A math equation becomes a set of step-by-step rules a computer can follow thousands of times faster than any human. Writing programs in the Scheme language, you translate first-order and then second-order differential equations into those rules. That speed makes it practical to model how atoms spread through a solid — a calculation that would be impossible to do by hand.

A program can test thousands of cases that would take years to draw by hand. Using the Logo programming language, you write a simulation of a ball bouncing inside a rectangular table and run 6,300 paths. A clear rule emerges: a ball reaches a corner only when the tangent of its launch angle is rational.

When you encode decision rules precisely, a machine follows them perfectly every time without forgetting or hesitating. Here, roughly 300 if-then statements define a smart player — each statement countering a move that could lead to a loss. Tested against three human players, the program never lost, showing that a complete set of rules can produce unbeatable play.

Abstract personality traits can be translated into concrete rules a machine follows during a game. Two AI players are programmed with different personality profiles, then the computer runs 300 games between them. The results show that personality matters most in three-player games, while two-player games rely almost entirely on probability.

Programming handles repetitive calculations so you can focus on finding patterns in the results. Using programs written on a graphing calculator, you generate two new sequences from the natural numbers by flipping or reversing binary digits. Analyzing at least 1,000 terms reveals surprising patterns — some obvious, some hidden — that you then explain with mathematical formulas.