Folks often seem surprised to hear I didn’t get a Computer Science degree. Back in the late 80s, CS was still considered (at least at my school) mostly a math discipline and, despite apparent expectations, I am decidedly not a fan of advanced math. I handled this by combining two things I do love (CS and psychology) into a custom degree. I’ve (almost) never needed the math, and psych has proved useful again and again. Well played!
Still, it kind of grates at me when I know something is out there that others “get” that I don’t. A sampling of my (copious) kryptonite: logic puzzles, think-ahead games like Go and Chess, 2D drawing and 3D sculpting, playing both hands on a piano. There are some obvious commonalities in that list, like maybe somebody in the nursery poked their thumb into a very specific part of my brain. I guess we’ll never know, will we, Nurse Brenda?
Sudoku December
Anyways, a couple of months ago I got to catch up with Thomas Snyder, a guy I was lucky to work with back at Adaptive Biotech. One of the smartest people I know, Thomas is a three-time world Sudoku champion and recently started a gig building puzzles for LinkedIn. The LinkedIn puzzles fill a great niche; quick but entertaining — I run though Zip, Mini Sudoku, Tango and Queens most mornings before getting up.
Inevitably the conversation turned to Sudoku, and in particular my lament that I’m “just not good at it.” While acknowledging that he sees patterns more easily than most, he also implied (not his words, he’s more polite than this) that perhaps I was just being a whiny little baby. Practice is a powerful thing, and I resolved to spend a few weeks trying to knock the Sudoku monkey off of my back.
If you’re one of my few regular readers, you may remember a similar experiment I did a few years ago with the NYT Crossword Puzzle. In that case, I actually became pretty proficient; let’s see what happened with this one.
The Basics
Most folks know the basic rules of Sudoku. Each digit one through nine must appear in each row, column and 3×3 box in a 9×9 grid. The easiest puzzles can be solved entirely or almost entirely by looking for “Unique Candidates” — groups where there is only one open place for a number, like the red three in the puzzle below. There must be a three in the bottom-left 3×3 box, and every other cell is either already filled or is blocked by the presence of an existing three. Simple enough.
Solving Strategies
Of course, those puzzles get boring fast — the answers are just too obvious. The next step up is what the NYT and other newspapers publish as “Medium” or “Hard.” These require the identification of more subtle patterns that are more difficult to catch by eye. A couple of simple examples:
The existing yellow six below, together with the full column in the red box, means that there are only two places for a six in the bottom-left box (blue shading). We don’t know which one, but we DO know that this excludes a six from in the left column of the left-middle box. Together these eliminate all cells but one, so we can place the green six. Nice!
Most advanced techniques require notations indicating the “candidates” that could possibly appear in each cell. What I’ve added below is “full notation,” which I’ll talk about more later. There are a number of more abbreviated “notation” styles, including a popular one invented by Thomas called Snyder Notation.
In this puzzle, the only numbers that can go in the red box are eight and six. This is called a “naked pair,” which helps us eliminate candidates from all the other cells in its row — none of those (circled in red) can be eight or six. Removing those leaves us with only a seven in the yellow cell. And bonus, by placing the seven we know the one right above it must be a six. Progress!
There are dozens of these strategies, increasingly complex and with great names like “Swordfish,” “Finned X-Wing” and “BUG +1” (check out a big list here). The more esoteric are only needed for seriously difficult puzzles, which are beyond what is considered “Newspaper Hard.”
My Results
I decided to use the NYT as my testing ground; they release Medium and Hard puzzles each day. I did these pretty much every morning through December and early January. using their online version because adding and removing notation is easier that way. I did not use “auto” candidate or other helpers that felt like cheating (one caveat to this I’ll explain below).
My initial strategy was to work in three passes:
- Fill in the unique candidates.
- Use Snyder Notation to identify pairs and other basic patterns.
- Add full notation and sweat it out.
Two things seemed to be working against me. First, I would just straight up make mistakes — typically missing things in visual scans. It’s weird to think that in such a bounded puzzle I could miss things, but it happened a lot. And unfortunately, Sudoku errors don’t generally reveal themselves until you’ve gone further down the road, and winding them back is really hard and frustrating.
This is where I took advantage of one “helper” feature that feels a bit cheaty — the “Check Puzzle” option just highlights errors, and from there I could “undo” back until the board was clean again. As I’ve gotten more proficient I do this less and less, but I think I may have just quit in the early stages without it.
The second issue has proven more difficult to practice away — I just can’t keep a bunch of arbitrary things in my head at once. Great solvers can see dependencies and patterns with little or no notation, for example “forcing chains” that start with an assumption and follow it along a path from cell to cell. By the time I get to the third element in a chain I have absolutely forgotten the assumption from the first one.
The only way I was able to get beyond this was by writing it all down. Since I found that for most puzzles I always ended up at full notation anyways, I started doing that first thing — fill it all in, 3-5 minutes of busywork, and go from there. This was a turning point — not only did full notation give me anchor points to start complicated patterns, it made others just leap off the page. For example, I get a ton of mileage out of naked or hidden triples and quads, but seeing those without notation takes a level of visualization I will simply never possess.
Except Not Quite
At this point I can consistently solve Newspaper Medium/Hard puzzles in 15-25 minutes, and my kit of strategies is such that I rarely feel “stuck” for more than a couple of minutes. They’re fun to do and I’ve continued to play. This is lightyears beyond where I started, so that’s cool. BUT.
First, entering full notation for the puzzle at the beginning is super-annoying busywork. It’s totally mechanical, but it just takes time — so even if it wasn’t boring, it’s a built-in handicap as to how fast I can solve compared to folks that use more abridged notation. Many of the online tools have “auto” modes where the candidates are managed for you, dynamically updating as you put in solves, but that absolutely feels like a cheat. I’d just like an initial autofill that I can then work manually. Easy feature.
The second is more problematic. I keep talking about “Newspaper” puzzles — unlike in the crossword case, the NYT Hard Sudoku is in no way considered the pinnacle of the form. There are many much more difficult puzzles out there, and that’s where the “real” Sudoku aficionados live.
I’ve done enough to prove to myself that if I really spent the time, I could probably at least get “ok” at these, but there’s a built-in arbitrary-ness that I’m struggling to get past. Sure, crosswords may be easier or harder, but that scale is less black and white (ha). Very occasionally I just won’t have the vocabulary (or opera knowledge) to fill in that last square, but somehow that’s OK. When I start a Sudoku but get stuck because it (invisibly) requires that one specific strategy I don’t know — it feels like a waste.
Where to Next?
Would I find it more engaging if, for example, the puzzle could tell me the “minimum” strategy required to take the next step? Maybe, but I’m not sure if that’s even feasible.
But that has me thinking about puzzle design in general — I’ve only been a casual consumer of this stuff, but there is actually part-art-part-science hiding under the covers. Of course Thomas pops up when I start looking for good books on this, but I’m going to start with something a bit broader: A Theory of Fun for Game Design. I swear this world is just full of the coolest stuff ever, always something new to learn. More to come!