A basic graph of 3-Cycle
Graphs are useful for representing all kinds of real-world problems.
If there is a greedy algorithm that will traverse a graph, selecting the largest node value at each point until it reaches a leaf of the graph, what path will the greedy algorithm follow in the graph below?
John lives in the Trees of Ten Houses, and it is a most ideal and idyllic place for him and the other dwellers up in the canopy. They have invested a tremendous amount of time in engineering these houses, and to ensure no house felt isolated from the others, they built a fresh, finely crafted bridge between each and every house!
Unfortunately, the Trees of Ten Houses were not immune to thunderstorms, nor were the bridges well engineered. The night was treacherous, howling with wind and freezing with rain, so the odds for the bridges were not good--each bridge seemed just as likely to survive as to be shattered!
Fortunately, as there were so very many bridges in the Trees of Ten Houses, when John did wake the following morning, he found he was able to make his way to each and every house using only the existing bridges, though round-about routes may have been necessary. As they began rebuilding, John became curious... what were the chances that they'd all be so lucky?
More formally, if \(P\) is the probability that, after the storm, John is able to traverse to each and every house, what is \(\big\lfloor 10^{10} P \big\rfloor?\)
Details and Assumptions:
Image credit: http://hdscreen.me/wallpaper/2645876-bridges-fantasy-art-landscapes-mountains.
Main Article: Probability
A probability is a number, between 0 and 1 inclusive, that represents the likelihood of an event. Discrete probability is a probability based on discrete sets of outcomes. The most basic type of probability is a uniform probability . If each outcome in a set is equally likely, then the probability of an event is equal to a ratio of cardinalities.
Let \(S\) be a sample space of outcomes. If each outcome in this set is equally likely, then the probability of an event \(A\) in \(S\) is \[P(A)=\frac{|A|}{|S|}.\]
What is the probability of getting 53 Sundays in a leap year?
Many of the rules of probability are analogous to the rules of combinatorics. The probabilistic rules of product , sum , and complement work similarly to those same rules from combinatorics. In addition, the structure of the probabilistic principle of inclusion and exclusion is the same as PIE for sets.
A biased coin is tossed repeatedly. Assume that the outcomes of different tosses are independent and the probability of heads is \(\frac{2}{3}\) for each toss. What is the probability of obtaining an even number of heads in 5 tosses?
A discrete probability distribution is a function that takes a numerical outcome as an argument and gives a probability as a result. Discrete probability distributions can be created using the rules and guidelines described above. There are also some discrete probability distributions that show up in many problems:
Although basic probabilities are based on discrete sets, the concept of probability can be extended to continuous sets by using concepts from calculus .
A point is marked at random on a unit line segment.
Find the expected value of the sum of the squares of the lengths of the two parts.
Main Article: Statistics
A statistic is a number used to describe a set of data or a probability distribution. Statistics is widely used in many fields outside of mathematics, from biology to politics to sports. The power of statistics lies in taking a massive, varied set of data and making sense out of it. Furthermore, statistics has the power to quantify confidence in those findings. Of course, the usefulness of statistics is not without controversy, but an understanding of its theoretical underpinnings can help one avoid its misuse.
One major kind of statistic is a measure of central tendency. A measure of central tendency is a number which describes what a value of a probability distribution or data set will tend to. An expected value is the theoretical long-run average outcome of a probability experiment when it is performed many times.
A game costs $150 to play. In this game, you roll a fair six-sided die repeatedly until each of all the six numbers has been rolled at least once. You are then paid 10 times the number of rolls you made.
For example, if the rolls were 3, 5, 4, 3, 2, 5, 1, 4, 1, 3, 6, then you would get \((10)(11) = 110\) dollars.
Including the price to play, what is your expected value in this game?
Somewhat related to the expected value is the mean. The mean is the average value of a set of numerical data.
What is the mean of the first 100 positive integers?
