Mysterious Numbers

Take any 3 digit number whose digits are not all the same. Numbers like 027, 889, and 534 are all valid examples. Now let us just start with 027. We are going to play a game. First rewrite 027 by ordering the numbers in descending order, 720. Then subtract from it the number made by […]

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Paradoxes: The Birthday Problem

As living beings we rely heavily day to day on our intuition. And most of the time it does not let us down. However, sometimes our intuition fails us and we feel compelled to call that event a paradox. This is the case with the birthday problem. I pose the following question. How many random […]

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Summing the Natural Numbers

Infinite sums are weird. Some converge and some do not. One might think it easy to conclude that if we sum the natural numbers, we would get . But is this correct? Believe it or not, we can rearrange the sum of the natural numbers to be equal to . We will give the proof […]

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The Pigeonhole Principle

  Imagine that two people are playing “Paper, Rock, Scissors” and one wins the first game.  Then the loser chimes in and declares “best two out of three!” This is a common phrase heard in win/lose games and it utilizes a very important and subtle principle known as the pigeonhole principle. We know that two out […]

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Paradoxes: Gabriels Horn

Gabriel’s horn is the function on the interval rotated around the x-axis. Intuitively, it should require less paint to fill the horn than to paint the surface (imagine painting a box or a ball). With Gabriel’s Horn this is not the case. In fact, There is an infinite surface area and only a finite volume. […]

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Paradoxes: Zeno of Elea

Zeno of Elea was a Greek philosopher famous for paradoxes which assert that motion is nothing but an illusion. It is assumed that Zeno created the paradoxes in support of Plato’s Parminedes.  We will present two of these paradoxes along with a refutation.   1. Achilles and the Tortoise Achilles was challenged to a race by […]

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The Road to Larger Infinities

Georg Cantor was a German mathematician responsible for the invention of set theory. Before him, the concept of infinity was not well studied or considered by mathematicians. The importance of Cantor’s development of set theory is very clear as it is the basis for much of modern mathematics. We will take a look at Cantor’s […]

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