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Solution below.

Check out this Problem of the Week.

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Solution below.

Be sure to let us know how you solved it in the comments below or on social media!

Solution below.

Check out this Problem of the Week.

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Solution below.

Be sure to let us know how you solved it in the comments below or on social media!

Solution below.

Check out this Problem of the Week.

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Solution below.

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Solution below.

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Solution below.

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Solution below.

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Solution below.

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Check out this logic based Think Thursday Problem!

This problem was originally posted by MAA online.

Solution below.

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Solution below.

Check out this logic based Think Thursday Problem!

Solution below.

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Solution below.

Welcome to our first Think Thursday Problem!

This series aims to introduce logic based problems, puzzles, and other tricky brain teasers. The problems featured here are Math related, but do not require a extensive knowledge of Mathematics to solve. We hope you enjoy this new series!

Solution below.

Check out this Problem of the Week and enjoy this math joke.

*Because the constant was incapable of change.*

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Solution below.

Why did the variable break up with the constant?

Solution below.

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The history of ‘e’ is a tangled one, one which would warrant an entire dedicated book to parse through mathematics to the original conception of the transcendental number. Even before e’s enigmatic beauty was fully unearthed, people using mathematics to solve real world problems encountered the number many times, and understood it enough to work it into their solutions. A good example of this is when e shows up in compound interest. Bankers found out that as the number of times one took annual compound interest grew to infinity, the rate of growth approached e! Watch the video to see two mathematical proofs of our statement, using two definitions of e.

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Solution below.

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Solution below.

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Oh no, back to school is right around the corner!

We know it can be hard to jump straight back into Math classes during the first few days after a summer away. Lucky for you we have arranged some of of our Youtube channel videos into a helpful guide to make sure you are on your game in the first week of class. Check them out below!

We know it can be hard to jump straight back into Math classes during the first few days after a summer away. Lucky for you we have arranged some of of our Youtube channel videos into a helpful guide to make sure you are on your game in the first week of class. Check them out below!

Check out this Algebraic Problem of the Week.

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Solution below.

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Solution below.

Back in 300 BC, Euclid proved that there were an infinite number of primes. He used line segments to show that some line lengths could only be made up from single-unit line lengths and not lines with lengths of 2, 3, etc. These line lengths represented prime numbers. This proof has the same principle but is a little different than Euclid's and uses proof by contradiction. Take a look at this simple proof which shows that primes are infinite!

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Solution below.

Check out this Algebraic Problem of the Week.

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Solution below.

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Solution below.

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The mythology behind this fairly simple proof is what makes it one of the most popular proofs in math classes across the world. The story follows a young Carl Friedrich Gauss, whose first grade teacher asked the class to add up the numbers 1 to 100 in order to pass a good amount of time. Before the teacher had time to start grading papers, Gauss handed in his assignment. Watch the video to find out Gauss’ observation that is now one of the most famous math proofs out there.

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Check out this Mathematics inspired Alphametic Problem of the Week

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The two basic rules for solving alphametics are as follows:

Each letter must be represented by a different digit. If the letter is used more than once, it must be represented by the same digit.

Once you substitute digits for all your letters, you must end up with an accurate addition problem.

Solution below.

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The two basic rules for solving alphametics are as follows:

Each letter must be represented by a different digit. If the letter is used more than once, it must be represented by the same digit.

Once you substitute digits for all your letters, you must end up with an accurate addition problem.

Solution below.

This problem of determining the area of a circle, or better defined as the area inside of a circle, was a huge dilemma in the field of mathematics. It was not until the mid 200's BC when Archimedes began to anticipate modern calculus and analysis though concepts of infinitesimals and exhaustion, which he used to solve this major challenge of finding the area of a circle.

Archimedes' method of finding the area is described as "squaring the circle", which is trying to find the square that has the same enclosed area as a circle of a given radius. Using this and also using a method where he approximated the area of a circle with other, known shapes such as squares and hexagons, Archimedes was able to determine the area inside of a circle. Take a look at the proof to see how Archimedes came up with the formula we know today:

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Solution below.

Check out this Problem of the week about Geometry and triangles within a circle. If you're interested in learning more about how you draw circles and what it says about your cultural background, read this article:

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Solution below.

## How do you draw a circle? We analyzed 100,000 drawings to show how culture shapes our instincts

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The advent of graph theory, from the mind of Leonhard Euler, came from a long-standing problem for the people of Königsberg. The problem was that no couple had a long and happy marriage, if they were married in Königsberg. As tradition dictated, a newlywed couple had one chance to travel across Königsberg’s four land masses using each of the seven bridges once and only once. If the two lovers could complete this seemingly simple task, their marriage would be long and happy. Years went by and nobody could complete to task, until Euler constructed a mathematical object that broke the curse of Königsberg… a graph!

Watch the proof proposed by Euler below to learn how mathematical abstraction created a whole new field of math, which is now regarded as an important predecessor to topology. Euler’s invention itself is remarkable, but the implications to mathematical philosophy reveals something very deep in the heart of mathematics. Namely, the art of abstraction to gain a better understanding of certain truths inherent in life’s situations.

The Königsberg Bridge Problem, and its solving:

This week’s Top Pop Math Chop comes from Georg Cantor, who first solved this piece of set theory in 1891. He presented this as a mathematical proof which showed it was impossible to link infinite sets with an infinite set of the natural numbers. This is known today as Cantor’s diagonal argument, which he proved using binary numbers.

Cantor showed that if he has a list of binary numbers, takes one digit from each going diagonally, produces a new number, and swaps every single digit with a corresponding 1 or 0 (if is a 1 it becomes 0 and vice versa), that the number will be different than every other binary number listed before it. This is because in the first number the first digit is different, so it’s definitely different than the new number; in the second number the second digit is different than the second digit in the new number and so on.

You can do this same thing with real numbers, and produce infinite decimals between 0 and 1. This shows the real numbers are uncountable.

Check out the video below explaining Georg Cantor’s proof:

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