Another major kind of statistic is a measure of variation. A measure of variation is a number which describes the distribution of a probability distribution or data set. The standard deviation of a probability distribution is a number that represents how much the outcomes differ from the expected value. Likewise, the standard deviation of a data set is a number that represents how much the elements of the set differ from the mean.
The elements of set \(S\) of real numbers with cardinality \(n\) form an arithmetic progression with common difference \(d\). Express the population standard deviation of set \(S\) in terms of \(n\) and \(d\).
An example of set \(S\) is \(\text{{2, 5, 8, 11, 14, 17}}\). It has cardinality 6, and its elements form an AP with common difference 3.
Although discrete statistics are based off of discrete events and probability distributions, these same concepts can be extended to continuous events and probability distributions using concepts from calculus .
Main Article: Bijection, Injection, and Surjection
A bijection is a relationship between two sets such that each element in a set is paired with exactly one element in the other set, and vice versa. Bijections can be applied to problem solving by establishing a bijection between a set that is difficult to enumerate and a discrete stucture that is well understood. By establishing a bijection, one can take advantage of the known formulas and theorems that the discrete structure affords.
The three Molloy siblings, April, Bradley, and Clark, have integer ages that sum to 15. How many possible distribution of ages are there? Note : It is possible that an age can be 0, which means that the child was just born. One can establish a bijection between the set of distributions of ages and a set of combinations. Consider the arrangement of stars and bars below: \[\star \star \mid \star \star \star \star \mid \star \star \star \star \star \star \star \star \star\] This arrangement corresponds to the following distribution of ages: April - 2, Bradley - 4, Clark - 9. Note that there are 15 stars and 2 bars in the arrangement above. This gives a total of 17 objects, 2 of which are bars. Placing the bars in different spots among the 17 placements will give a new distribution of ages. Thus, a bijection can be established between the set of distributions of ages and the set of combinations of 2 objects out of 17. The number of distributions of ages is \[\binom{17}{2}=136.\ _\square\]
A parking lot has 10 empty spaces in a row.
6 cars arrive, each of which fills exactly 1 parking spot, chosen at random from among the available spaces. Robbie then arrives in his pick-up truck, which requires 2 empty adjacent spaces to park.
If the probability that Robbie will be able to park is \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers, then what is \(a+b?\)
Main Articles: Logical Puzzles Propositional Logic Boolean Algebra
A proposition is a statement that can either be true or false. Propositional logic aims to outline the rules of how these statements can be altered and combined.
Which of the following statements are true and which are false, knowing that the entire set is uncontradictory?
S1. Statements 2 and 3 are either both true either both false. S2. Exactly one of the statements 4 and 5 is true. S3. Exactly one of the statements 4 and 6 is true. S4. Exactly one of the statements 1 and 6 is true. S5. Statements 1 and 3 are of the same type (both true or both false). S6. Exactly one statement from statements 2 and 5 is true.
Write the answer as the concatenation of the digits 1 and 0 for the truth values of the statements (true and false) starting from S1 to S6, where for the value true corresponds 1 and for the value false corresponds 0. For example, if the first 2 statements would be true and the rest false, the answer would be 110000.
If the correct answer begins with some number of leading 0s, remove it from writing the answer. For example, if the answer is 001100, write 1100 anyway.
Similarly, Boolean algebra outlines the operations defined on variables that can take the values of true (1) or false (0). Boolean algebra is used to design computer circuits through logic gates , which take signal(s) as inputs and return a signal as an output.
The AND gates in the diagram will output a 1 if both inputs are also 1.
If A = 1, B = 1, and C = 0, what will the final output be?
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Oscar Levin
Voronoi diagrams allow students and teachers to explore a technique that is used in a variety of applications, while at the same time employing critical thinking skills and geometric concepts.
These types of diagrams allow you to map out the areas in a given space that are closest to one specified point or another. For example, if there are 17 ice cream shops of equal quality in your town, a Voronoi diagram can show you which one is the closest for each region of town. This example is shown in the picture below.
