute the sum of the integers 1 through 100?
for (int i = 1; i <= N; i++) { int sum = 0; sum = sum + i; } System.out.println(sum);

Write a program Hurricane.java that
that takes the wind speed (in miles per hour) as an integer commandline argument
and prints out whether it qualifies as a hurricane, and if so, whether
it is a Category 1, 2, 3, 4, or 5 hurricane. Below is a table of the
wind speeds according to the
SaffirSimpson scale.
Category Wind Speed (mph) 1 74  95 2 96  110 3 111  130 4 131  155 5 155 and above 
What is wrong with the following code fragment?
double x = 32.2; boolean isPositive = (x > 0); if (isPositive = true) System.out.println(x + " is positive"); else System.out.println(x + " is not positive");
Answer: It uses the assignment operator = instead of the equality operator ==. A better solution is to write if (isPositive).

Change/add one character so that the following program prints out
20 x's. There are two different solutions.
int i = 0, n = 20; for (i = 0; i < n; i) System.out.print("x");

What does the following code fragment do?
if (x > 0); System.out.println("positive");
Answer: always prints positive regardless of the value of x because of the extra semicolon after the if statement.
 Boys and girls. A couple beginning a family decides to keep having children until they have at least one of either sex. Estimate the average number of children they will have via simulation. Also estimate the most common outcome (record the frequency counts for 2, 3, and 4 children, and also for 5 and above). Assume that the probability p of having a boy or girl is 1/2.

What does the following program do?
public static void main(String[] args) { int N = Integer.parseInt(args[0]); int x = 1; while (N >= 1) { System.out.println(x); x = 2 * x; N = N / 2; } }
 Boys and girls.
Repeat the previous question, but assume the couple keeps having children
until they have another child which is of the same sex as the first child.
How does your answer change if p is different from 1/2?
Surprisingly, the average number of children is 2 if p = 0 or 1, and 3 for all other values of p. But the most likely value is 2 for all values of p.

Given two positive integers a and b, what result does the
following code fragment leave in c
c = 0; while (b > 0) { if (b % 2 == 1) c = c + a; b = b / 2; a = a + a; }
Answer a * b.

Write a program using a loop and four conditionals to print
12 midnight 1am 2am ... 12 noon 1pm ... 11pm

What does the following program print out?
public class Test { public static void main(String[] args) { if (10 > 5); else; { System.out.println("Here"); }; } }
 Alice tosses a fair coin until she sees two consecutive heads. Bob tosses another fair coin until he sees a head followed by a tail. Write a program to estimate the probability that Alice will make fewer tosses than Bob? Answer: 39/121.
 Rewrite DayOfWeek.java from Exercise 1.2.29 so that it prints out the day of the week as Sunday, Monday, and so forth instead of an integer between 0 and 6. Use a switch statement.
 NumbertoEnglish. Write a program to read in a command line integer between 999,999,999 and 999,999,999 and print out the English equivalent. Here is an exhaustive list of words that your program should use: negative, zero, one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, hundred, thousand, million . Don't use hundred, when you can use thousand, e.g., use one thousand five hundred instead of fifteen hundred. Reference.
 Gymnastics judging. A gymnast's score is determined by a panel of 6 judges who each decide a score between 0.0 and 10.0. The final score is determined by discarding the high and low scores, and averaging the remaining 4. Write a program GymnasticsScorer.java that takes 6 real command line inputs representing the 6 scores and prints out their average, after throwing out the high and low scores.
 Quarterback rating.
To compare NFL quarterbacks, the NFL devised a
the quarterback rating
formula based on the quarterbacks number of completed passes (A),
pass attempts (B),
passing yards (C), touchdown passes (D), and interception (E) as follows:
 Completion ratio: W = 250/3 * ((A / B)  0.3).
 Yards per pass: X = 25/6 * ((C / B)  3).
 Touchdown ratio: Y = 1000/3 * (D / B)
 Interception ratio: Z = 1250/3 * (0.095  (E / B))
 Decimal expansion of rational numbers. Given two integers p and q, the decimal expansion of p/q has an infinitely repeating cycle. For example, 1/33 = 0.03030303.... We use the notation 0.(03) to denote that 03 repeats indefinitely. As another example, 8639/70000 = 0.1234(142857). Write a program DecimalExpansion.java that reads in two command line integers p and q and prints out the decimal expansion of p/q using the above notation. Hint: use Floyd's rule.
 Friday the 13th.
What is the maximum number of consecutive days in which no Friday
the 13th occurs?
Hint: The Gregorian calendar repeats itself
every 400 years (146097 days) so you only need to worry about
a 400 year interval.
Answer: 426 (e.g., from 8/13/1999 to 10/13/2000).
 January 1.
Is January 1 more likely to fall on a Saturday or Sunday?
Write a program to determine the number of times each occurs
in a 400 year interval.
Answer: Sunday (58 times) is more likely than Saturday (56 times).

