KenKen: A Simple Puzzle That Goes Deep

In the conclusions I wrote for Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, I mentioned KenKen*. Our copy editor asked what it was. As I searched for a good reference to put in a footnote, a memory began to surface of an article I read long ago. Memory is funny. It turns out I had read the article over four years ago, and I still remembered a particular word from it - midnight. That led me to the article I had read, but it's unfortunately behind a paywall. Luckily, I had downloaded it the first time I read it, and finally found it on my own computer.

I had saved it back then because I was using KenKen with the kids at Wildcat Community FreeSchool. The puzzles I was sharing with the kids seem pretty easy, but to solve them requires holding addition facts in your head while also thinking logically about the relationships. This is a good way to deepen your hold on those facts. I wanted the parents to understand how valuable this simple puzzle was, so I copied the article.

'Midnight' was in the first paragraph, in the dramatic opening of the story written by Leo Lewis for The Times of London:

At one minute to midnight every September 30, the decrepit, cluttered schoolroom of Tetsuya Miyamoto stands frozen in time. Breaking the sepulchral silence of the Yokohama side street, the clock ticks over into the first day of October and a fax machine in the corner shudders to life. 

Throughout the rest of the night, page after page spews out of the machine, each one representing a different seven-year-old child, each one an application form pregnant with parental hopes and fears. 

The class these parents so desperately wanted their kids in consisted of puzzle-solving sessions. Tetsuya Miyamoto provided the KenKen Puzzles he had invented, and the children would then work alone for 40 minutes on up to three puzzles. The first is a 4 by 4 grid, the next is a harder 5 by 5 grid, and the third one is harder yet. After they've worked on the puzzles alone, the group works together on a puzzle Tetsuya Miyamoto puts on the board. He calls on a student for a number, then says right or wrong. That's it (according to Leo).

Before I go any further, I'd better share a KenKen puzzle with you.  Here's today's puzzle from the New York Times. If you like it, go there for more puzzles. Since this puzzle is 4 by 4, each row and column will have the numbers 1 to 4 in it. The clue on the middle top, 6X, means that the two numbers for that outlined box must multiply to 6. We know we can't use 1x6, because 6 won't be used in this puzzle. Is that enough to get you started?

Tetsuya Miyamoto designed his KenKen puzzles to draw students in and get them sweating:

Every puzzle, says Mr Miyamoto, contains a “trick, a discovery – a story”. The puzzle works in his classroom, he says, only because the children want to root out the clues and persevere with the discovery process. “As the feeling of achievement increases, so too does the level of concentration,” he says.
...
By combining the four main mathematical functions of addition, subtraction, multiplication and division, the brain is forced to dart between competing theories. The puzzle, he says, is impossible to solve without the scientific process of trial and error. 
...
The puzzle, Mr Miyamoto says, draws out the primal, self-starting learning instinct of human beings – an instinct that is notoriously suppressed by the fact-cramming teaching methods of the Japanese education system, but which he says needs to be encouraged in people of all backgrounds.

I think U.S. schools are headed toward that same sort of fact-cramming. It was never a good idea, but it seems clear to me that we need particular facts less than ever with the internet at our side. What we need are understandings of how it all fits together.

Mr Miyamoto’s theory is that the brain – of a child or adult – is failed by conventional teaching. By concentrating on a “third way” of problem-solving, he believes that the mind becomes a more potent tool for dealing with the rest of life...

I wish I knew what second way is implied here. I'm assuming fact-cramming is the first way.

For both children and adults, runs Mr Miyamoto’s theory, the brain feeds on what it has worked out for itself rather than what it has been told to focus on.

This important idea has been stated many ways by many excellent teachers. I am reminded of the quote the Kaplan's use to define their math circle philosophy:

"What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises."    --G. C. Lichtenberg

There are other ways to make arithmetic challenging and appealing, some of which you'll find in Playing With Math. But KenKen is a particularly easy one to bring into your life. Enjoy!




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*The name KenKen is trademarked. Because of this, you can also find these puzzles under other names. Calcudoku seems to be the most common.