Introduction: What made these games so Intriguing
While researching the mathematics behind the iconic waffle game and NYT’s Wordle, I have reviewed multiple articles and attempted to understand it myself first, to make it easier for you.
The perfect unscrambling means you have 11 steps to solve the puzzle(with at least one of length 1, on the 21 squares).
Word games have been in the news for a long time. In the year 1942, the NYT introduced crosswords initially to divert attention from the bombing of Pearl Harbor, and later just for fun. Two of the most popular word games that came out of the NYT were Wordle(2021) and Connections (2023). They became popular at the start of the COVID-19 pandemic, also claimed to improve brain function(unscientifically).
But in this article, we will be exploring the other elephant in the room, and we are talking about Waffle. These games were both created by software developers(Josh Wardle created Wordle in 2021, and James Robinson created Waffle in 2022).
In Wordle, your goal is simple: guess a single five-letter English word in six tries. And for the Waffle Game, it is slightly more complex. Here, you need to arrange six five-letter words correctly on a 5×5 grid that looks like a waffle because it has four empty spaces (holes). The letters start scrambled, and you fix them by swapping pairs of letters. You have up to 15 swaps, but if you solve it in 10 swaps, that’s a perfect game.
Just like in Wordle, the colors in Waffle act as clues.
- Green means the letter is already in the correct position.
- Yellow means the letter is in the word, but not in the right spot.
- Gray means the letter does not belong in that word at all.
These colors guide players toward the solution, but in this paper, we’ll take things further. Let’s assume you’re a perfectionist—someone who wants that coveted perfect score of 10 swaps—or you’re simply curious about the elegant math behind achieving it.
Understanding Perfect Unscrambling Using Cayley’s Theorem
In simple words, you need three step to obtain a perfect score in Waffle.
Here are three steps as follows:
(S1): Identify the six five-letter words hidden in the grid
(S2): Determine the exact letter arrangement that restores the correct solution (a perfect unscrambling).
(S3): Execute exactly 10 swaps to reach that arrangement.
Among these, Step 3 is straightforward once you know the right approach, which we’ll discuss later. However, Step 2 becomes tricky when letters repeat, and Step 1 becomes difficult when all letters are distinct because there are fewer obvious patterns. While making small mistakes can make both steps easier, our goal here is perfection: zero errors and exactly 10 swaps for a full 5-star score.
Let’s dive into the Math behind it
The Math Behind It
Think of the scrambled grid as a permutation of 21 letter positions. Each swap you make exchanges two letters—this is a transposition in math terms. Cayley’s Lemma gives us the formula for the minimum number of swaps needed: Minimum swaps=n−c\text{Minimum swaps} = n – c.
Where:
- n=21n = 21 (the number of lettered squares)
- cc = the number of cycles in the permutation
A cycle is a group of letters that rotate among positions. For example, if A → B’s spot, B → C’s spot, and C → A’s spot, that’s one cycle of length 3.
For a perfect Waffle solve in 10 swaps: 21−c=10 ⟹ c=1121 – c = 10 \implies c = 11
So the scrambled arrangement must consist of 11 cycles, with at least one being a single fixed letter (a green tile).
3. Why Coloring Matters (and Guarantees Unique Solutions)
Waffle uses color clues similar to Wordle:
- Green = letter in the correct position
- Yellow = letter is in the word but in the wrong position
- Gray = letter is not in the word
These clues make solving possible because they reduce the number of valid permutations dramatically. Without coloring, figuring out the correct six words would be almost impossible for most boards. We explore the math behind coloring in Section 4.
4. What Makes a Waffle Puzzle Hard?
Not all Waffle puzzles are created equal. Difficulty depends on several factors:
- Number of green letters – More greens = more fixed points = fewer cycles = easier.
- Repeated letters – Make Step (2) (unscrambling) harder because multiple positions can share the same letter.
- All unique letters – Make Step (1) (finding words) harder because the player can’t rely on duplicates for hints.
- Cycle structure – A permutation with large cycles requires more planning; smaller cycles make swaps easier.
The hardest scenario? One green letter, no yellows, all other letters unique. Then the 20 non-green letters must form 10 disjoint 2-cycles. The number of possible arrangements is astronomically large (~108.810^{8.8}), making brute force impossible without a strategy.
5. Probability of Perfect Waffles and Why They Are Rare
How likely is it that a scrambled Waffle board allows a perfect score in 10 swaps? To answer this, we use the math of permutations and cycles.
Step 1: Total Possible Arrangements
If all 21 letters were distinct, there would be: 21!≈5.1×101921! \approx 5.1 \times 10^{19}21!≈5.1×1019
possible ways to arrange them. (That’s 51 followed by 18 zeros—a mind-boggling number.)
Step 2: Perfect Score Condition
A perfect score requires exactly 11 cycles in the permutation.
The number of permutations of nnn items with exactly kkk cycles is given by the unsigned Stirling number of the first kind, denoted: c(n,k)c(n, k)c(n,k)
For our case: n=21,k=11n = 21, k = 11n=21,k=11.
The exact value is huge, but for context: c(21,11)≈4.4×10
c(21,11)≈4.4×1015
Step 3: Probability
The probability that a random permutation of 21 letters has exactly 11 cycles is: P=c(21,11)21!≈4.4×10155.1×1019≈8.6×10−5P = \frac{c(21, 11)}{21!} \approx \frac{4.4 \times 10^{15}}{5.1 \times 10^{19}} \approx 8.6 \times 10^{-5}P=21!c(21,11)≈5.1×10194.4×1015≈8.6×10−5
That’s about 1 in 11,600 permutations.
And this is before considering other constraints like:
- Letters forming real words,
- Grid layout rules,
- No invalid letter placements.
When those constraints are included, the probability becomes astronomically smaller—far less than 1 in a billion. That’s why the Waffle generator carefully designs puzzles rather than picking random permutations.
Effect of Repeated Letters
Repeats increase the number of valid permutations (since some positions become indistinguishable) but also reduce uniqueness. While this can slightly raise the chance of forming 11 cycles, it still remains tiny because of the word constraints.
Why Perfect Waffles Feel Rare
Even with designed puzzles, only some allow exactly 10 swaps. Many require 11 or 12, making a perfect score an achievement. In short:
- Random chance ≈ is practically zero
- Designed chance = enough to make it fun, but still challenging
Conclusion
The beauty of Waffle lies not just in wordplay but in the elegant mathematics behind it. A perfect game depends on the cycle structure of a permutation, as described by Cayley’s Lemma. Coloring provides essential guidance, while letter patterns determine difficulty. In short, Waffle is more than a casual game—it’s a fun application of combinatorics hiding in plain sight.
Here is our other blog where we wrote about Why Wordle Changed the Web Word Game Landscape
:(Internet’s poster boy for the web-based word puzzle game).
