What kinds of patterns can be used as wallpaper? What are their groups of symmetries, and how can we classify them? How many are there? We will attempt to answer some of these questions and learn how to use Thurston's "orbifold notation" for wallpaper patterns.

Did you ever want to communicate with your friends in a secret way that only those who knew the secret could understand? A cipher is a tool for writing and reading secret messages. We will study three ciphers this time: the reverse cipher, Polybius cipher, and Casear cipher. You will find a template for making a Caesar cipher disk at the link below.

Did you ever want to communicate with your friends in a secret way that only those who knew the secret could understand? A cipher is a tool for writing and reading secret messages. We will study three ciphers this time: the reverse cipher, Polybius cipher, and Casear cipher. You will find a template for making a Caesar cipher disk at the link below.

To start out the quarter, we will be developing logic skills by looking at problems with hats and doors. The goal of this handout is to learn how to make an assumption, test the assumption, and readjust the original assumption if necessary.

To start out the quarter, we will be developing logic skills by looking at problems with hats and doors. The goal of this handout is to learn how to make an assumption, test the assumption, and readjust the original assumption if necessary.

Today we will be looking at how to optimally divide cakes among any amount of people by using an ancient Egyptian technique of breaking fractions into their unit parts but doing so in a specific way, which we will work with and then prove to be optimal.

Today we will be looking at how to optimally divide cakes among any amount of people by using an ancient Egyptian technique of breaking fractions into their unit parts but doing so in a specific way, which we will work with and then prove to be optimal.

We will discuss a few properties concerning representations of nonnegative integers and rationals in a general base (such as divisibility criteria), and particularly in base 10. Then we will apply this knowledge to several Olympiad-type Number Theory problems.

We will continue to answer some of the questions posed about wallpaper symmetries. For those that finish the worksheet on wallpaper symmetries, we started discussing the origins and early properties of the p-adic numbers.

An anagram is a word or phrase formed by rearranging letters of another word or phrase. Most often, all the original letters are used once. For example, the words "silent" and "listen" form an anagram. This time, we will have anagrams for a warm-up and then study two ciphers: pigpen cipher and rail fence cipher.

An anagram is a word or phrase formed by rearranging letters of another word or phrase. Most often, all the original letters are used once. For example, the words "silent" and "listen" form an anagram. This time, we will have anagrams for a warm-up and then study two ciphers: pigpen cipher and rail fence cipher.

We will discuss board-tiling problems-- is it possible to completely cover a region with a particular set of tiles without overlap? -- and interesting mathematical problems that arise from these puzzles.

We will discuss board-tiling problems-- is it possible to completely cover a region with a particular set of tiles without overlap? -- and interesting mathematical problems that arise from these puzzles.

We will solve a variety of geometry problems involving the computation of a length or an area, or using notions about areas to prove an identity. The problems range in difficulty from introductory to fairly challenging.

We will study the symmetries of frieze patterns, especially those of unimodular frieze patterns. We work toward an interesting result proved by Conway and Coxeter about polygonal structures in frieze patterns.

We will study the symmetries of frieze patterns, especially those of unimodular frieze patterns. We work toward an interesting result proved by Conway and Coxeter about polygonal structures in frieze patterns.

We will finish the study of the rail fence cipher and take a quiz on ciphers. Remember, we do not quiz students, we quiz teachers. If you get a low grade, it means we didn't do a good job. Please study for the quiz - don't let us down!

We will finish the study of the rail fence cipher and take a quiz on ciphers. Remember, we do not quiz students, we quiz teachers. If you get a low grade, it means we didn't do a good job. Please study for the quiz - don't let us down!

We will continue with different ways to multiplying 2 numbers. This week we will look at Russian Peasant Multiplication, which, surprisingly, has no relation to Russia or Peasants. However, this will be another good way to show the students how to write numbers as sum of powers of 2

We will continue with different ways to multiplying 2 numbers. This week we will look at Russian Peasant Multiplication, which, surprisingly, has no relation to Russia or Peasants. However, this will be another good way to show the students how to write numbers as sum of powers of 2

This week we begin on a very very very important topic in mathematics; Modular Arithmetic. Although the importance of this topic is not fully realized until a course in Abstract Algebra, it is still a very important topic because it can instill an understanding that the mathematics or arithmetic that we see commonly is not the only way of operating and that mathematics can be far more general and far reaching.

