ProblemSolving - Mr. L's Math

KenKen and Number Sense

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Mar 072012

KenKen 6x6 Number Sense

KenKen is a self-motivating puzzle that builds number sense. As an added bonus, using this teaching and learning tool builds creative logical thinking and patterns of self-reliance. It also develops strong focus for an extended period of time, a necessary skill in mathematical problem solving. This article spotlights some of the prime factorization, number trees, and number combinations found in KenKen.

KenKen and Sudoku share two of the same rules, which makes an easy transition for teachers and students: numbers in each row and column must not repeat. The third rule sets KenKen apart as being more mathematical than Sudoku. In KenKen there are Cages, which are heavily outlined areas with specific numerical properties.

The most commonly used puzzles for classroom use are 4×4, 5×5, and 6×6. As with Sudoku, KenKen comes in several levels of difficulty. This allows the instructor to introduce the puzzle at the elementary level, and then gradually ramp up the challenge level as students are ready.

Here is a 6×6 KenKen from the 3-7-12 issue of the NY Times at the Medium difficulty level:

What I suggest solvers do is write the possible numbers at the top or bottom as on a number line. This gives a sense of number magnitude and provides a great help when solving.

A. Look at the 9+ cage: How many different sets of three numbers add up to nine? Hint: there is more than one way.

B. Check out the 4+ cage: How many different sets of three numbers multiply to 4? Again, there is more than one way.

C. Now let your eyes rest on a similar region, the 24+ cage, with even more possibilities. Hint: what is the prime factorization of 24, and how many ways can you make a factor tree for 24?

By looking at numerical possibilities, along with other clues from rows and columns, it’s possible to solve this KenKen by logic and reasoning alone, with no guesswork. This is the hallmark of great mathematical thinking!

Every teacher that has talked/emailed me after using KenKen with students is enthusiastic in support of this methodology. Students always look forward to doing KenKens, and it’s always a great addition to a Friday workday, or useful as extra credit. Give it a try, and you will be a hero with your students!

KenKen puzzles can be found for free on several websites. Here are my two favorites, which publish six new puzzles daily: www.nytimes.com/ref/crosswords/kenken and www.kenken.com

The second site also has allows participants to receive regularly via email a nine page PDF file with KenKen puzzles in sizes from 3×3 through 6×6, including answers – something for everybody!

KenKen – a Video Introduction to the 3×3 Puzzle

Mr. L's Videos No Responses »

Jul 222010

Here is a four minute video that introduces the 3×3 KenKen. It can be played in HD (720p) for use in classrooms with projectors.

KenKen is a logic and mathematical puzzle which is similar to Sudoku. As a math teacher, I use KenKen to teach and reinforce logical, deductive reasoning and to help with number concepts such as prime factorization.

The most common puzzles are 4×4 and 6×6 grids. I show the 3×3 puzzle in this video simply as an intro to the structure of KenKen. The goal is to fill the grid in with the digits 1 through 3 such that:

Each row contains exactly one of each digit
Each column contains exactly one of each digit
The numbers in each heavily outlined set of squares, called cages, must combine (in any order) to produce the target number in the top corner using the mathematical operation indicated.

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Games and Puzzles that Reach the Kids and Teach the Standards

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Nov 272009

The title of this post is from a conference session at the California Math Council conference at Asilomar in December, 2009. You may download the handout, with instructions for some activities, through the following link:

Games and Puzzles that Reach the Kids and Teach the Standards – download file

Puzzles and Games are a great way to build enthusiasm, excitement, and skills in a math classroom. When these are used properly, there is always an underlying mathematical theme that is being explored or reinforced. Students learn the value of logical thinking, proper planning, and long-term focus.

Mathematical games can develop a lot of creativity and number sense in students. Many times a game or puzzle at first looks overwhelming, but becomes manageable after focusing on the key concept. Motivation is almost always inherent in a game or puzzle; this creates conceptual emotion which is a key to successful learning.

Opportunities for transitioning from number sense to algebra thinking abound with puzzles. Number properties show their strength in supporting algebra throughout puzzle activities. It’s always a pleasure to watch the enthusiasm and skills grow in students!

