Ryan Collison

A tree is just a connected graph with no cycles — but that constraint produces rich structure. Trees appear in file directories, Huffman compression, decision trees, evolutionary phylogenies, and parse trees inside every compiler. Understanding them is foundational to almost all of computer science.

In 1736, Euler showed you can't walk Königsberg's bridges without retracing — not by trial and error, but by pure logic about node degrees. That proof was the birth of graph theory, the mathematics of networks.

Expanding (a+b)¹⁰ by hand is a nightmare. Pascal's triangle makes it instant. The binomial theorem explains why those triangular numbers keep showing up.

You're forming a committee. You need three people from a group of ten. Alice, Bob, and Carol volunteer. That's your committee. Bob, Carol, and Alice volunteer. Same committee. Carol, Alice, and Bob. Still the same committee. The order you pick them doesn't matter. You&

How many ways can 10 runners finish a race? How many 4-digit PINs exist? These are permutation problems — arrangements where sequence is the whole point. Master the factorial logic once and counting problems become systematic.

Counting every possibility by hand stops being feasible after about twenty items. Combinatorics replaces exhaustive listing with structural reasoning — and the results are often astonishing.