Phil’s Puzzle Primer: Kakuro

As discussed before, I’ve met several people who have been scared away from Sudoku due to the numbers. “I’m bad at math,” they say. “I just don’t have a head for it.” And I usually explain to them that, no, Sudoku doesn’t really use math, at least not in the sense that they mean, but in the back of my mind I’m thinking: don’t touch any kakuro, then.

A sample kakuro puzzle from Wikipedia. (Licensed CC-BY-SA by Octahedron80.)

Kakuro are sometimes referred to as “math crosswords.” That’s a name that better fits Figure Logics, a puzzle type I’ve never quite wrapped my head around, but it’s a reasonable stab at describing these gridded challenges as well. The rules of kakuro are very simple, if (almost necessarily) more complex than sudoku:

  • Each row and column sums up to the number on its top or left edge1,
  • Cells contain only the digits one through nine, and
  • A digit cannot repeat in a given row or column.

It’s that last rule that makes kakuro such an interesting design. For example, a clue of 4, which in a well-formed kakuro is always clued by two boxes rather than just one–they almost universally follow “American-style” crossword construction rather than Japanese- or British-style, with their looser cross-hatching–is always a 1 and a 3, because two 2s make for an invalid puzzle. If a 3 and a 4 cross each other, the only shared digit is 1, which gives you an easy “break in” to the puzzle.

And let me not sell this short: while all you have to do is a bunch of addition, you do a lot of addition when solving a kakuro. The math is relentless. The smallest number that fits in a four-cell row is 10 (1 2 3 4) and the largest is 30 (6 7 8 9), and there are sums other than the obvious ones that have hard-set requirements; 7-in-3 is always 1 2 4, for example, which is something you’ll have drilled into your head after you’ve solved a half-dozen of these whether you like it or not. I still count on my fingers sometimes to double-check my math, and I count on my fingers a lot while solving kakuro.

The solution to the example kakuro. Also from Wikipedia, of course. (Licensed CC-BY-SA by Octahedron80.)

Like sudoku, kakuro originated in the pages of Dell’s puzzle magazines here in the US, and like sudoku it blew up in Japan well before it became anything like a phenomenon in the rest of the world. That said, while kakuro are reasonably popular here and elsewhere, they’ve never reached anything like the same level of popularity as sudoku, and honestly they never will. Their fundamental reliance on math will always make them a little too scary for a large segment of the population, which is definitely a shame.

Kakuro’s format doesn’t lend itself to quite as much in the way of variation as sudoku, although there are definitely several tweaked versions available. I’ve seen ones that use multiplication rather than addition as the core operation–lots of big numbers!–and some that have rectangles drawn over portions of the puzzle that act like the rows and columns, their contents summing to a particular value with no repeated digits. But I suspect the format inherently lends itself to fewer wacky alternate takes than sudoku, never mind the fact that it’s a less-popular type and so inherently has less incentive to come up with interesting tweaks to the formula.

As for me? I’ve been seeing kakuro for decades, thanks to my early fascination with puzzles–they were called Cross Sums back then, although nowadays Dell fully embraces the kakuro name–and they always intimidated the heck out of me, even though I’m actually pretty good at math. Hey, look, I was part of the problem! I admit it. But about a year ago I finally buckled down and decided to Git Gud2 at them, and now I actually quite enjoy the type. It’s not my favorite, to be sure; I’m not always in the mood to show the world (or at least the ants in my living room) that I still have to count on my fingers whilst verging into middle age, and they can feel a little same-y, so even when I am in the mood I often intersperse them with other puzzle types. But there’s a good reason they’re one of the most popular puzzle types in Japan, and they’re definitely worth trying out.

Even if you don’t like math.

Getting started with kakuro

The most important part about getting going with kakuro is understanding some simple, but vital, solving techniques. The puzzle type simply isn’t as intuitive as sudoku, and so it’s a bit harder to break into. Fortunately there are plenty of excellent tutorials and guides online. I’ll point you to Krazydad’s superb step-by-step solving guide, which covers both basic and advanced techniques, but a quick Google search will turn up many, many options, including interactive formats if that’s your style.

