# Untying The Gordian Knot via Experimental Mathematics

## TL;DR

This paper advocates for an experimental mathematics approach in enumerative combinatorics, demonstrating that computational data and heuristic guessing can lead to rigorous solutions for complex problems.

## Contribution

It introduces a methodology where computational experiments guide the discovery and proof of combinatorial results, emphasizing the value of heuristic and experimental techniques.

## Key findings

- Computer-generated data can suggest correct solutions.
- Heuristic 'guess and verify' methods can be rigorously justified.
- The approach simplifies solving complex enumerative problems.

## Abstract

In this methodological article on experimental-yet-rigorous enumerative combinatorics, we use two instructive case studies, to show that often, just like Alexander the Great before us, the simple, "cheating" solution to a hard problem is the best. So before you spend days (and possibly years) trying to answer a mathematical question by analyzing and trying to 'understand' its structure, let your computer generate enough data, and then let it guess the answer. Often its guess can be proved by a quick 'hand-waving' (yet fully rigorous) 'meta-argument'. Since our purpose is to illustrate a methodology, we include many details, as well as Maple source-code.