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.)

  1. Some kakuro lean fully into the crossword format and separate the clues into Across and Down text. Those kakuro are silly.
  2. Or, at least, Git Reesunable.

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