XWord Info Flow Analysis
Sunday, February 23, 2025 by Daniel Grinberg and Rafael Musa
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Solvers and constructors have an intuitive idea of what makes a crossword puzzle "flow" well. Mathematically, flow is a measure of how easily information can propagate through the grid from the clues to the answers. A puzzle with high flow allows solvers to leverage known answers to deduce unknown answers more easily, leading to a more enjoyable solving experience.
Mathematician Fritz Juhnke developed a method to quantify this concept using graph theory. He focused on algebraic connectivity, which measures how well-connected a graph is. (Jump to the scary math section for details).
The results can be surprising, but if you examine the grids closely, you can usually see why certain patterns lead to higher or lower flow.
These calculations first appeared on Crosserville. The algorithm has been tweaked to better match solver intuition, and now Crosserville and XWord Info provide the same results.
Unlike Freshness Factor, Flow values are independent of previous values, day of the week, or grid size or shape.
Flow values for Modern Era crosswords range from 0.0 (puzzle has disconnected regions) to 219.2, with a median value of 31.8.
Some interesting results:
- Crosswords with the highest flow (list, thumbs)
- By day of week, say, Tuesday (list, thumbs)
- Crosswords with the lowest flow (list, thumbs)
- Constructors with the highest average flow
Scary Math
Here are the steps you can use at home to calculate your own grid flow:
- This grid has 140 words, so generate a 140x140 Laplacian matrix that completely describes how each word is connected to every other word.
- Next, calculate the eigenvalues for this matrix. The second-smallest eigenvalue, also called the Fiedler value, measures how well-connected the graph is. For this puzzle, it's 0.09229633.
- Then, multiply that by the number of checked squares in the grid, 371 here.
- The final result is 34.241939, and that's our flow value.