A Super Factorial, Double Factorial, Hyper Factorial?!

Happy New Year everyone! You probably know what a factorial is. $$n!=n \left(n-1\right)\left(n-2\right)\left(n-3\right)\times3\times2\times1$$ Or, if you like calculus: $$x!=\int_{0}^{\infty} t^x{e^{-t}} dt$$ Like, for example, 8! = 8\(\times\) 7\(\times\)6\(\times\)5\(\times\)4\(\times\)3\(\times\)2\(\times\)1 = 40,320. However, what if we put two exclamation marks? Or a dollar sign? Well today, I am going to show you. Double Factorial The name comes… Continue reading A Super Factorial, Double Factorial, Hyper Factorial?!

Proof Writing (Part 1): An introduction to a couple types of proof.

Hello, and welcome back, and today we’re going to be pausing our discussion of Pascal’s triangle temporarily, and discussing some proof writing! (Don’t worry, this will only be a 2 part series, so we will be back on to Pascal’s triangle in no time!) What is proof? A proof is a series of statements, each… Continue reading Proof Writing (Part 1): An introduction to a couple types of proof.

Pascal’s Triangle (Part 3):

Hello, and welcome to the 3rd episode in this 5 part series! In this blog we will be formally writing the Hockey-stick Principle showcased in the last blog, see what it has to do with multi-dimensional triangles, and finish our study on this pattern of Pascal’s triangle. Formal writing of the Hockey-stick Principle To begin,… Continue reading Pascal’s Triangle (Part 3):

Recamán Sequence

The Recamán Sequence might look like a jumble of numbers at first glance. But then it becomes clear. THE BIG PICTURE The 0th term is 0. To find the \(n\)th term, you usually subtract, \(a(n)-n\), however if that results in a negative number or a number already used in the sequence, you add, \(a(n)+n\). So… Continue reading Recamán Sequence

Erwin Schrödinger and the Half-Alive Cat

Austrian physicist Erwin Schrödinger is one of the founders of quantum mechanics. In 1935, he posed this problem: imagine you take a cat and place it in a sealed box along with a radioactive device that had a 50% chance of killing the cat. In an hour, is the cat dead or alive? What do you think?

Pascal’s Triangle (Part 2)

Hello everyone! Welcome back to my blog series on Pascal’s Triangle. Today we will be looking at the first pattern covered in the last blog–triangular numbers, and their relation to the Hockey Stick Principle. A short proof To begin this blog, we will first prove the property of the triangular numbers shown in part 1:… Continue reading Pascal’s Triangle (Part 2)