This week we begin on a very very very important topic in mathematics; Modular Arithmetic. Although the importance of this topic is not fully realized until a course in Abstract Algebra, it is still a very important topic because it can instill an understanding that the mathematics or arithmetic that we see commonly is not the only way of operating and that mathematics can be far more general and far reaching.

Our third week of probability includes a handout on random variables and expected value, as well as some more interesting conditional probability problems.

Our third week of probability includes a handout on random variables and expected value, as well as some more interesting conditional probability problems.

We continue our study of the symmetries of unimodular frieze patterns with some challenge problems. We introduce the result proved by Conway and Coxeter connecting polygonal triangulations to frieze patterns.

We continue our study of the symmetries of unimodular frieze patterns with some challenge problems. We introduce the result proved by Conway and Coxeter connecting polygonal triangulations to frieze patterns.

A continuation of next week. Please use this week's sheet as a a set of problems that build stronger arithmetic skills and provide insight into other ways of "counting."

A continuation of next week. Please use this week's sheet as a a set of problems that build stronger arithmetic skills and provide insight into other ways of "counting."

We will be working through a fun application of probability and graph theory to electrical circuits. No knowledge of physics is necessary, everything needed is contained in the handout.

We will be working through a fun application of probability and graph theory to electrical circuits. No knowledge of physics is necessary, everything needed is contained in the handout.

This week we are working with some logic problems in the guise of "real world" sentences. The hope is that students can see that a basic logical structure is embedded in common speech, and that logical structure is the basis for mathematics and is taken to its most pure form and to its greatest lengths. Please use these worksheets as a warm up for the logical skills needed for induction next week.

This week we are working with some logic problems in the guise of "real world" sentences. The hope is that students can see that a basic logical structure is embedded in common speech, and that logical structure is the basis for mathematics and is taken to its most pure form and to its greatest lengths. Please use these worksheets as a warm up for the logical skills needed for induction next week.

We dive further into the connection between random walks and electrical circuits, deriving series and parallel circuits. We then discuss electrical circuits on an integer lattice and its connection to Polya's problem.

We dive further into the connection between random walks and electrical circuits, deriving series and parallel circuits. We then discuss electrical circuits on an integer lattice and its connection to Polya's problem.

This week we are starting induction! If these problems are challenging (they most likely are) do not panic. These are meant to be attempted to the best of everyone's ability and asking questions is extremely encouraged.

This week we are starting induction! If these problems are challenging (they most likely are) do not panic. These are meant to be attempted to the best of everyone's ability and asking questions is extremely encouraged.

We start a new unit: Complex Numbers! We will see that while complex numbers might seem unnatural, they are actually extremely useful and can simplify problems that have seemingly nothing to do with the square root of -1.

We start a new unit: Complex Numbers! We will see that while complex numbers might seem unnatural, they are actually extremely useful and can simplify problems that have seemingly nothing to do with the square root of -1.

This week we are continuing on induction proofs. It is fully expected that these problems are challenging, but they are extremely important to attempt, not necessarily get correct. The point of these two weeks is to 1. prepare students for induction proofs (the basis of discrete math and many proofs about finite objects) and 2. to challenge them to see the power of mathematical logic to prove an infinite amount of statements in a finite time.

This week we are continuing on induction proofs. It is fully expected that these problems are challenging, but they are extremely important to attempt, not necessarily get correct. The point of these two weeks is to 1. prepare students for induction proofs (the basis of discrete math and many proofs about finite objects) and 2. to challenge them to see the power of mathematical logic to prove an infinite amount of statements in a finite time.

The Euclidean Algorithm is a way to find the greatest common divisor of two numbers. Using what we've learned in algebra in the past weeks, we will investigate how this method works.

