On Stable Matchings

Suppose every student ranks colleges and every college ranks students. Can they be paired up so that no student and college would both rather be with each other than with the partners they were assigned? I introduce the stable matching problem through such blocking pairs, show why naive approaches fail, and present the Gale–Shapley algorithm, in which one side proposes in order of preference and the other holds on to the best offer received so far. I explain why the process always ends in a stable matching, and why this classic algorithm matters across mathematics, economics, and computer science.