One of the things you’ll want to have a copy of, at least until you internalize most of its contents, is the “forced number” chart that you’ll see in Krazydad’s guide (and most of the others as well). You’ll quickly remember off the top of your head that 29-in-4 has to be 5 7 8 9, but getting to that point takes some practice, and having that chart will make it easier in the early going.

Krazydad’s site also has a bunch of kakuro of all sizes to try. Here’s a link to a nice web-based implementation he wrote, starting with the smallest puzzles. You can of course find a whole bunch of other sites full of computer-generated puzzles online, at pretty much any size and difficulty you like.

Getting good kakuro

As is the case with sudoku, the best hand-made kakuro come from Japan. Nikoli has a long history of publishing the puzzle type, and they have tons of volumes of Kakuro that you can pick up off of their website or from I’m also partial to 頭脳全開足し算クロス, a magazine you can snag from Amazon Japan that contains nothing but kakuro. It recently started including several of the “rectangle restriction” variation puzzles that I mentioned earlier, and in the back has a pair of puzzles on an enormous sheet of newsprint that will keep you busy for weeks, but the opening puzzles are easy enough for even a beginning kakuro solver to tackle.

That said, I definitely get less of a “feel” in terms of difference between hand-made and computer-generated kakuro than I do with sudoku. Maybe it’s because I’ve solved less, or maybe it’s inherent to the format, but in any event there are plenty of books and websites chock-full with more kakuro. Amazon has the “Martial Arts” series of books, Djape puts out a whole bunch, I’ve already linked Krazydad’s site… you should be able to find something that suits your timeframe and level of competence without too much effort. (Sadly, Simon Tatham’s Portable Puzzle Collection doesn’t have kakuro in it.)

Phil’s Puzzle Primer: Sudoku

Sudoku gets a bad rap.

On the face of it, that might seem like an absurd thing to say.  Crosswords are the only type of pencil puzzle with more widespread appeal than sudoku, and they’ve been around a hell of a lot longer.  (Word searches are certainly a thing, but most serious puzzle people don’t take them very, well, seriously, for reasons we may explore some day.)  But in the same way that I was once told by someone I otherwise respected that they had “moved on” from things like ’80s pop music to more serious auditory pursuits, it’s easy to find people who–while they may have, once, seen the appeal–find no interest in the 9×9 grid of numbers that comprise this most common of logic puzzles.

Like Mr. Too-Good-For-Depeche-Mode, they are wrong.

Anyone surprised that the first image-heavy post on my blog (and the second to ever have images at all) happens to be about puzzles doesn’t know me very well.

Sudoku looks like it’s about numbers, but it really isn’t.  I had a discussion with my mail carrier a week or so ago; I had ordered some sudoku magazines that shipped in clear plastic wrap, so you could see the cover, and she admitted to me that she had never done a sudoku, had never even tried, because “it has something to do with math?”  You know exactly the face that she was making while she said that, too.  So much of our population is genuinely intimidated by math, which is a damn shame, but that’s yet another subject.  Anyhow, I explained to her how it actually works, and she lit up.  “I understand how it works now!”  She said that with the sort of shocked pleasure usually reserved for finding a forgotten fifty-dollar bill in a jacket pocket.

So, just in case, here: To solve a sudoku, you must put the digits 1-9 in the empty cells of the grid, such that each row, each column, and each 3×3 box marked out with the thick black lines has each digit exactly once.

That’s it.  That’s all there is to sudoku.

Of course, that’s not even remotely true.  Sudoku can be trivially simple, an exercise of writing in numbers as quick as you see the missing values, or it can be extremely difficult, requiring convoluted logic (and, in the worst cases, just flat-out brute force “let’s try everything” methods) to make even the slightest bit of progress.  And that’s not counting the roughly 1.21 giga-variations of the puzzle.  Those range from the most basic (a 6×6 grid, say, or a 16×16 ones) to bewildering mashups of multiple complicated rulesets.  Interestingly, while vanilla sudoku make no use of the fact that the symbols are the digits 1 through 9, many variations actually rely on that fact… bringing math back into the whole thing.