This technique is used in biology, chemistry, geology, forestry, and more, as well as in resource planning and placement. This last topic is easily familiar to students as it can include determining placement for a new cell phone tower. or a new pizza place!
These diagrams are constructed using perpendicular bisectors, but students can approach these either strictly through geometric constructions, or through a more algebraic approach for more advanced students.
A discrete quantity can only take on certain values. For example, the number of students in a classroom can only be a positive integer and is thus a discrete quantity since the range of possible values is not continuous . On the other hand, the average height of the students in a classroom can take on any real value and is thus a continuous variable since its possible range of values is continuous.
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Richard rusczyk.
Richard Rusczyk founded Art of Problem Solving (AoPS) in 2003 to create interactive educational opportunities for avid math students. Richard is one of the co-authors of the Art of Problem Solving classic textbooks, author of Art of Problem Solving's Introduction to Algebra, Introduction to Geometry, and Precalculus textbooks, co-author of Art of Problem Solving's Intermediate Algebra and Prealgebra, one of the co-creators of the Mandelbrot Competition, and a past Director of the USA Mathematical Talent Search. He was a participant in National MATHCOUNTS, a three-time participant in the Math Olympiad Summer Program, and a USA Mathematical Olympiad winner (1989). He graduated from Princeton University in 1993, and worked as a bond trader for D.E. Shaw & Company for four years. AoPS marks Richard's return to his vocation: educating motivated students.
David Patrick has been an author and curriculum developer at Art of Problem Solving since 2004. He is the author of Art of Problem Solving's Introduction to Counting & Probability, Intermediate Counting & Probability, and Calculus textbooks, and co-author of AoPS's new Prealgebra textbook. He is also an instructor in AoPS's widely-acclaimed online school for high performing middle and high school students.
Dave earned the sole perfect score on the American High School Mathematics Examination (AHSME) in 1988 and was a USA Mathematical Olympiad winner that year. He attended the Research Science Institute (RSI) in 1987, and the Math Olympiad Summer Program in 1988, where he first met fellow student Richard Rusczyk. He also finished in the top 10 on the Putnam exam in 1991. Dave graduated from Carnegie Mellon in 1992 with a BS in Mathematics/Computer Science and an MS in Mathematics. He went on to earn his Ph.D. in mathematics from MIT in 1997. He was an acting Assistant Professor at the University of Washington from 1997 to 2001. Dave is originally from Western New York and is an alumnus of the SUNY Buffalo Gifted Math Program.
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Discrete mathematics. Discrete math is very broadly the area of mathematics that deals with discrete objects; that is, objects that be enumerated. Another very broad description is that branches of mathematics considered discrete math are primarily concerned with sets, whereas branches of math known as analysis (such as calculus) are primarily ...
Discrete math is the mathematics of computing. The mathematics of modern computer science is built almost entirely on discrete math, in particular combinatorics and graph theory. This means that in order to learn the fundamental algorithms used by computer programmers, students will need a solid background in these subjects.
Art of Problem Solving offers two other multifaceted programs. Beast Academy is our comic-based online math curriculum for students ages 6-13. And AoPS Academy brings our methodology to students grades 2-12 through small, in-person classes at local campuses. Through our three programs, AoPS offers the most comprehensive honors math pathway ...
Since 1993, Art of Problem Solving has helped train the next generation of intellectual leaders. Hundreds of thousands of our students have gone on to attend prestigious universities, win global math competitions, and achieve success in highly competitive careers. ... Explore number theory and discrete mathematics, manipulate dot diagrams, and ...
Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ ... Discrete is an adjective meaning, "not continuous, not the same, dissimilar, ... Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. Beast Academy. AoPS Academy. About.
By David Patrick. An intermediate textbook in counting and probability for students in grades 9-12, containing topics such as inclusion-exclusion, recursion, conditional probability, generating functions, graph theory, and more. Related course: Intermediate Counting and Probability. VIEW DETAILS. I think this course was really fun and challenging.