What do the following two code fragments do?
for (int i = 0; i < N; i++) for (int j = 0; j < N; j++) if (i != j) System.out.println(i + ", " + j); for (int i = 0; i < N; i++) for (int j = 0; (i != j) && (j < N); j++) System.out.println(i + ", " + j);

Determine what value gets printed out without using a computer.
Choose the correct answer from 0, 100, 101, 517, or 1000.
int cnt = 0; for (int i = 0; i < 10; i++) for (int j = 0; j < 10; j++) for (int k = 0; k < 10; k++) if (2*i + j >= 3*k) cnt++; System.out.println(cnt);
 Rewrite CarLoan.java from Creative Exercise XYZ so that it properly handles an interest rate of 0% and avoids dividing by 0.

Write the shortest Java program you can that reads
in an integer N from the command line and
print true if (1 + 2 + ... + N) ^{2} is equal
to (1^{3} + 2^{3} + ... + N^{3}).
Answer: always print true.
 Modify Sqrt.java so that it reports an error if the user enters a negative number and works properly if the user enters zero.
 What happens if we initialize t to x instead of x in program Sqrt.java?
 Sample standard deviation of uniform distribution. Modify Exercise 8 so that it prints out the sample standard deviation in addition to the average.
 Sample standard deviation of normal distribution. that takes an integer N as a command line argument and uses Web Exercise 1 from Section 1.2 to print N standard normal random variables, and their average value, and sample standard deviation.
 Loaded dice. [Stephen Rudich] Suppose you have three, three sided dice. A: {2, 6, 7}, B: { 1, 5, 9}, and C: {3, 4, 8}. Two players roll a die and the one with the highest value wins. Which die would you choose? Answer: A beats B with probability 5/9, B beats C with probability 5/9 and C beats A with probability 5/9. Be sure to choose second!
 ThueMorse sequence. Write a program ThueMorse.java that reads in a command line integer N and prints the ThueMorse sequence of order N. The first few strings are 0, 01, 0110, 01101001. Each successive string is obtained by flipping all of the bits of the previous string and concatenating the result to the end of the previous string. The sequence has many amazing properties. For example, it is a binary sequence that is cubefree: it does not contain 000, 111, 010101, or sss where s is any string. It is selfsimilar: if you delete every other bit, you get another ThueMorse sequence. It arises in diverse areas of mathematics as well as chess, graphic design, weaving patterns, and music composition.
 Program Binary.java prints the binary representation of a decimal number N by casting out powers of 2. Write an alternate version Program Binary2.java that is based on the following method: Write 1 if N is odd, 0 if N is even. Divide N by 2, throwing away the remainder. Repeat until N = 0 and read the answer backwards. Use % to determine whether or not N is even, and use string concatenation to form the answer in reverse order.

What does the following code fragment do?
int digits = 0; do { digits++; n = n / 10; } while (n > 0);
Answer the number of bits in the binary representation of a natural number n. We use a dowhile loop so that code output 1 if n = 0.
 Write a program NPerLine.java that takes a commandline argument N and prints the integers from 10 to 99 with N integers per line.
 Modify NPerLine.java so that it prints out the integers from 1 to 1000 with N integers per line. Make the integers line up by printing the right number of spaces before an integer (e.g., three for 19, two for 1099, and one for 100999).
 Suppose a, b, and c are random number uniformly distributed between 0 and 1. What is the probability that a, b, and c form the side length of some triangle? Hint: they will form a triangle if and only if the sum of every two values is larger than the third.
 Repeat the previous question, but calculate the probability that the resulting triangle is obtuse, given that the three numbers for a triangle. Hint: the three lengths will form an obtuse triangle if and only if (i) the sum of every two values is larger than the third and (ii) the sum of the squares of every two side lengths is greater than or equal to the square of the third.