This is far from the most complicated multi-variant sudoku I’ve ever seen, but it’s pleasantly indicative of the format.  It was also made before some terms came to dominate the language around the puzzle.  That top-right one would now be called a “jigsaw,” for example.

If there are tragedies behind the worldwide explosion in popularity of the puzzle type, they have nothing to do with the form of the puzzle itself, which can be infinitely fascinating even in its most-basic form.  The first and ultimate tragedy is that the creator of the puzzle type, a Mr. Howard Garns, passed away in 1989 before the craze became multi-national.  He would no doubt have been delighted at the proliferation of the puzzle on bookshelves and in magazines around the world.

The second tragedy is more insidious, and is part and parcel of many modern puzzle types: most of the sudoku produced in the world are computer-generated, and it shows.  They have boring solving paths, or ones that are too tightly constrained, requiring one specific deduction to make any progress at all.  And there is an enormous glut of these boring puzzles, crowding out the good ones from easy availability.  It’s always cheaper for a computer to churn out a thousand puzzles than to pay a human to make twenty, after all.

The very first sudoku (well, Number Place) puzzles, dating from 1979.  These are laughably easy if you’ve ever solved more than a couple of modern ones, but of course the format was brand new then.  Also note that even the first puzzle was actually a mild variant!

 Here’s a confession: I used to look down on sudoku.  Although I strive to be an anti-hipster in all of my tastes, I was–I say this with shame–a puzzle hipster.  Sudoku was the super-popular one; my personal favorites, like Slitherlink and Fillomino, went basically ignored.  So clearly they were superior; clearly sudoku was for weak-minded fools who couldn’t bother to learn how to do interesting puzzles.

Don’t worry.  I got better.

That said, it’s very common for people to like it and, well, no other puzzle types, really.  I personally think that that’s something of a shame–there are so many good puzzle types out there in the world–but if you’re going to pick a puzzle type to focus on, you could do much worse than sudoku.

Getting started with sudoku

If you’re totally intimidated by the format and you want a physical book, start with Djape’s Sudoku for Kids (Amazon).  They’re computer-generated, but they start out at 4×4 and get bigger from there.  There are also several variations in the book (even-odd and X, primarily), which is a good primer on some of the more common alternate sudoku types you’ll see in the wild.

Alternately, install Simon Tatham’s Portable Puzzle Collection (Google Play Store | Apple App Store | Desktop) and you can dial the puzzle difficulty however you like it, from trivial 4x4s to diabolical 16x16s.  They’re computer-generated, so will never be great, but they have the distinct benefit of being free.

Getting good sudoku

…is actually quite easy despite the enormous numbers of cookie-cutter computer generated puzzles out there in the market.

Nikoli’s Original Sudoku series is easily available in the US (Amazon) at any major book-seller.  They are the company that originally popularized the puzzle, first in their native Japan and then elsewhere, and all of their puzzles are hand-made.

If you’ve done a ton of vanilla sudoku and want to stretch out a bit, Djape’s Loco Sudoku (Amazon) and its sibling Cuckoo Sudoku are full of variety, including 5-puzzle “Samurai” grids like the one above.  I’m also particularly partial to Thomas Snyder and Wei-Hua Huang’s Tight Fit Sudoku (Amazon), which has the distinct advantage of being cheap and small enough to fit in a purse.

If you feel like you’ve tapped out the vein of regular sudoku for being too easy, well, have I got the puzzle magazine for you: 超難問ナンプレ&頭脳全開数理パズル (, which roughly translates to “Super Difficult Sudoku & Math Puzzles.”  The first puzzle in a typical issue starts at 4-star difficulty (out of five), and they go all the way to 6.  No, don’t ask me how that works.

The best overall sudoku-focused magazine that I know of is also Japanese; it’s ナンプレファン (, or Nanpure Fan; for reasons I don’t want to get into, sudoku is called “Nanpure” in Japan by everyone but Nikoli.  Each issue has a good mix of classic sudoku and a bunch of different variants, along with a dash of random other logic puzzles.