Sunday, November 27, 2016

The Allure of KENKEN

This is KENKEN, a puzzle that piggybacked on Sudoku's popularity over a decade ago. I personally fizzled out on Sudoku pretty early on when I heard about editors merely deleting some of the numbers off the opening Sudoku puzzles to make them harder (never mind that the puzzles had been designed to be solved through logic alone). Disillusioned with a puzzle that could so easily be tampered with, I turned to KENKEN.

KENKEN, like Sudoku, requires the same numbers to be entered into every horizontal row and vertical column. In this particular puzzle, because the grid is 6x6, every row and column require the numbers 1 through 6 exactly once each.

What makes KENKEN different is that it incorporates basic math. Every bold shape (usually a rectangle or square, though sometimes a funky looking piece) has either a number alone or a number and an arithmetic symbol. The squares that have single numbers have no puzzle attached and can be filled in, as below:

After these giveaways, a solver must look to the other bold shapes, or cages, to solve the puzzle. Most cages have a number and symbol which is the clue to that cage's numbers. For example, the long rectangle next to the 1 in the upper left corner has "30 x" in it. That means the three numbers in that cage, when multiplied, equal 30. Two sets of numbers makes this true: 6, 5, 1 and 5, 3, 2. Since 1 is represented in that row, the set 6, 5, 1 is eliminated, leaving the other. However, the solver doesn't know what order to put those numbers in, so the puzzling continues. 

Like Sudoku, you have to sort out exactly what you can logically deduce from the information given, fill in a square or two, then repeat. Do this enough times, and you solve the puzzle.

The reason I linked to this particular puzzle is because it beautifully illustrated what I like about KENKEN. Even though these examples are time-stamped within a few minutes of starting, I actually had to spend nearly half an hour to make my first move!

I first went around the grid, trying to learn what I could. After figuring out part of the "30 x" cage, I knew the upper right corner had the numbers 6, 4 because they were the last two numbers for the row AND the equation made sense (6 - 4 = 2). But I still couldn't write anything. I moved on wondering about all the "2 ÷"s and feeling a little hopeless about them because they could all be solved with 2, 1 or 4, 2 or 6, 3. They represented too many variables to sort out. I looked at the "10 +" and the other "10 +". The second one can be problematic because it wraps around a corner, just like the "15 +" below. Because of this wraparound, they obey slightly different rules. These corner cages can have the same number in them so long as those numbers don't fall in the same row or column. In fact, several of my first KENKEN puzzles were ruined because I didn't understand this fact. The corner "10 +" could be 3, 4, 3, as long as the 3s didn't touch. Anyway, I looked at all these potential avenues for breaking into the puzzle, and though I collected bits of information, it wasn't enough to write anything down.

Finally, I had a breakthrough.

One thing I had overlooked was that certain cages COULDN'T have certain numbers in them. For example, all of the "2 ÷"s could never have a 5 in them. Once I figured this out, then I was able to bounce to the row with the 4 in it. Since it had two cages with "2 ÷", I knew that the 5 had to be in the "3 -" cage. This meant that the two numbers were 5, 2 (because 5 - 2 = 3), though I didn't know the order. That in turn meant that the rest of the row had to be filled in with 1, 3, 6. Only two of these numbers (6, 3) could fit into the far right "2 ÷" cage and make sense in an equation.

So after all of that, I knew definitively that the 1 had to go in the yellow square below. My first written square after the freebies!

Yes, that one square took me about half an hour. If I did these puzzles more regularly, I'm sure I would have found it quicker, but I'm out of practice. After the 1, I could fill in the square below it with 2 since that's the only equation that can work (2 ÷ 1 = 2).

Then it was back to the drawing board. Once again, you have to analyze the cages and figure out what information is there. With the "15 +" cage, I knew that it had to have a 6 in it. No combination of numbers adds up to 15 without a 6. But where did the 6 go?

If you'll recall, the top right cage has a 6 in it, and the cage to the right of the 4 has a 6 in it, so the 6s in both of those columns are spoken for, so to speak. You don't know which square they belong to, but you can black out those columns. And for our purposes, that's enough information!

With one square left in the "15 +" cage, we know definitely that the 6 goes in the yellow square below:

This is your introduction to KENKEN. I love it and I hope you enjoyed your introduction to it. How it fits in this blog is perhaps somewhat puzzling, but I leave it to you to figure it out. Any ideas?

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