The Euclidean Algorithm is a way to find the greatest common divisor of two numbers. Using what we've learned in algebra in the past weeks, we will investigate how this method works.

We define and discuss basic concepts from trigonometry, including the law of sines and the law of cosines. We then apply these notions to solving several Olympiad-type geometry problems.

This week we will be looking at what it means for shapes to be similar, as well as exploring how we can add on to a shape to create another that is similar to the original.

This week we will be looking at what it means for shapes to be similar, as well as exploring how we can add on to a shape to create another that is similar to the original.

This week we are working on finding the perimeter of some interesting shapes. These problems are designed to prepare students to quickly recognize the shapes they are looking at and how they can break them into easily solvable pieces.

This week we are working on finding the perimeter of some interesting shapes. These problems are designed to prepare students to quickly recognize the shapes they are looking at and how they can break them into easily solvable pieces.

We start off the new (calendar) year with something you probably haven't seen before: tropical geometry. Throw out everything you know about addition and multiplication; we define new operations and explore polynomials under the new rules.

We start off the new (calendar) year with something you probably haven't seen before: tropical polynomials. Throw out everything you know about addition and multiplication; we define new operations and explore polynomials under the new rules.

We motivate the study of metrics by introducing the taxicab metric (the distance traveled by a taxi in a city with a grid layout). How would you define a circle, line segment between points, or a parabola with a new notion of distance?

We introduce a new notion of distance - the taxicab metric. We investigate how geometry behaves with this new distance, and try to find similarities and differences between this and the Euclidean metric.

Students were asked to study the 8th packet, More Solids, during the Winter break. Since the packet is rather challenging for the age, we will through it class.

We started a warm-up discussing auction theory. We will begin our exploration of game theory by starting with the example of subtraction games of varying subtraction sets.

We started a warm-up discussing auction theory. We will begin our exploration of game theory by starting with the example of subtraction games of varying subtraction sets.

This week we will be looking at geometrical numbers and the successive difference found in the sequence of these numbers. Interestingly, these types of numbers have been studied for thousands of years ever sense the Ancient Greeks. These sorts of problems are very useful to build strategies for induction proofs and pattern spotting.

This week we will be looking at geometrical numbers and the successive difference found in the sequence of these numbers. Interestingly, these types of numbers have been studied for thousands of years ever sense the Ancient Greeks. These sorts of problems are very useful to build strategies for induction proofs and pattern spotting.

We continue our study of metrics, this time considering more abstract examples. We discuss sequences, and we learn that with regards to sequences, the taxicab metric and the Euclidean metric are equivalent.

This week we started off class discussing "I cut you choose." We then continued will be continuing our discussion of subtraction games, in particular the Game 21. With 21 sticks, each player can either take away 1,2, or 3 sticks each turn, what is the winning strategy if you don't want to take the last stick? If you want to take the last one? How are these two related?

This week we started off class discussing "I cut you choose." We then continued will be continuing our discussion of subtraction games, in particular the Game 21. With 21 sticks, each player can either take away 1,2, or 3 sticks each turn, what is the winning strategy if you don't want to take the last stick? If you want to take the last one? How are these two related?

This week we will be working on some very interesting logical problems. These sorts of problems, although not generalization to higher mathematical theory, are very useful in practicing mathematical logic and intuition.

This week we will be working on some very interesting logical problems. These sorts of problems, although not generalization to higher mathematical theory, are very useful in practicing mathematical logic and intuition.

First, we will go through a few problems from the 9th packet. Then students will take a quiz on polygons and solids. Then we will finish discussing the packet.

First, we will go through a few problems from the 9th packet. Then students will take a quiz on polygons and solids. Then we will finish discussing the packet.

In the second part of the sequence, we explore more bijections and prove several surprising facts about the cardinalities about the naturals, integers, and the reals.

Intro: How many days are in a year? How are leap years counted? Why do we include every 400 years, but not 100, 200, or 300? Hint: a year is technically about 365.25 days - 11 minutes.

In this handout we examine how to systematically perform calculations to find the day of the week (Sunday, Monday, Tuesday, etc.) a particular date is, e.g., your birthday

Intro: How many days are in a year? How are leap years counted? Why do we include every 400 years, but not 100, 200, or 300? Hint: a year is technically about 365.25 days - 11 minutes.

