What is Aftermath?
Most of the resources on this page were created to support an after-school mathematics club called Aftermath. The goal of Aftermath is to provide students with an opportunity to explore intriguing mathematics through collaboration and play. Although the Aftermath content is largely extracurricular, much of it can be used to enrich the learning of curriculum expectations. The following topics, problems, and tools are excellent starting points for sparking curiosity, uncovering math’s rich history, and fueling enjoyment in doing mathematics.
Topics and Resources
Click a topic to jump to its section.
Hundred Chart
Exploring patterns and properties of whole numbers? Click the image below for an interactive hundred chart.
COLLATZ CONJECTURE
(3x+1 PROBLEM)
Take any positive integer and apply the following rule:
- If the number is even, divide it by 2.
- If the number is odd, multiply it by 3 and add 1.
Repeatedly apply this rule to each resulting number. Did you eventually end up in 4-2-1 loop? The Collatz Conjecture, named after Lothar Collatz (1910 – 1990), states that this outcome will always occur, but it has yet to be proven! Click the image below to explore this conjecture using a Python program.
A Daily Puzzle
Based on DragonFjord’s A-Puzzle-A-Day, the instructions for this challenge are simple: Place the pieces on the board such that only the current month and date are visible. This puzzle also provides a fun way to talk about transformations. Click the following image to play a digital version.
Prime Numbers
A prime number is a whole number greater than 1 that has exactly two factors (itself and 1). The Fundamental Theorem of Arithmetic states that every whole number greater than 1 can be factored into a product of primes in exactly one way. For this reason, prime numbers are considered to be the building blocks of the whole numbers.
Is It A Prime Number?
The following Python program can be used to check if a given number is prime. Click the image to check it out!
Primes in an Interval
The Python program below can be used to investigate the prime numbers in a given interval. Click the image to explore!
How many Prime Numbers Exist?
Euclid’s proof that there are infinitely many prime numbers is arguably the most famous proof in history! Click the following image for an activity that can assist in exploring Euclid’s method.
Sieve of Eratosthenes
The Sieve of Eratosthenes is a simple and efficient filtering algorithm for finding all prime numbers up to a specified value. Check out the following video and Python program to learn more!
What's Special About 7 & 8?
The relationship between 7 and 8 is truly unique. Check out the following video to see why!
What's Special About 69?
The number 69 has a property that no other number possesses. Watch the following video to learn all about it!
Goldbach Conjecture
Named after Christian Goldbach (1690 – 1764), the Goldbach Conjecture states that every even integer greater than 2 be expressed as a sum of two prime numbers. For example,
28 = 11 + 17 and 486 = 229 + 257.
The Goldbach Conjecture has yet to be proven and remains one of the greatest unsolved problems in mathematics!
Explore it with Coding
The following Python program takes an even number entered by the user and expresses it as the sum of two prime numbers. Click the image to try it out!
π
We usually first encounter π (pi) as the ratio of a circle’s circumference to its diameter. Have you ever wondered how we’re able to come up with a good approximation of this famous irrational number, or how else we can represent it?
APPROXIMATING π
Archimedes of Syracuse (287 BC – 212 BC) used circumscribed and inscribed polygons of up to 96 sides to find an approximate value of π. Click the following images to investigate this idea using an interactive applet or a printable activity.
Interactive Applet
Printable Activity
π AS INFINITE SERIES
π can be represented as the sum of several different infinite series. Click the image on the right to see some examples. For each of these series, the more terms that are added, the better the approximation of π. Click the image below to explore this idea using the Madhava-Gregory-Leibniz series in a Python program.
The Early Commuter Problem
A commuter arrives at his suburban train station each evening exactly at five o’clock. His wife always meets the train and drives him home. One day he takes an earlier train, arriving at the station at four o’clock. The weather is pleasant, so instead of calling home he starts walking along the route always taken by his wife. They meet somewhere on the way. He gets into the car and they drive home, arriving at their house ten minutes earlier than usual. Assuming that the wife always drives at a constant speed, and that on this occasion she left just in time to meet the five o’clock train, how long did the husband walk before he was picked up?
Click the image below to explore this problem using an interactive applet.