What is the value of s after executing
the following code?
int M = 987654321; String s = ""; while (M != 0) { int digit = M % 10; s = s + digit; M = M / 10; }

What is the value of i after the following
confusing code is executed?
int i = 10; i = i++; i = ++i; i = i++ + ++i;
Moral: don't write code like this.
 Formatted ISBN number. Write a program ISBN2.java that reads in a 9 digit integer from a commandline argument, computes the check digit, and prints out the fully formatted ISBN number, e.g, 0201314525.
 UPC codes.
The Universal Product Code
(UPC)
is a 12 digit code that uniquely specifies a product.
The least significant digit d_{1}(rightmost one)
is a check digit which is the uniquely determined by making
the following expression a multiple of 10:
(d_{1} + d_{3} + d_{5} + d_{7} + d_{9} + d_{11}) + 3 (d_{2} + d_{4} + d_{6} + d_{8} + d_{10} + d_{12})
As an example, the check digit corresponding to 04850000102 (Tropicana Pure Premium Orange Juice) is 8 since
(8 + 0 + 0 + 0 + 5 + 4) + 3 (2 + 1 + 0 + 0 + 8 + 0) = 50
and 50 is a multiple of 10. Write a program that reads in a 11 digit integer from a command line parameter, computes the check digit, and prints out the the full UPC. Hint: use a variable of type long to store the 11 digit number.
 Write a program that reads in the wind speed (in knots) as a command line argument and prints out its force according to the Beaufort scale. Use a switch statement.
 Making change.
Write a program that reads in a command line integer N (number of pennies)
and prints out the best way (fewest number of coins)
to make change using US coins (quarters, dimes, nickels, and pennies only).
For example, if N = 73 then print out
2 quarters 2 dimes 3 pennies
Hint: use the greedy algorithm. That is, dispense as many quarters as possible, then dimes, then nickels, and finally pennies.

Write a program Triangle.java that takes a
commandline argument N and prints an NbyN triangular pattern like the one below.
* * * * * * . * * * * * . . * * * * . . . * * * . . . . * * . . . . . *

Write a program Ex.java that takes a commandline
argument N and prints a (2N + 1)by(2N + 1) ex like the one below.
Use two for loops and one ifelse statement.
* . . . . . * . * . . . * . . . * . * . . . . . * . . . . . * . * . . . * . . . * . * . . . . . *

Write a program BowTie.java that takes a commandline
argument N and prints a (2N + 1)by(2N + 1) bowtie like the one below.
Use two for loops and one ifelse statement.
* . . . . . * * * . . . * * * * * . * * * * * * * * * * * * * . * * * * * . . . * * * . . . . . *

Write a program Diamond.java that takes a commandline
argument N and prints a (2N + 1)by(2N + 1) diamond like the one below.
% java Diamond 4 . . . . * . . . . . . . * * * . . . . . * * * * * . . . * * * * * * * . * * * * * * * * * . * * * * * * * . . . * * * * * . . . . . * * * . . . . . . . * . . . .
 Write a program Heart.java that takes a commandline argument N and prints a heart.