Some of these activities are from Teacher to Teacher Press books:

Simply Great Math Games

Simply Great Math Activities: Algebra Readiness

Simply Great Math Activities: Number Sense

Put these activities, puzzles, games, and ideas into regular practice, you will find increased skills and motivation in your students and more joy in your teaching.

Best wishes,

Bill Lombard (a.k.a. Mr. L)

What Are the Odds of Losing Your Lunch? – A Probability Adventure

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May 172009

Here’s a neat problem recently under discussion in our school district: “Four kids, each with their own lunch, go on a trip. When it’s time to eat, the four identical-looking lunches are mixed up and the kids randomly choose one. what is the probability that exactly one kid gets his own lunch?”

This problem is nice because it can be solved several ways, allowing students to see how others solve the problem. By relating the different methods of solution, students build and strengthen their own problem-solving abilities.

It’s also a nice problem because it reminds us that probability is a subject full of “slippery concepts”; just when you think you’ve solved a problem, it can elude you. Many probability problems are simple, but can be made more difficult by adding/changing just a few words in the posing of the problem. Consider how many high-level mathematicians were fooled by the Monty Hall problem, made famous by Marilyn vos Savant.

One of my KenKen posts asks the question, “How Many Possible Puzzles Are There?” This has resulted in many reader responses and a lively discussion. The question in today’s post may elicit many solutions as well – will one of them be yours?

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Average Rate of Speed – a Simple Solution to a Confusing Problem

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Mar 302009

Ask people the following problem, and almost all of them will get it wrong: “If a car goes up a hill at 20 m.p.h. and comes down the same hill at 30 m.p.h., what is the average rate of speed for the whole trip?”

The “obvious” answer is 25 m.p.h., but that would be wrong. Take a quick look at a slideshow of some of the projects my students have done to see some nice explanations of confounding problems like these.

View more presentations from blombard.

Simply stated, since the car spends more time traveling at the slower speed, this has the effect of reducing the average rate of speed below the average of 20 and 30. The surprising answer to this poser is 24 m.p.h.

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Friday Afternoon Lifesavers – the Teacher’s Best Friend; Part 7, Twins, an Exploration in Congruence and Symmetry

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Feb 262009

Friday Afternoon Lifesavers are exactly what they sound like: at the end of the week when you and your students are running out of gas, these activities come to the rescue. They are mathematical, engaging, fun, and give the teacher a much-needed break.

One way to help kids become better problem solvers is to pose spatial visualization puzzles like this one. Students are given non-standard shapes and asked to divide them into two congruent parts. Kids love the challenge of finding how to create two identical shapes; this activity engages all students and gives the teacher the opportunity to circulate the room and give encouragement and advice as needed without student management issues.

This activity provides the teacher the option to teach/reinforce the concepts of congruence, line symmetry, and rotational symmetry. Also available here are area, distance, and perimeter teaching opportunities.

After students solve as many shape puzzles as they can, the teacher has the chance to take them to the next level. Hand out some blank grid paper and encourage them to create their own Twins. By asking them to design these types of puzzles, students can appreciate the skill it takes to create a good puzzle by turning two congruent shapes into a larger shape and “disguising the line of separation.”

The downloadable pdf file includes the Student Puzzle Page, a Solution Page, and a blank grid page. Another dissection puzzle can be found at Friday Afternoon Lifesavers, Part 8, Triples.

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KenKen News – Will Shortz Adds Puzzle to Daily New York Times Puzzle Page

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Feb 092009

The New York Times has added KenKen to its Puzzle Page, next to their crossword puzzle, and publish it daily Monday-Saturday. This puzzle launch is accompanied by an article by the NYTimes Puzzle Editor, Will Shortz. Here are some excerpts from Will’s article:

“KenKen, which starts in The New York Times today next to the crossword puzzle, is a new numerical logic puzzle from Japan. The name means loosely “cleverness squared.

KenKen shares some properties with sudoku. Each is a pure logic challenge in which numbers are filled in the squares of a grid. Unlike sudoku, though, in which the numbers act solely as symbols (letters or pictures would work as well), KenKen requires arithmetic.

The rules are simple: Fill the grid with digits so as not to repeat a digit within any row or column, and so the digits within each heavily outlined box (called a cage) go together using the arithmetic operation shown to make the target number indicated.“

Will goes on to give a few simple tips to new KenKen solvers, as well as some history behind KenKen. Here is the link to the Will Shortz article.

