The Method(!) of "Guess and Check"

TL;DR

This paper discusses the guess-and-check methodology in combinatorics, illustrating its use through lattice walk enumeration problems and advocating it as a metaphor for future mathematical approaches.

Contribution

It introduces the guess-and-check method as a practical approach for complex enumeration problems and discusses its philosophical implications for future mathematics.

Findings

Enumerating lattice walks with constraints can be approached via guess-and-check.

Rigorous proofs are possible but often impractical in these contexts.

The paper advocates for the guess-and-check methodology as a future mathematical paradigm.

Abstract

The problems of enumerating lattice walks, with an arbitrary finite set of allowed steps, both in one and two dimensions, where one must always stay in the non-negative half-line and quarter-plane respectively, are used, as case studies, to illustrate the `naive' methodology of guess-and-check, where rigorous proofs are possible, but not worth the trouble. We argue that this is a metaphor for future math.