The latest statistic captured by XWord Info is called “Flow”. It’s intended to quantify how well the grid shape (the arrangement of black squares) contributes to the solving experience.
I asked the inventor of this metric, Fritz Juhnke, to tell its origin story. The rest of this post is his response.
Thinking about flow
One day in the summer of 2023, my wife, Katie Hoody, started making crosswords, which became a fun adventure for me as well as for her. I eventually dipped a toe in the water as a constructor too, but from the start I distracted myself with related mathematical questions, engaging with numbers instead of words whenever possible. For example, when Katie switched to mostly themeless construction, I delighted in generating grids with fun geometrical properties for her to rack her brain filling and cluing.
The grid for Katie’s 7/12/25 crossword reflected my particular interest in connectivity. Pick any two open squares of that puzzle, even in opposite quadrants, and there will be four separate paths between those two squares. The idea of counting up non-intersecting paths is half of the classical definition of connectivity according to graph theory. The other half is to use the lowest number of separate paths among all possible starting and ending points; only the worst constriction counts. If there are no paths between some pair, the connectivity of the whole is zero, i.e. the graph is disconnected.
Some initial stumbles
The trouble with applying the classical definition to crosswords is that there are only two ways out of any corner square, so there can be at most two separate paths between opposite corners. The connectivity of a crossword is either 0, 1, or 2, with no bonus for more paths in the interior. This way of counting doesn’t recognize the special feature I admired about the 7/12/25 grid. I didn’t feel seen.
I hit upon the idea of calculating the average number of separate paths between all pairs of squares, instead of the minimum. I floated this way of measuring to Jim Horne, but he was simultaneously underwhelmed (with my metric) and overwhelmed (with what else was going on at XWord Info). I realized that to persuade anyone what a brilliant idea I had, I needed to be able to quickly calculate the average number of paths within an arbitrary grid, so that I could show by example what superb numbers popped out of my formula. But other interests intervened, and it was over a year later that I finally turned to the #code-chat channel in Crosscord for help with the programming. There I confessed that my central objective was to impress Jesse Goldberg with my connectivity metric so that he would add it to Crosserville, my favorite crossword construction software.
Little did I know that Jesse was lurking in #code-chat. He seemed intrigued, but pressed me for justification, and it wasn’t long before I doubted the awesomeness of averaging the number of separate paths. In particular, it began to seem dubious to measure how squares are connected instead of how slots are connected. If someone hands you one letter of a crossword for free, it doesn’t just give you a hint about the four adjacent letters: it gives you a hint about the entire entry in both directions, no matter how far those entries extend. Solutions don’t just ooze from one square to the next; they can leap the entire length of a slot.
Concurrently, I scoured Wikipedia for other possible ways to measure the connectivity of a graph. I was reminded of a second classical metric, which corresponds to counting the minimum number of squares that need to be blacked out to disconnect a crossword. Patrick Berry touted this method in his book, saying (essentially) that a crossword is weakly connected if one block could disconnect it. Unfortunately, this metric also produces scores of only 0, 1, or 2, because blacking out the two squares adjacent to a corner square always disconnects that corner. Grids that aren’t disconnected (0) or “weakly connected” (1), all rate a 2; there is no bonus for, say, interior pools of open squares.
The breakthrough
Much to my surprise, however, I stumbled on a modern metric that was introduced by Miroslav Fiedler in 1973. He called it “algebraic connectivity”; most present-day practitioners call it the “Fiedler value” in his honor. My Ph.D. is in numerical differential equations, so I know relatively little about graph theory, but Fiedler’s calculation of connectivity is shockingly reminiscent of numerical simulation of heat flow, something with which I am more intimately familiar.
It dawned on me by degrees that the Fiedler value is an exact answer to this differential equations homework problem: Suppose every slot of a crossword is a superconducting rod of uniform temperature. Randomly assign a different initial temperature to each rod. Suppose that wherever two slots intersect (i.e. two rods touch each other), heat flows from the warmer rod to the colder rod at a rate proportional to the difference in temperature. The temperature distribution will exponentially decay towards all rods having the same temperature. What is the asymptotic rate of that exponential decay? (Yep, it’s the Fiedler value; students please show all steps for full marks!)
It is linguistically serendipitous that we speak not only of the connectivity of a crossword but also the “flow” of a solving experience, and those two concepts are related in vaguely the same way that the algebraic connectivity of a graph is related to a hypothetical heat flow. In order to experience a whoosh of one answer leading to the next to the next in a crossword, the grid must be well-connected, just as for heat to whoosh through a lattice of superconducting rods, that lattice must be well-connected.
