The Divisor Functions. The Prime Divisor Functions. Certified Signatures. Primitive Roots. Knapsack Encryption. Quadratic Residues. Fast Transformations and Kronecker Products. Quadratic Congruences. Pseudoprimes, Poker and Remote Coin Tossing. Generating Functions and Partitions. Cyclotomic Polynomials. Linear Systems and Polynomials.
Polynomial Theory. Galois Fields. Spectral Properties of Galois Sequences. Random Number Generators. Waveforms and Radiation Patterns. Back Matter Pages About this book Introduction "Beauty is the first test: there is no permanent place in the world for ugly mathematics. 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.
Your program should take an integer command-line argument 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. Euler's sum-of-powers conjecture. Write a program Euler. Use the long data type. Web Exercises Write a program RollDie. Write a program that takes three integer command-line arguments a, b, and c and print the number of distinct values 1, 2, or 3 among a, b, and c.
Write a program that takes five integer command-line arguments and prints the median the third largest one. How can I create in an infinite loop with a for loop? Solution : for ;; is the same as while true. What's wrong with the following loop? By defining it inside the loop, a new variable sum is initialized to 0 each time through the loop; also it is not even accessible outside the loop. Write a program Hurricane. Below is a table of the wind speeds according to the Saffir-Simpson scale.
Category Wind Speed mph 1 74 - 95 2 96 - 3 - 4 - 5 and above What is wrong with the following code fragment? A better solution is to write if isPositive. There are two different solutions. 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.
What does the following program do? 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. But the most likely value is 2 for all values of p. Write a program using a loop and four conditionals to print 12 midnight 1am 2am 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? Rewrite DayOfWeek. Use a switch statement. Write a program to read in a command line integer between ,, and ,, and print 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. Gymnastics judging. A gymnast's score is determined by a panel of 6 judges who each decide a score between 0. The final score is determined by discarding the high and low scores, and averaging the remaining 4. Write a program GymnasticsScorer.
Quarterback rating. Write a program QuarterbackRating. Use your program to compute Steve Young's record-setting season As of , the all-time single-season record is Decimal expansion of rational numbers. We use the notation 0. Write a program DecimalExpansion. 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 years days so you only need to worry about a year interval.
Solution : e. 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 year interval. Solution: Sunday 58 times is more likely than Saturday 56 times.
What do the following two code fragments do? Choose the correct answer from 0, , , , or Solution : Always print true. Modify Sqrt. What happens if we initialize t to -x instead of x in program Sqrt. Sample standard deviation of uniform distribution. Modify Exercise 8 so that it prints the sample standard deviation in addition to the average. Sample standard deviation of normal distribution. Loaded dice. Two players roll a die and the one with the highest value wins. Which die would you choose?
Be sure to choose second! Thue—Morse sequence. Write a program ThueMorse. The first few strings are 0, 01, , 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 cube-free : it does not contain , , , or sss where s is any string. It is self-similar : if you delete every other bit, you get another Thue—Morse sequence.
It arises in diverse areas of mathematics as well as chess, graphic design, weaving patterns , and music composition. Program Binary. Write an alternate version Program Binary2. Divide n by 2, throwing away the remainder. What does the following code fragment do? Write a program NPerLine. Modify NPerLine. Make the integers line up by printing the right number of spaces before an integer e. 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? Formatted ISBN number. Write a program ISBN2. UPC codes. Write a program that reads in a 11 digit integer from a command line parameter, computes the check digit, and prints 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 its force according to the Beaufort scale. Making change. Write a program that reads in a command line integer N number of pennies and prints the best way fewest number of coins to make change using US coins quarters, dimes, nickels, and pennies only.
That is, dispense as many quarters as possible, then dimes, then nickels, and finally pennies. Write a program Triangle. Use two for loops and one if-else statement. Write a program Heart. What does the program Circle.
Write a program Season. Write a program Zodiac. Write a program that reads in the weight of a Muay Thai kickboxer in pounds as a command-line argument and prints their weight class. In 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 by exhaustive computer search. 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.
Write a program Blackjack. 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 if-else 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 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.
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. Can you find another one using Newton's method? Sorting networks. Write a program Sort3. How many if statements does your program use? 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? Solution : Sort4. The 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.
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. Solution : Sort6. Optimal non-oblivious 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 Weather balloon. Etter and Ingber, p. What is its maximum height? Will the following code fragment compile?
If so, what will it do? In Java, the result of this statement is an integer, but the compiler expects a boolean. As a result, the program will not compile.
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