Mathematics of the NYT daily word game Waffle

TL;DR

This paper explores the combinatorial structure of the NYT Waffle game, revealing key permutation properties, and introduces algorithms for solving and designing challenging game instances.

Contribution

It characterizes the permutation orbits necessary for perfect unscrambling and provides practical algorithms for solving and creating Waffle puzzles.

Findings

A perfect unscramble requires exactly 11 orbits with at least one fixed point.

The paper describes algorithms for solving Waffle games.

It offers methods to generate Waffle puzzles with extreme difficulty properties.

Abstract

This note investigates the combinatorics of permutations underlying the NYT daily word game Waffle. It helps to solve Waffle games and helps to understand why some games are easy to solve while others are very hard. It shows that a perfect unscrambling must have precisely 11 orbits, with at least one of length 1, on the 21 Waffle squares. It also describes practical algorithms for solving Waffle games and creating new games with extreme properties.