Sophia Liao

Latest Posts

Sum Ways to Simplify a Some

Many moons ago, I was met with this incredible fact: \begin{aligned} \frac1x + \frac1{x+1} + \cdots + \frac1{x+p-1} \equiv \frac{-1}{x^p-x} \quad (\text{mod } p),\end{aligned} where p is any prime. It’s so surprising that it’s stuck with me until now, so I figured I should probably talk about it. How would anyone go about…

Diagonals of an n-gon

What happens if we take the product of the lengths of all of the diagonals which stem from a single vertex of a regular n-gon? For example, for this regular pentagon (below), we’d want to calculate the product of all of the green segments. Now, we can set the length from the center of…

Dividing a Square into Equal-Area Pieces

Here’s a fun little problem. It’s from the latest MIT magazine: M/J3. Tom Harriman wants you to divide a square of side length 2 into four equal-area pieces so that the sum of the lengths of the boundaries is minimized. Hint: It is easy to get four side =1 squares with total boundary…

Fibonacci Numbers and Finite Fields

I found a cool application of finite fields to prove an intriguing result related to Fibonacci numbers last fall. The theorem goes as follows: If p\equiv 1,4 \text{ (mod 5)}, then F_{p-1} is divisible by p,if p\equiv 2,3 \text{ (mod 5)}, then F_{p+1} is divisible by p,and F_5…