Original graph research
Inside the four-letter word ladder graph
We measured every link and every shortest route in Puzzled’s largest four-letter word component. The result shows where ladders stay flexible, where they narrow, and why changing four letters can take far more than four moves.
The shape of the graph
One huge neighborhood with honest gaps
Each word is a node. Two words share an undirected link when they differ in exactly one letter position. That simple rule produces one component containing 2,500 words, plus smaller islands and isolated entries.
Being in the dictionary does not guarantee a route to every other word. The graph records Puzzled’s reviewed vocabulary boundary; it is not a claim that every English word belongs here.
- Reviewed words
- 2,661
- One-letter links
- 11,902
- In the largest component
- 93.95%
- Isolated words
- 91
A neighborhood you can inspect
CARE has 23 one-move exits
The diagram groups every accepted neighbor by the position that changes. The exact word table follows it, so the relationship remains available without relying on shape or color.
| Changed position | Accepted neighbors | Count |
|---|---|---|
| First letter | BARE, DARE, FARE, HARE, MARE, PARE, RARE, TARE, WARE | 9 |
| Second letter | CORE, CURE | 2 |
| Third letter | CAFE, CAGE, CAKE, CAME, CANE, CAPE, CASE, CAVE | 8 |
| Fourth letter | CARD, CARP, CARS, CART | 4 |
Finding 1
The first letter carries almost half the links
Changing a word’s first position accounts for 44.56% of every one-letter link. The second position is far less flexible. This imbalance helps explain why some visually similar endpoints branch very differently.
| Position | Links | Share |
|---|---|---|
| First letter | 5,304 | |
| Second letter | 1,471 | |
| Third letter | 2,779 | |
| Fourth letter | 2,348 |
Finding 2
Four changed letters can still require eighteen moves
Letter difference is only a lower bound. Across all 3,123,750 unordered pairs in the largest component, the average shortest route uses 5.41 moves and the median uses 5. Only 19.72% of pairs reach the lower bound exactly.
| Moves | Pairs | Share |
|---|---|---|
| 1 | 11,827 | 0.38% |
| 2 | 70,847 | 2.27% |
| 3 | 261,332 | 8.37% |
| 4 | 596,657 | 19.10% |
| 5 | 821,993 | 26.31% |
| 6 | 640,269 | 20.50% |
| 7 | 396,900 | 12.71% |
| 8 | 194,534 | 6.23% |
| 9 | 76,447 | 2.45% |
| 10 | 28,133 | 0.90% |
| 11 | 10,156 | 0.33% |
| 12 | 4,714 | 0.15% |
| 13 | 3,802 | 0.12% |
| 14 | 3,482 | 0.11% |
| 15 | 1,987 | 0.06% |
| 16 | 554 | 0.02% |
| 17 | 101 | 0.00% |
| 18 | 15 | 0.00% |
| Detour | Pairs | Share |
|---|---|---|
| At the letter-difference lower bound | 616,118 | |
| One extra move | 979,789 | |
| Two extra moves | 726,211 | |
| Three or more extra moves | 801,632 |
The diameter pair ANEW–EDIT is one of 15 pairs at 18 moves. “Diameter” means the longest shortest route, not the only possible route.
Finding 3
A hub is not always a bridge
A hub has many immediate exits. A bridge word is stricter: removing it disconnects part of the component. The graph contains 135 such articulation words, but the most connected word is WINE with 26 links.
| Rank | Word | Links |
|---|---|---|
| 1 | WINE | 26 |
| 2 | CORE | 25 |
| 3 | LINE | 25 |
| 4 | PATS | 25 |
| 5 | WARE | 25 |
| 6 | BARE | 24 |
| 7 | BARS | 24 |
| 8 | BATS | 24 |
| 9 | BOTS | 24 |
| 10 | CAPS | 24 |
Finding 4
Flexibility is common, but not evenly shared
One third of the dictionary has at least 12 immediate links, while 17.96% has only one to three. The current word matters as much as the number of letters still different from the goal.
| Immediate links | Words | Share |
|---|---|---|
| No links | 91 | |
| 1–3 links | 478 | |
| 4–7 links | 603 | |
| 8–11 links | 599 | |
| 12 or more links | 890 |
Exact methodology
How these numbers were produced
- Source. Use the same pinned American-English SCOWL/Hunspell-derived, general-audience-reviewed four-letter dictionary as Word Chain and the solver.
- Links. Connect two different words only when exactly one of their four positions differs. Count each undirected link once.
- Distances. Run breadth-first search from every word in the 2,500-word largest component, then count each unordered pair once.
- Bridges. Identify articulation vertices: words whose removal increases the number of connected pieces in the largest component.
- Reproducibility. Generate this committed record from the reviewed dictionary. The release build recomputes every value and fails if the page data drifts.
What the analysis does not measure: familiarity, definition difficulty, cultural context, or which route a human notices first. Those remain editorial and human-solving questions.
Use the graph three ways
Study the shape, solve a pair, or protect the daily surprise
The strategy guide turns graph ideas into a human method. The solver checks arbitrary endpoints. Daily Word Chain keeps today’s route hidden while you play.