## Friday, January 16, 2009

### KenKen

Every night before we go to bed, Andrea and I wind down by doing a new Japanese puzzle called KenKen. You may be familiar with Sudoku, and KenKen is similar, but better. It was invented by a Japanese math teacher and actually involves math (unlike Sudoku, which could just as easily use the letters A through I as the digits 1 through 9). Anyway, here's a sample KenKen grid.

Each row and each column has to contain the digits 1 through 6. So far, sounds just like Sudoku. The twist is this: each area bounded in bold (sometimes called a "cage") has to contain digits that give the numerical result in the upper left hand corner using the operation indicated (addition, subtraction, multiplication or division). That sounds complicated, but it's actually not. Look at this cage:

Because "3-" is in the upper left hand corner, the two digits that go in this cage have to fit in the following formula:

___ - ___ = 3

In other words, it gives you the operation and the result, and you have to provide the "operands," to use a math geek term. In this case, there are 3 possible pairs (6,3) (5,2) and (4,1):

6 - 3 = 3
5 - 2 = 3
4 - 1 = 3

The order doesn't matter. Sometimes a cage has only one square, and no operation indicated, like this:

All this means is that they are providing a "gimme," and you just fill in the square with the number shown, like this:

Why don't they just fill in the square for you? Beats me. Anyhow, you have to use logic to eliminate possibilities and fill in the whole grid. For instance, consider this cage:

You know that the two digits have to satisfy the formula

___ ÷ ___ = 3

The only possibilities (using digits 1 through 6) are

6 ÷ 2 = 3
3 ÷ 1 = 3

But because the digit 3 appears in the same column as the cage, you know the cage can't contain another 3 (remember each row and column must contain the digits 1 through 6). So the two digits in the cage have to be 6 and 2. Since order doesn't matter, it could look like this:

or like this:

You don't know yet which of those is correct, so you can't fill in the squares definitively yet. But knowing that those are the possibilities helps you with the cage right below that:

This is the one I talked about before that had three possibilities:

6 - 3 = 3
5 - 2 = 3
4 - 1 = 3

But now you can eliminate the first two of those possibilities, and you know that the cage has to contain 4 and 1 (in either order). Now look at the entire column.

You know the top square in the column contains 3, the upper cage contains 6 and 2, and the lower cage contains 4 and 1, so the digit in the bottom square has to be 5 (again, to satisfy the rule that each column and row must contain the digits 1 through 6). So you can definitively fill in that square with a 5, like so:

Now you're on your way to solving the whole puzzle, because the other number in the cage with the 5 can only be 1.

5 - 1 = 4

So, you'd like to try one for yourself? Andrea and I have two sources for our puzzles. The one I've used here comes from the New York Times puzzle page. They have interactive KenKens at differing skill levels. You can even try a 4 x 4 one to get the hang of it. We also do the daily puzzle from the UK Times online, which we just print out and do the old-fashioned way--with pencils.

Kristin said...

I do KenKen every night too! I've just been using a will shortz book I got in my stocking, but I'm almost through so now I know where to get more. thanks!

Melinda said...

You'd be a good math teacher, Dan.

LL said...

Oooo.... Games for smart people! I like it. If I could figure out where my brains went I'd give it a try. Or maybe by trying I'll come across my missing brains...

Leslie said...

I'm a big Sudoku geek, so I am excited to try this...

Janie said...

Oooo, this sounds like something I could waste lots of time doing. Andrea keeps talking about it, thanks for the step by step directions. I get it now!

Michelle and Byron said...

Dan, way to tempt Byron into another addiction. We might have to stay away from your blog altogether.