What does the program Circle.java print
out when N = 5?
for (int i = N; i <= N; i++) { for (int j = N; j <= N; j++) { if (i*i + j*j <= N*N) System.out.print("* "); else System.out.print(". "); } System.out.println(); }
 Seasons.
Write a program Season.java that takes two command
line integers M and D and prints the season corresponding
to month M (1 = January, 12 = December) and day D in the
northern hemisphere. Use the following table
SEASON FROM TO Spring March 21 June 20 Summer June 21 September 22 Fall September 23 December 21 Winter December 21 March 20
 Zodiac signs.
Write a program Zodiac.java that takes two command
line integers M and D and prints the Zodiac sign corresponding
to month M (1 = January, 12 = December) and day D. Use the
following table
SIGN FROM TO Capricorn December 22 January 19 Aquarius January 20 February 17 Pisces February 18 March 19 Aries March 20 April 19 Taurus April 20 May 20 Gemini May 21 June 20 Cancer June 21 July 22 Leo July 23 August 22 Virgo August 23 September 22 Libra September 23 October 22 Scorpio October 23 November 21 Sagittarius November 22 December 21  Muay Thai kickboxing.
Write a program that reads in the weight of a
Muay Thai kickboxer (in pounds) as a command line argument
and prints out their weight class.
Use a switch statement.
CLASS FROM TO Flyweight 0 112 Super flyweight 112 115 Bantamweight 115 118 Super bantamweight 118 122 Featherweight 122 126 Super featherweight 126 130 Lightweight 130 135 Super lightweight 135 140 Welterweight 140 147 Super welterweight 147 154 Middleweight 154 160 Super middleweight 160 167 Light heavyweight 167 175 Super light heavyweight 175 183 Cruiserweight 183 190 Heavyweight 190 220 Super heavyweight 220   Euler's sum of powers conjecture. In 1769 Euler generalized Fermat's Last Theorem and conjectured that it is impossible to find three 4th powers whose sum is a 4th power, or four 5th powers whose sum is a 5th power, etc. The conjecture was disproved in 1966 by exhaustive computer search. Disprove the conjecture by finding positive integers a, b, c, d, and e such that a^{5} + b^{5} + c^{5} + d^{5}= e^{5}. Write a program Euler.java that reads in a command line parameter N and exhaustively searches for all such solutions with a, b, c, d, and e less than or equal to N. No counterexamples are known for powers greater than 5, but you can join EulerNet, a distributed computing effort to find a counterexample for sixth powers.
 Blackjack. Write a program Blackjack.java that takes three command line integers x, y, and z representing your two blackjack cards x and y, and the dealers faceup card z, and prints out the "standard strategy" for a 6 card deck in Atlantic city. Assume that x, y, and z are integers between 1 and 10, representing an ace through a face card. Report whether the player should hit, stand, or split according to these strategy tables. (When you learn about arrays, you will encounter an alternate strategy that does not involve as many ifelse statements).
 Blackjack with doubling. Modify the previous exercise to allow doubling.
 Projectile motion. The following equation gives the trajectory of a ballistic missile as a function of the initial angle theta and windspeed: xxxx. Write a java program to print the (x, y) position of the missile at each time step t. Use trial and error to determine at what angle you should aim the missile if you hope to incinerate a target located 100 miles due east of your current location and at the same elevation. Assume the windspeed is 20 mph due east.
 World series. The baseball world series is a best of 7 competition, where the first team to win four games wins the World Series. Suppose the stronger team has probability p > 1/2 of winning each game. Write a program to estimate the chance that the weaker teams wins the World Series and to estimate how many games on average it will take.
 Consider the equation (9/4)^x = x^(9/4). One solution is 9/4. Can you find another one using Newton's method?

Sorting networks.
Write a program Sort3.java
with three if statements (and no loops)
that reads in three integers a, b, and c
from the command line and prints them out in ascending order.
if (a > b) swap a and b if (a > c) swap a and c if (b > c) swap b and c

Oblivious sorting network.
Convince yourself that the following code fragment rearranges the
integers stored in the variables A, B, C, and D so that
A <= B <= C <= D.
if (A > B) { t = A; A = B; B = t; } if (B > C) { t = B; B = C; C = t; } if (A > B) { t = A; A = B; B = t; } if (C > D) { t = C; C = D; D = t; } if (B > C) { t = B; B = C; C = t; } if (A > B) { t = A; A = B; B = t; } if (D > E) { t = D; D = E; E = t; } if (C > D) { t = C; C = D; D = t; } if (B > C) { t = B; B = C; C = t; } if (A > B) { t = A; A = B; B = t; }

Optimal oblivious sorting networks.
Create a program that sorts four integers using only
5 if statements, and one that sorts five integers
using only 9 if statements of the type above?
Oblivious sorting networks are useful for implementing sorting algorithms
in hardware.
How can you check that your program works for all inputs?
Answer: Sort4.java sorts 4 elements using 5 compareexchanges. Sort5.java sorts 5 elements using 9 compareexchanges.
The 01 principle asserts that you can verify the correctness of a (deterministic) sorting algorithm by checking whether it correctly sorts an input that is a sequence of 0s and 1s. Thus, to check that Sort5.java works, you only need to test it on the 2^5 = 32 possible inputs of 0s and 1s.

Optimal oblivious sorting (challenging).
Find an optimal sorting network for 6, 7, and 8 inputs, using
12, 16, and 19 if statements of the form in the previous problem,
respectively.
Answer: Sort6.java is the solution for sorting 6 elements.
 Optimal nonoblivious sorting. Write a program that sorts 5 inputs using only 7 comparisons. Hint: First compare the first two numbers, the second two numbers, and the larger of the two groups, and label them so that a < b < d and c < d. Second, insert the remaining element e into its proper place in the chain a < b < d by first comparing against b, then either a or d depending on the outcome. Third, insert c into the proper place in the chain involving a, b, d, and e in the same manner that you inserted e (with the knowledge that c < d). This uses 3 (first step) + 2 (second step) + 2 (third step) = 7 comparisons. This method was first discovered by H. B. Demuth in 1956.
 Weather balloon. (Etter and Ingber, p. 123) Suppose that h(t) = 0.12t^{4} + 12t^{3}  380t^{2} + 4100t + 220 represents the height of a weather balloon at time t (measured in hours) for the first 48 hours after its launch. Create a table of the height at time t for t = 0 to 48. What is its maximum height? Answer: t= 5.