In this handout we examine how to systematically perform calculations to find the day of the week (Sunday, Monday, Tuesday, etc.) a particular date is, e.g., your birthday

This week we will be playing a game with the Math Kangaroo Practice tests. The Math Kangaroo is a math test for young mathematicians and is usually held in the middle of the year.

The game we are playing this week will encourage team work, communication, and dieligent justification of answers.

This week we will be playing a game with the Math Kangaroo Practice tests. The Math Kangaroo is a math test for young mathematicians and is usually held in the middle of the year.

The game we are playing this week will encourage team work, communication, and dieligent justification of answers.

This week we start on the topic of Graph Theory! This topic considers a particular object of mathematics, the Graph, which is defined by its vertices and the edges connecting them. We will be considering the degree of the vertices as well a particular type of Graph, the Bipartite Graph.

We will introduce the generating function, a creative, combinatorial tool that can simply solve many interesting problems. By the end, we will have used generating functions to study the Fibonacci sequence, dice games, and ways to pay the unlucky cashier with coins.

We will introduce the generating function, a creative, combinatorial tool that can simply solve many interesting problems. By the end, we will have used generating functions to study the Fibonacci sequence, dice games, and ways to pay the unlucky cashier with coins.

This week we start on the topic of Graph Theory! This topic considers a particular object of mathematics, the Graph, which is defined by its vertices and the edges connecting them. We will be considering the degree of the vertices as well a particular type of Graph, the Bipartite Graph.

This week we completed the rest of the Graph Theory packet, going over topics such as the coloring of a graph, which is the least number of colors required to fully "color" a graph. An interesting theory associated with this is that if one considers ANY map that separates the area into counties, states, cities, etc. then one can color that map with four colors such that no two adjacent counties, states, cities, etc. have the same color. This theorem was proved using computers!

This week we will be solving the instant insanity puzzle. This puzzle is a mathematicians favorite as it is difficult to solve by “brute force,” but permits a very simple and elegant mathematical solution. We will begin by trying brute force tactics and when that fails us we will find a way to use graph theory to solve the puzzle very simply.

here is the amazon link to the puzzle: Winning Moves Games Instant Insanity https://www.amazon.com/dp/B004KCN6EQ/ref=cm_sw_r_cp_api_i_QwSuEb6C13BXG

This week we will be solving the instant insanity puzzle. This puzzle is a mathematicians favorite as it is difficult to solve by “brute force,” but permits a very simple and elegant mathematical solution. We will begin by trying brute force tactics and when that fails us we will find a way to use graph theory to solve the puzzle very simply.

here is the amazon link to the puzzle: Winning Moves Games Instant Insanity https://www.amazon.com/dp/B004KCN6EQ/ref=cm_sw_r_cp_api_i_QwSuEb6C13BXG

Warm-Up: If I give you two numbers, like 998 and 992, and I ask you to multiply them together using conventional math techniques, you end up writing a lot of numbers to generate the answer. But notice that 998 is just 2 shy of 1000, and 992 is just 8 shy of 1000. If you multiply 2 times 8, you get 16. And if you take 8 away from 998, or you take 2 away from 992, you get 990. And guess what? The correct answer is 990016. Similarly, if I ask you to multiply 990 times 991, you could work it out ... or you could recognize that 990 is 10 below 1000, 991 is 9 below, the product of 9 and 10 is 90, and 990 minus 9 is 981, and 991 minus 10 is also 981. The answer: 981090.

The insight: if you rewrite 998 as (1000-2) and 992 as (1000-8), multiply the two we get 1000*1000 - 2*1000 - 8*1000 + 8*2. Hence we get (1000-2-8)*1000 + 16 = 990,016.

We will be continuing our topic on Polyhedra from last week!

