2. Four types of problem

‘Four types of problem’ explains that combinatorics is concerned with four types of problem: existence problems (does x exist?); construction problems (if x exists, how can we construct it?); enumeration problems (how many x are there?); and optimization problems (which x is best?). Existence problems discussed include tilings, placing dominoes on a chess board, the knight’s tour problem, the Königsberg bridges problem, the Gas–Water–Electricity problem, and the map-colour problem. Construction problems include solving mazes, and the two types of enumeration problems considered are counting problems and listing problems. Examples of an optimization problem include the minimum connector problem and the travelling salesman problem. The efficiency of algorithms is also explained.