Will the following code fragment compile? If so, what will it do?
int a = 10, b = 18; if (a = b) System.out.println("equal"); else System.out.println("not equal");
Answer: It uses the assignment operator = instead of the equality operator == in the conditional. In Java, the result of this statement is an integer, but the compiler expects a boolean. As a result, the program will not compile. In some languages (notably C and C++), this code fragment will set the variable a to 18 and print equal without an error.
 Gotcha 1.
What does the following code fragment do?
boolean a = false; if (a = true) System.out.println("yes"); else System.out.println("no");
 Gotcha 2.
What does the following code fragment do?
int a = 17, x = 5, y = 12; if (x > y); { a = 13; x = 23; } System.out.println(a);

What does the following code fragment do?
int income = Integer.parseInt(args[0]); if (income >= 311950) rate = .35; if (income >= 174700) rate = .33; if (income >= 114650) rate = .28; if (income >= 47450) rate = .25; if (income >= 0) rate = .22; System.out.println(rate);
 Application of Newton's method. Write a program BohrRadius.java that finds the radii where the probability of finding the electron in the 4s excited state of hydrogen is zero. The probability is given by: (1  3r/4 + r^{2}/8  r^{3}/192)^{2} e^{r/2}, where r is the radius in units of the Bohr radius (0.529173E8 cm). Use Newton's method. By starting Newton's method at different values of r, you can discover all three roots. Hint: use initial values of r= 0, 5, and 13. Challenge: explain what happens if you use an initial value of r = 4 or 12.
 Pepys problem. In 1693, Samuel Pepys asked Isaac Newton which was more likely: getting at least one 1 when rolling a fair die 6 times or getting at least two 1's when rolling a fair die 12 times. Write a program Pepys.java that uses simulation to determine the correct answer.

What is the value of the variable s after
running the following loop when N = 1, 2, 3, 4, and 5.
String s = ""; for (int i = 1; i <= N; i++) { if (i % 2 == 0) s = s + i + s; else s = i + s + i; }
Solution: Pal.java.
 Body mass index.
The body mass index (BMI)
is the ratio of the weight of a person (in kilograms) to the square of the height (in meters).
Write a program BMI.java that takes two commandline arguments, weight
and height, computes the BMI, and prints out the corresponding BMI category:
 Starvation: less than 15
 Anorexic: less than 17.5
 Underweight: less than 18.5
 Ideal: greater than or equal to 18.5 but less than 25
 Overweight: greater than or equal to 25 but less than 30
 Obese: greater than or equal to 30 but less than 40
 Morbidly Obese: greater than or equal to 40
 Reynolds number. The Reynolds number is the ratio if inertial forces to viscous forces and is an important quantity in fluid dynamics. Write a program that takes in 4 commandline arguments, the diameter d, the velocity v, the density rho, and the viscosity mu, and prints out the Reynold's number d * v * rho / mu (assuming all arguments are in SI units). If the Reynold's number is less than 2000, print laminar flow, if it's between 2000 and 4000, print transient flow, and if it's more than 4000, print turbulent flow.
 Wind chill revisited. The wind chill formula from Exercise 1.2.14 is only valid if the wind speed is above 3MPH and below 110MPH and the temperature is below 50 degrees Fahrenheit and above 50 degrees. Modify your solution to print out an error message if the user types in a value outside the allowable range.
 Point on a sphere. Write a program to print the (x, y, z) coordinates of a random point on the surface of a sphere. Use Marsaglia' method: pick a random point (a, b) in the unit circle as in the dowhile example. Then, set x = 2a sqrt(1  a^2  b^2), y = 2b sqrt(1  a^2  b^2), z = 1  2(a^2 + b^2).
 Powers of k. Write a program PowersOfK.java that takes an integer K as command line argument and prints all the positive powers of K in the Java long data type. Note: the constant Long.MAX_VALUE is the value of the largest integer in long.
 Square root, revisited.
Why not use the loopcontinuation condition (Math.abs(t*t  c) > EPSILON)
in Sqrt.java instead of
Math.abs(t  c/t) > t*EPSILON)?
Answer: surprisingly, it can lead to inaccurate results or worse. For example, if you supply SqrtBug.java with the commandline argument 1e50, you get 1e50 as the answer (instead of 1e25); if you supply 16664444, you get an infinite loop!