Practice

Discrete Mathematics Warmups

If there are only a handful of objects, then you can count them with a moment's thought, but the techniques of combinatorics can extend to quickly and efficiently tabulating astronomical quantities.

Rule of Sum and Rule of Product

Seek simple and succinct solutions in these systems by sussing-out the significant symmetries.

Complements

You are in the complement of the set of people who are not Brilliant.

Permutations

As head of Brilliant's cybersecurity, you need to know how many different passwords can be created by rearranging the digits 1, 2, 3, 4, 5, and 6. Can you count them all?

Combinations

If a coin is flipped twenty times, how many ways are there to get exactly ten tails and ten heads?

Grid Walking

How many 5 block paths can take you home if walking 2 blocks north then 3 blocks west got you here? This should feel familiar if you're from NYC, Chicago, Barcelona, or Toronto!

Set Notation

When Cantor introduced his classification of multiple infinities, he was vehemently rejected by most mathematicians. Ye be warned: contemplating the continuum hypothesis can drive anyone a little mad!

Set Operations

Familiarize yourself with the foundational notation, tools and concepts for the operations that are applied to sets.

Principle of Inclusion and Exclusion

Count to 100. How many of those numbers are odd or multiples of 5? Since 50 are odd and 20 are multiples of 5, at first glance the answer is 70...

Bijections

Bijections, surjections, and injections are three types of functions which associate the elements between two sets. For example, each word in this sentence can be mapped to exactly one in the last.

Pigeonhole Principle

If you have 12 pigeons and there are only 11 roosts, then at least one roost will be quite cozy.

Distribution into Bins

How can five candies be distributed among 3 friends? Well, the answer depends on if you can tell the candies apart!

Understanding Data

Whether you're trying to analyze a political poll, the results of a scientific study, or just your sleeping habits, data is the best tool for understanding the world around you. Dive in and see how!

Data Presentation

Data is only as good as how it is presented. How do you take hundreds or thousands of data points and create something a human can understand? Check out the world of charts, graphs, and more.

Discrete Probability

How often will a die come up "4"? How likely is it to rain tomorrow? Probability is one of the most powerful frameworks for modeling the world around us.

Conditional Probability

Conditional probability is the art of updating probabilities based on given information. What is the probability that the sidewalk is wet? And what if we know that it rained a few hours ago?

Expected Value

If you play roulette, are you going to break-even? Maybe sometimes, but not if you play forever, because the expected value is negative. The expected value finds the average outcome of random events.

Variance

Would you rather get paid $2 for flipping heads, or $6 for rolling a "1"? The expected value is the same ($1)...but the bets are different! Variance and standard deviation add color to probability.

Geometric Probability

Sometimes, probability questions can be interpreted geometrically, from simple examples like throwing darts to surprising applications like catching the bus!

Discrete Random Variables

How many heartbeats do you have each minute? How many points will your favorite team score in their game tonight? These any many other real-world values can be modeled by discrete random variables.

Continuous Random Variables

When will that bus finally arrive? How hot is it going to be outside today? These any many other real-world values can be modeled by continuous random variables.

Discrete Probability Distributions

What is the probability that the sum of two dice will be less than 5? How many throws will it take you to hit your first "bullseye" on a dart board? Use discrete probability distributions to find out!

Continuous Probability Distributions

How much variation should you have in your blood pressure? How likely is that stock price to double by the end of the year? Use continuous probability distributions to find out!

Statistical Testing

Markov Chains

From weather conditions to baseball scores to stock performances, many probabilistic real-world systems can be modeled with Markov chains.

Binomial Theorem

If a coin comes up heads you win $10, but if it comes up tails you win $0. How likely are you to win exactly $30 in five flips?

Multinomial Theorem

It's like the Binomial Theorem, but better!

Introduction to Recursion

The Fibonacci sequence is not the only one that is defined recursively. Learn how to solve combinatorics problems with recursion, and how to turn recurrence relations into closed-form expressions.

Fibonacci Numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... They show up in nature, they show up in math, and they've got some beautiful properties.

Linear Recurrence Relations

Take your recursion skills to the next level. If you've got a recurrence relation but no computer, how can you find a closed form? What about asymptotic behavior? How fast do rabbits reproduce?

Pigeonhole Principle

If you have 12 pigeons and there are only 11 roosts, then at least one roost will be quite cozy.

Combinatorial Games

Whether you’re a master of games or just playing around, learn how combinatorial ideas can be used to analyze and solve games such as Nim.

Problem Solving Tactics

Learn about advanced problem-solving tactics such as Construction, the Extremal Principle, and the Invariant Principle, and you'll be crushing tricky problems in no time.

Graph Theory

Any connected group of things can be represented as a graph: cities and roads, people and friendships, and more. Learn why an even number of people have an odd number of friends.

Generating Functions

This combinatorial tool is used for solving recurrence relations and other difficult problems.

Coloring

If coloring sounds pretty fun, it's because it is! See how coloring can turn a complex combinatorial problem into a one-line solution.