Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mathematics of the NYT daily word game Waffle

S. P. Glasby

TL;DR

This work maps the NYT Waffle puzzle to a permutation problem on $21$ colored squares, showing that a perfect unscrambling corresponds to a $g\in S_{21}$ with $s(g)=10$ and $c(g)=11$, making direct solutions rare due to the typical low cycle count of random permutations (about $3.65$ cycles). By invoking Cayley’s lemma and the Cayley graph distance, the paper connects puzzle solvability to cycle structure and fixed points, and develops practical algorithms that use color constraints and dictionary information to identify valid unscramblings. It also analyzes the factors governing puzzle difficulty, highlighting how the numbers of green and yellow squares, $N_g$ and $N_y$, influence search complexity, with extreme hard cases arising when $N_g=1$ and $N_y=0$, where the residual $g'$ decomposes into ten disjoint $2$-cycles and the search space is enormous. The findings inform both puzzle design (e.g., choosing $N_g$ to balance difficulty) and algorithmic solving strategies, including parity-based coloring, non-uniqueness considerations, and frequency-informed pruning. Overall, the paper provides a rigorous combinatorial framework for understanding Waffle hardness and practical methods for solving or designing perfect-scoring games.

Abstract

We investigate the combinatorics of permutations underlying the the daily word game Waffle, and learn why some games are easy to solve while extreme games are very hard. A perfect unscrambling must have precisely 11 orbits, with at least one of length 1, on the 21 squares. We also describe practical algorithms for studying Waffle.

Mathematics of the NYT daily word game Waffle

TL;DR

This work maps the NYT Waffle puzzle to a permutation problem on colored squares, showing that a perfect unscrambling corresponds to a with and , making direct solutions rare due to the typical low cycle count of random permutations (about cycles). By invoking Cayley’s lemma and the Cayley graph distance, the paper connects puzzle solvability to cycle structure and fixed points, and develops practical algorithms that use color constraints and dictionary information to identify valid unscramblings. It also analyzes the factors governing puzzle difficulty, highlighting how the numbers of green and yellow squares, and , influence search complexity, with extreme hard cases arising when and , where the residual decomposes into ten disjoint -cycles and the search space is enormous. The findings inform both puzzle design (e.g., choosing to balance difficulty) and algorithmic solving strategies, including parity-based coloring, non-uniqueness considerations, and frequency-informed pruning. Overall, the paper provides a rigorous combinatorial framework for understanding Waffle hardness and practical methods for solving or designing perfect-scoring games.

Abstract

We investigate the combinatorics of permutations underlying the the daily word game Waffle, and learn why some games are easy to solve while extreme games are very hard. A perfect unscrambling must have precisely 11 orbits, with at least one of length 1, on the 21 squares. We also describe practical algorithms for studying Waffle.
Paper Structure (4 sections, 4 equations, 5 figures)

This paper contains 4 sections, 4 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Perfect unscramblings and Cayley's lemma
  3. Coloring and uniqueness
  4. What makes some games hard?

Figures (5)

  • Figure 1: (a) scrambled game #2 (b) unscrambled (c) numbered squares (d) parities
  • Figure 2: $\hbox{$g=(x_1)(x_2,x_3)(x_4,x_5,x_6)(x_7,x_8,x_9,x_{10})(x_{11},x_{12},x_{13},x_{14},x_{15})(x_{16},x_{17},x_{18},x_{19},x_{20},x_{21})$}$
  • Figure 3: (a) (S,T) gives BOOTS, NASAL, (b) (A,I,O) gives BISON, SACKS, (c) (O,U,Y) gives SUCKS, DOING, DYKES and (d) (O,S,D,N,A) gives LANDS, NOISE
  • Figure 4: (a) archived Waffle game # 280 (b,c) two valid solutions
  • Figure 5: $N_g=1$ (a) easy scrambled, (b) unscrambled, (c) hard, (d) unscrambled