Discrete mathematics is the study of mathematical structures that are countable or otherwise distinct and separable. Examples of structures that are discrete are combinations, graphs, and logical statements. Discrete structures can be finite or infinite. Discrete mathematics is in contrast to continuous mathematics, which deals with structures which can range in value over the real numbers, or ...
logic, this text emphasizes problem solving and the application of theory as it carefully guides the reader from basic to more complex topics. Discrete Mathematics is an ideal resource for discovering the fundamentals of discrete math. Discrete Mathematics, Second Edition is designed for an
0 Introduction and Preliminaries. 1 Counting. 2 Sequences. 3 Symbolic Logic and Proofs. 4 Graph Theory. 5 Additional Topics.
Discrete Mathematics is a rapidly growing and increasingly used area of mathematics, with many practical and relevant applications. ... It's an excellent tool for improving reasoning and problem-solving skills, and is appropriate for students at all levels and of all abilities. Teachers have found that discrete mathematics offers a way of ...
Problem 3 For every integer n 0, Xn i=0 i2 = n(n+ 1)(2n+ 1) 6: Prove. Problem 4 Prove that every integer greater than 1 is a product of prime numbers. Proof: We will use strong induction. Base Case: 2 is a product of prime numbers, since 2 itself is a prime. Inductive Step: Let n be an arbitrary integer greater than 1.
Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚. Books for Grades 5-12 Online Courses
1. There is one and only one perfect square in the form. where and are prime. Find that perfect square. 2. and are positive integers. If , find . 3.The fraction, where and are side lengths of a triangle, lies in the interval , where and are rational numbers.
Live virtual math and language arts classes for motivated students in elementary and middle school. Art of Problem Solving AoPS Online. Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ...
We also introduce students to the rich field of complex numbers and to important functions and concepts in discrete math. Art of Problem Solving AoPS Online. Math texts, online classes, and more for students in grades 5-12. ... The textbook for the course is Introduction to Algebra by Art of Problem Solving. Hardcopy and lifetime digital access ...
The Art of Problem Solving (AoPS) w ebsite, 1established in 2003, has. grown to over 29 000 members.2We believe that it is the largest website. of its kind in the English-speaking world, with ...
Discrete math is very broadly the area of mathematics that deals with discrete objects; that is, objects that be enumerated. Another very broad description is that branches of mathematics considered discrete math are primarily concerned with sets, whereas branches of math known as analysis (such as calculus) are primarily concerned with functions.
The Art of Problem Solving Math 479 - Dr. Monks Contemplation within activity is a million times better than contemplation within stillness. -Hakuin 1 The Way of Problem Solving • Art: Problem solving is an art. Like any art it requires proper attitude, practice, creativity, and passion to master.
The view of discrete mathematics has evolved from a course in pure mathematics to a preparatory course for programming-related topics. There is an emphasis on explaining concepts using intuitive terms, to look at mathematical statements visually, to reach audiences with varied learning abilities, and to encourage programming at a younger age.
A discrete quantity can only take on certain values. For example, the number of students in a classroom can only be a positive integer and is thus a discrete quantity since the range of possible values is not continuous.On the other hand, the average height of the students in a classroom can take on any real value and is thus a continuous variable since its possible range of values is continuous.
He is co-author of the Art of Problem Solving, Volumes 1 and 2 and Intermediate Algebra, and author of Introduction to Algebra, Introduction to Geometry, and Precalculus. He was a national MATHCOUNTS participant in 1985, a three-time participant in the Math Olympiad Summer Program, and a USA Mathematical Olympiad winner in 1989.
Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ ... Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. Beast Academy. AoPS Academy. About. About AoPS. Our Team. Our History. Jobs. AoPS Blog. Site Info. Terms.
First year calc problems that appear difficult are much easier with tons of practice. The difference between a seasoned mathematician and a novice is the way the problem is approached. With practice comes new and beautifully creative ways of attack, the majority of people who can do this have it worked their peers. 2.