Warm-Up: If I give you two numbers, like 998 and 992, and I ask you to multiply them together using conventional math techniques, you end up writing a lot of numbers to generate the answer. But notice that 998 is just 2 shy of 1000, and 992 is just 8 shy of 1000. If you multiply 2 times 8, you get 16. And if you take 8 away from 998, or you take 2 away from 992, you get 990. And guess what? The correct answer is 990016. Similarly, if I ask you to multiply 990 times 991, you could work it out ... or you could recognize that 990 is 10 below 1000, 991 is 9 below, the product of 9 and 10 is 90, and 990 minus 9 is 981, and 991 minus 10 is also 981. The answer: 981090.

The insight: if you rewrite 998 as (1000-2) and 992 as (1000-8), multiply the two we get 1000*1000 - 2*1000 - 8*1000 + 8*2. Hence we get (1000-2-8)*1000 + 16 = 990,016.

We will be continuing our topic on Polyhedra from last week!

This week we will be working with probability! We will be learning how to calculate the expected value of a given game. This sort of "average" thinking is used all the time by everyone unconsciously and explicitly.

This week we will be working with probability! We will be learning how to calculate the expected value of a given game. This sort of "average" thinking is used all the time by everyone unconsciously and explicitly.

We will continue our study of random variables by introducing the Bernoulli, geometric, and binomial distributions. As a final surprising example, we construct a famous example of a non-measurable set.

For our final (virtual) meeting of the quarter, we will be hosting a Kahoot competition over the Zoom application. Please make sure to download the app at https://zoom.us/download and join the meeting from 4-6 pm at the link sent via email. See you there!

This week we solved a mathematical trick using equivalence relations! Equivalence relations are extremely important in mathematics and allow mathematicians to formally categorize "objects" in a particular "Space."

This week we solved a mathematical trick using equivalence relations! Equivalence relations are extremely important in mathematics and allow mathematicians to formally categorize "objects" in a particular "Space."

This week we worked on some intuitive mathematics! By that I mean mathematics that can be reliably answered by the physical intuition we have all built up over our lives. This "weighty" way of thinking that we are all seemingly borne with was first formalized mathematically in Ancient Greece and this week we get to explore some of its implications.

Sperner's Lemma is an interesting result about the coloring of special graphs. We connect the result with a famous result, the Brouwer fixed-point theorem.

We will be continuing the idea of rates from last week, but this time applying to trade! What is the best way to optimize the outcome of both parties in trade? When is trade not a win-win situation?

We will be continuing the idea of rates from last week, but this time applying to trade! What is the best way to optimize the outcome of both parties in trade? When is trade not a win-win situation?

This week we will be taking on some challenging problems from the Russian math Olympiad! These are questions designed to test some of the smartest young mathematicians. I hope everyone is challenged and excited!

This week we will be taking on some challenging problems from the Russian math Olympiad! These are questions designed to test some of the smartest young mathematicians. I hope everyone is challenged and excited!

This week, we will continue our discussion on trade, learning about opportunity costs as a way of determining when trade is a win-win situation and when it is not, as well as finding a rate that both traders will be happy with

This week, we will continue our discussion on trade, learning about opportunity costs as a way of determining when trade is a win-win situation and when it is not, as well as finding a rate that both traders will be happy with

We introduce random variables, expectations, variance, independence and functions of random variables, then apply these notions to Olympiad-type problems.

We will help a mouse and two ants to travel through and around a cube. The journeys reveal a wealth of ideas and serve as an intro to studying nets of cubes.

We will help a mouse and two ants to travel through and around a cube. The journeys reveal a wealth of ideas and serve as an intro to studying nets of cubes.

We'll start off this summer with a lecture on basic test-taking techniques. Students will take the 2018 AMC 8 as a diagnostic test. You can access the problems and solutions from that exam on the AoPS website.

After week one's diagnostic test, we start the official AMC 8 prep portion of the course with an introductory lesson on combinatorics. See the attached documents below for resources used in class that can be helpful for future study.

This week, we'll be using the combinatorial skills from week two to solve more advanced problems in statistics and probability. All handouts and resources should be posted below.

This week, we'll once again be building on our previous knowledge by learning about number theory, the mathematical study of integers. All associated resources used in class should be posted below.