The Mutilated Chessboard Problem
If the top-left and bottom-right squares of an 8×8 chessboard are removed, can the remaining squares be perfectly covered using 2×1 dominoes?
Click the following image for an interactive applet that can be used to help solve this problem and other chessboard problems.
Perfect Numbers
A perfect number is a number whose proper divisors add to the number itself. For example, 6 and 28 are both perfect numbers because
6 = 1 + 2 + 3 and
28 = 1 + 2 + 4 + 7 + 14.
It is currently unknown whether there are infinitely many perfect numbers and whether any odd perfect numbers exist.
Click the image below to find the perfect numbers in an interval using a Python program.
Twin Prime Conjecture
Two prime numbers are called twin primes if there exists only one composite number between them. For example,
- 3 and 5 are twin primes.
- 17 and 19 are twin primes.
- 191 and 193 are twin primes.
The twin prime conjecture states that there are infinitely many pairs of twin primes, but this conjecture has yet to be proven!
Riemann Hypothesis
Named after Bernhard Riemann (1826 – 1866), the Riemann Hypothesis is the arguably the most important unsolved problem in mathematics! As one of the famous Millenium Prize Problems, a million dollars will be awarded to the first person who solves it. Click the image below to learn more about this notorius problem!
Hexaflexagons
Hexaflexagons are flat paper hexagons that can be “flexed” to reveal hidden faces. Made by folding a single strip of paper, the trihexaflexagon features three interchanging faces and the more complex hexahexaflexagon boasts six. Check out the following resources to explore this fascinating blend of math and origami!
Printable Templates
Interactive Folding Guide
Click the image below for a digital interactive folding guide for the trihexaflexagon.
Platonic Solids
Also known as a regular polyhedron, a Platonic solid is a polyhedron with the following properties:
- The polyhedron is convex.
- Every face is a regular polygon.
- All of the faces are congruent.
- Every vertex is surrounded by the same number of faces.
Click the following image to explore the five Platonic solids using an interactive applet.
Why Only Five?
Curious about why there aren’t more than five Platonic solids? Euclid (325 BC – 265 BC) reasoned that the answer can be found by considering the angle sum at each vertex. Click the image below for an interactive applet that can be used to investigate this idea.
Euler's Polyhedron Formula
Named after Leonhard Euler (1707 – 1783), Euler’s polyhedron formula states that for any convex polyhedron, the number of vertices minus the number of edges plus the number of faces always equals 2. That is,
V – E + F = 2.
Golden Ratio and Rectangle
Two quantities are in the golden ratio if the ratio of the larger to the smaller is equal to the ratio of the sum to the larger. This ratio is commonly found in nature and frequently used to create aesthetically pleasing designs. Do you think the golden rectangle is is more visually appealing than other rectangles? Click the image below to explore with an interactive applet.
Fibonacci and Golden Spirals
The Fibonacci spiral is created using squares that have areas following the Fibonacci sequence (1,1,2,3,5,8,….). The golden spiral comes from a golden rectangle repeatedly divided using the golden ratio. Although the two spirals are very similar, they are not identical! Click the following image to investigate using an interactive applet.
36° and 72°...
A Curious Relationship
36° and 72° are not typically considered to be special angles, so it may seem surprising that the difference between their cosines is exactly one half! Check out the following video for a visual explanation of this relationship through the geometry of right triangles and a regular pentagon.
SUM OF n·n!
Products of numbers and their factorials have an interesting property when added together. Check out the following video to see what it is and why it happens!
Pythagorean Theorem
Is there a 3D version of the famous Pythagorean Theorem, perhaps using rectangular prisms or cubes instead of squares? How is this idea related to Fermat’s Last Theorem? Click the image below to explore using an interactive applet!
Conic Sections
Circles, ellipses, parabolas and hyperbolas are called conic sections since they can each be generated by the intersection of a plane and a cone. Click the image below for an interactive applet to explore this concept.
The Tetris Rectangle Problem
The classic video game Tetris is based on the seven blocks shown above.
Using each piece exactly once and without overlapping, can you construct a solid rectangle with no empty spaces in its interior? If so, show an example. If not, prove that it can’t be done. Check out the following video to learn more about this intriguing problem!