# Introducing KenKen puzzles

### From MathAndMusic

Logic puzzles are serious work, not just "dessert." The Law School Admission Test contains many logic puzzles of exactly the kind that appear in schools as rainy-afternoon work after students are finished with their "real" work. IQ tests and job-aptitude tests also frequently include puzzles that schools often treat as dessert. The puzzlesarefun, but are extremely valuable ingredients in the "main diet," not just the dessert of school.

"KenKen" puzzles, also called "MathDoku" puzzles, exercise elementary school arithmetic -- addition, subtraction, multiplication, and division -- in ways that giveexcellentpractice and also generate the thinking and problem-solving skills children will need for success in high stakes tests and later in algebra.

Materials:

Download transparency to use in introducing KenKen.- Download six 4x4 puzzles. Easy puzzles for beginners.
- Download six 6x6 puzzles. Low-medium level 6x6 puzzles, suitable for first introduction to 6x6 size.
Other sources of puzzles of this kind:

- Download twelve 4x4 puzzles. Mixed difficulty level, but all 4x4.
- Download six 6x6 puzzles. Medium difficulty 6x6 puzzles.
- Online "KenKen" puzzles. One puzzle per day, various sizes from 4x4 (generally easy) to 9x9 (always hard).
- "MathDoku" Online puzzles of the same kind. Many puzzles available every day. Can select difficulty level.

Introduce KenKen puzzles as a classroom cooperative puzzle.

This page is for your background, to help you prepare. It is not a lesson plan or a prescription for teaching. You'll find your own style, but keep the focus on having the children puzzle things out.

You have two very important roles as a teacher the first time you introduce these puzzles:Of course, you'll also have to help them learn the rules, and you'll want to model the "puzzling it out" process, but both of those can come as

- Your first task is mostly to be the "secretary," helping the children learn how to keep track of what they
doknow. You'll see examples of what this means below.- You also have the job of teaching them the
patience not to guessat what theydon'tknow. Again, examples below.partof playing the first time, because it's hard to understand and remember all the rulesbeforeyou start playing with the puzzles.

## Contents |

## Two good starter puzzles

## "Social solving"

*Introduce* the idea of solving the puzzle *with* another person, in a kind of turn-taking way. As you introduce the puzzles to the class, model the idea of looking around for a good place to start, or to go next, by scanning yourself for a place to go, and waiting a bit for student ideas. *Do* feel free to contribute ideas yourself -- after all, that's part of the turn-taking -- but not much, and keep to the simplest ideas, so that students have the chance to solve "harder" situations.

Some students may eventually prefer solving their own puzzles. Have a supply ready to play with.

## Rules for KenKen

It's hard to learn the rules all at once before interacting with the puzzle, so just the briefest introduction to the rules makes sense before diving in. Start with a 4x4 puzzle, like the one shown here.

- The only numbers you may write are 1, 2, 3, or 4. (A 6x6 puzzle requires 1 through 6.)
- No numbers may appear more than once in any row or column. (That is, all required numbers must appear in every row and column.)
- Each "cage" (region bounded by a heavy border) contains a "target number." If there's more than one cell in the cage, the target is also accompanied by an arithmetic operation. You must fill that cage with numbers that, using only the specified arithmetic operation, produce the target number. (Numbers
*may*be repeated within a cage, if necessary, as long as they do not repeat within a single row or column.) - In a one-cell cage, just write the target number in that cell.

## Figuring out where to start

You can download this puzzle to project on a smartboard (or overhead projector) to introduce KenKen puzzles.

**Special message #1: "Be lazy!"** Look for the *easiest* place to start!

Students who are not used to solving puzzles may not know that there is no special order for working through puzzles like these, no "rule" for it. Finding the easiest places to start is part of what *makes* this a puzzle.

If a puzzle has single-cell regions, as this one does, they are obviously the "easiest" places to start. That number is the goal, no operation is needed, so we just write the number.

Now what?

- Ask for ideas, but recognize that students are not yet likely to expect that solving a puzzle is about
*deduction*not guessing. For example, they might guess that 3 and 1 could go in the first two cells of the first row. This fits the rules -- the goal is to make 2 using subtraction, and 3 - 1 = 2 -- but so would three other pairs of numbers: (2, 4), (4, 2), and (1, 3). We don't yet know which is correct. - As students make suggestions, you might fairly regularly ask "How did you figure that out?" Alternatively, if students make suggestions that are arithmetically correct -- like suggesting (3, 1) in those top left two cells because 3-1=2 -- you might also ask "do you know that those
*must*be the numbers, or are you just saying that they*might*be?" This helps distinguish deduction from guessing.

## Modeling how to "puzzle" out a solution

After waiting long enough for students to have a chance, feel free to model "finding an easy place" by pointing to an "easy place" that *you* see. There are two good candidates in this puzzle.

We could start in either place.

Suppose we start with the "3,+" region. Ask students what might go in the two cells. (It *must* be filled with 1 and 2 -- no other pair of numbers adds up to 3.) When students give the numbers *point out* that we don't know which *order* to write the numbers.

**Special message #2: "Be bold!"** Even though we don't know everything, we do know *something* so we should write it down!

**Special message #3: "Don't guess!"** But, since we don't know what *order* they are in, we write them in a way that doesn't specify the order.

We *know* what numbers are there, so we can write.

Even this partial information is important. Pointing to the third cell in that row, ask students what they can now figure out about the number that goes in that cell. (We now know *for sure* that 3 goes there, because 1, 2, and 4 are already used up in that row.)

Alternatively, suppose we start with the "9,x" region. What *three* numbers can we multiply to make 9? Students may have had little experience with "products of three numbers," so this may not feel obvious. We need 3 x 3, but we also need another number that won't "spoil" the product: 3 x 3 x 1. Do we know where to write them? Well, the 3s can't both go in the same row or column, so.

What can we do next?

*Now* we might look at the "3,+" region. What can go there? (Just 1 and 2.) Do we know what order to write them? (Yes! The 1 can't go in the same column with the other 1.)

Now, what can we fill in? (Just by looking for what's missing, we can completely fill the first two columns and the second row.

And where might we go next? By this point, students may well have ideas about how to proceed.

## Puzzles on their own

After children really "know how," let them play on their own. Give "starter" (easy) KenKen puzzles to pairs of students. You might choose to give the same puzzle to each pair, or have a few different ones. The only purpose is to let them try on their own what the class just did together as a group.

These puzzles should be a *regular* feature of your classroom, so that students see that *you* consider puzzles "serious work" (even though they're fun!).

You might use them officially two or even three times a week. A small collection of starter puzzles -- easy 4x4 puzzles for beginners, and easy 6x6 puzzles for after that -- can be found below. After children are comfortable with these, more can be found on the web at the sites listed below.

## Sources of KenKen puzzles

Prepare:

Download transparency to use in introducing KenKen.- Download six 4x4 puzzles. Easy puzzles for beginners.
- Download six 6x6 puzzles. Easy 6x6 puzzles, still suitable for relative beginners who want more.
- Download twelve 4x4 puzzles. Mixed difficulty level, but all 4x4.
- Download six 6x6 puzzles. Medium difficulty 6x6 puzzles.