# medium

These fun facts are at the medium level.

## Multiplication by 111

If you liked the Fun Fact Multiplication by 11, you’ll enjoy seeing how to take that idea one step farther. Here’s...

## i to the i is a Real Number

If you are familiar with complex numbers, the “imaginary” number i has the property that the square of i is -1....

## Face Derivatives and Computer Vision

One challenge in robotics is the problem of computer vision: how do you program a computer to interpret and “understand”...

## Square Root of Two is Irrational

An irrational number is a number that cannot be expressed as a fraction. But are there any irrational numbers? It was known to...

## Finding the N-th digit of Pi

Here is a very interesting formula for pi, discovered by David Bailey, Peter Borwein, and Simon Plouffe in 1995:Pi =...

## Ellipsoidal Paths

Given an ellipse, and a smaller ellipse strictly inside it, start at a point on the outer ellipse, and in a...

## Large Counterexample

A positive integer is said to be of even type if its factorization into primes has an even number of primes. Otherwise it is...

## Whiffle Ball

Does a ball take longer to come down than go up, or does it take the same amount of time...