My students have been doing KenKen puzzles for about two months now. It’s their favorite puzzle in all the years I’ve been teaching. They have matured into dedicated aficianados of KenKen, and I’ve watched their puzzle-solving abilities grow week by week. Their language skills have improved as well; they are more able now to explain “WHY” certain numbers must appear in squares. KenKen popularity has grown throughout my school, and now students from other classes visit me to get puzzles to work on!

Give this puzzle a try, and I’m sure you’ll agree with Will Shortz and me that it’s a great way to train your brain and have fun at the same time.

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Friday Afternoon Lifesavers – the Teacher’s Best Friend; Part 4, A Knight’s Tour

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Feb 062009

A Knight’s Tour is a motivating activity/puzzle that works with both over and under achievers, promotes critical thinking and strategy, and is a great introduction to chess. It has multiple solutions, is self-assessing, and can be extended to an 8×8 grid involving magic squares.

A knight on a chessboard moves in a unique fashion. It must move two squares in one direction, then one square in a right angle direction. The graphic above shows some examples of knight moves. A knight can move from the square marked with the horse to any of the squares with an “X”. A Knight’s Tour is a series of moves by a knight that visits each square of a chessboard exactly once.

Here is Level 1 of the Knight’s Tour challenge: How can a knight in the lower left hand corner of a 5×5 grid visit each square exactly once? The knight may not revisit a square. There are several different solutions. What helps students with this puzzle is having them number the squares as they visit them. In other words, the lower left hand corner square is numbered 1, then the next square visited is numbered 2, and so on. Here is a sample grid to work with:

Click here for a popup window showing the Knight’s Tour Solution

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KenKen Puzzles – Six New Puzzles for You Each Day from 4×4 to 9×9

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Jan 252009

I just received word that new daily KenKen puzzles are now available at this kenken.com website. You can choose from small puzzles, such as the 4×4, all the way to the mega-size 9×9 for real KenKen enthusiasts. All the puzzles are rated Medium.

This serves as a nice companion to the New York Times KenKen site, where there are only two sizes, 4×4 and 6×6, but you can choose from Easier, Medium, or Harder versions of each size. I like these new options for the classroom since I can now work with students at the 4×4 size, then gradually “graduate” them to the 5×5, then to the 6×6, and so on. Many of my students really look forward to these puzzles, and we do them on a regular basis. Sometimes the KenKen puzzles are for warmups, some are for homework, and they also appear on tests (whatever you want kids to study and master needs to be made part of your assessment).

Some traveling news: the creator of KenKen, Mr. Tetsuya Miyamoto, is coming to the USA for his first ever visit end of February. He intends to visit New York and San Francisco. I’ll keep you posted as I hear more.

KenKen Conundrum – How Many Possible Puzzles Are There?

Puzzle 17 Responses »

Jan 092009

kenken conundrum KenKen Conundrum – How Many Possible Puzzles Are There?

On the way to school yesterday an interesting question occurred to me. How many ways are there to arrange the numbers 1, 2, 3, 4 in a 4×4 grid according to KenKen rules? After driving a few more miles, I came up with “the answer” of 288. I shared this with my first class of the morning; unfortunately, at the end of my presentation I realized that my answer was wrong.

My analysis: Looking at the first row of the 4×4 grid, there are 4 ways to fill in the first square, but only 3 ways to choose the second number, 2 ways to pick the third number, and finally, only 1 way to fill in the last square. This is 4!, a common result in permutations.

Then the trouble started. There are certainly 3 ways to fill the first square in the second row since you cannot duplicate the number above it. But then I mistakenly thought that there were only 2 ways to choose the number for the second square since you could not repeat the numbers above and to the left. But in KenKen it can be possible for these numbers to be the same. OOPS!

So sometimes there are only 2 ways to choose a number for the second spot in row 2, and sometimes there are 3 ways. At this point the problem became more interesting and challenging. It seems that someone must have solved this problem already. I’ve emailed two KenKen sites, and so far received one reply which agrees with my second solution of 3,456 ways. But I’m not convinced this is the correct answer either.

I tried to contact Will Shortz, Puzzle Editor of the New York Times, but had no luck finding his email address. If any readers have a solution to this conundrum, please post a Comment to help us out.

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