Does the parallel in vocabulary illuminate anything? Jesse was willing to give it a try via displaying the Fiedler value as a new feature of Crosserville. He quickly noticed, however, that 21-by-21 puzzles had lower Fiedler values on average than 15-by-15 puzzles. Jesse thought we should just live with multiple different scales, similar to how we interpret word count differently on a Sunday than the rest of the week. I, in contrast, was keen on having a unified scale to make all puzzles comparable, and Jesse graciously let me have my way. He multiplied the Fiedler value by the length and width of the puzzle and we dubbed the product “Grid Flow”.
Tweaking the algorithm
After some months of Grid Flow appearing to be consistently useful, I came back to Jim with a far less flawed metric than the “brilliant” one I had proposed nearly two years prior. Furthermore, I could now use Crosserville’s calculations to fire off convincing real-life examples. Jim was rather more enthusiastic than on the previous go-around. However, before promulgating Grid Flow we made a small technical tweak. A two-dimensional physics simulation has a length scale and a width scale no matter what shape it is, but a crossword has a length and a width only because we shoehorn it into a rectangle. We switched from length times width to counting the number of checked squares regardless of their overall layout and shape. Jesse was amenable to the change, so the Grid Flow numbers in Crosserville match those on XWord Info, and hopefully additional places in the future.
To my eyes, Grid Flow appears highly relevant to crossword solving experience. If to your eyes Grid Flow appears too high or too low on a particular grid, then you may be counting paths or seeing how few blocks could disconnect the grid. Which is to say, you may be evaluating square-by-square while Grid Flow is evaluating slot-by-slot. Which of the two perspectives relates more strongly to solving experience is not obvious a priori; only time will tell.
High and Low flow extremes
A test case is the 12/21/2001 puzzle by William I. Johnston with a stratospheric Grid Flow of 133. Critics will highlight all the choke points; each corner can be cut off by a single block. I would counter that if you have solved that entire puzzle except for, say, the northeast corner, you are looking at {Simply accept} for ____ATFACEVALUE and {Where people take a crash course?} for DEMOLITIOND___. That doesn’t just give you the next adjacent letter E in the choke point; you get all of TAKE and the whole dang ERBY. A whopping six hints for the corner is, I claim, excellent flow. What matters is not just the number of ways to get between otherwise isolated regions; it’s also how far the straddling entries extend on either side of the divide.
At the other end of the scale is the 2/14/2004 puzzle by Frank Longo with an abysmal Grid Flow of 8. Each of its three regions has nearly perfect internal connectivity. One could argue from a square-based perspective that the choke points are no worse than the previous example, because in each instance a single block disconnects the grid. They’re both weakly connected per Berry. But here one could figure out the entire middle region, then solve _ENTAP_ from the clue {Flipped}, and still be stuck with only a single letter to get started on each of the other two regions.
These extreme cases highlight that Grid Flow is measuring something new and different. The 2/14/2004 grid is better on all prior XWord Info metrics: more grid-spanners (9 to 8), fewer words (62 to 70), more open squares (125 to 85), and greater average word length (6.29 to 5.40). Those are all interesting and useful comparisons, but none reveal that the puzzle is very close to being three separate puzzles that happen to be adjacent. A slot-based perspective reveals that the 12/21/2001 grid is far more interlocking. These examples also illustrate a general trend: marquee entries that cross each other contribute more to Grid Flow than stacked marquee entries.
The grid for Katie’s 7/12/25 puzzle, which I loved for its complete absence of choke points, also lacks crossing marquee entries, so it tips the Grid Flow scale at a modest 35. Does this correspond to solving experience? I had the opportunity to check by watching multiple YouTube videos from solvers who daily record themselves tackling the NYT crossword. Yes, there were some instances of a solution in one quadrant pushing through the center to give hints on another quadrant, but most often new quadrants were broken into by directly laying down answers with no hints, exactly as one is required to do in a poorly-connected grid. Multiple central paths exist, to be sure, but they fight through a flurry of four-letter entries, which isn’t fantastic for flow.
The future of flow
I expect that ultimately the right way to measure the flow of a particular crossword puzzle depends more on the entries and clues than on the configuration of the grid. In the differential equations homework problem, each superconducting rod had a uniform temperature: one couldn’t be hot on the left side and cold on the right side. Is figuring out a crossword entry all-or-nothing in the same way? Not necessarily. In practice the clue for a grid-spanner might let you locally parlay a couple of letters into _________DESERT without yielding the entirety of THESAHARADESERT. In that instance a square-based perspective might trump a slot-based perspective. Similarly the clue {What is on my bedroom walls} wouldn’t shoot the solution from _____PAINT all the way across the slot. You would have to get GREEN from the local crossings, in which case square-based flow would again be the better way to look at it.
Is Grid Flow relevant to your actual experience of crossword-solving flow? I’m very curious to know. Please, email me with your stories at yangfuli@yahoo.com. I’d love to talk about it.