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 through the following link:

Games and Puzzles that Reach the Kids and Teach the Standards

Here’s the handout with instructions for some activities:

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)

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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.

Here’s another Knight’s Tour; this one was created by the famous Swiss mathematician, Leonhard Euler. It has the added property of being a magic square. If you like this puzzle, you may want to try last week’s Friday Afternoon Lifesaver featuring a 5×5 Knight’s Tour.

Over the years I’ve used this in class many times; students love the challenge and really get excited when they figure it out. Some students like to work together, while others prefer to “fly solo” and get all the glory. It’s a good idea to circulate the room to keep the kids motivated, as well as to answer any questions.

Here is the solution

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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 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.

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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1. Games and Puzzles that Reach the Kids and Teach the Standards

Shown in the graphic above is the solution to the Pythagorean Theorem Cut-Up problem posed last week as a Friday Afternoon Lifesaver. This puzzle gives teachers a chance to talk about the theorem and its applications. After printing the puzzle, you can talk about how the five pieces must fit so that one edge of the new square must align with the hypotenuse of the original triangle. By measuring accurately you can find a nice application of a² + b² = c².

Research shows that spatial reasoning puzzles build good problem-solving skills. Hope you and your students had fun with this one!

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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.

Here’s a puzzle that works anytime, but works especially well if your class is studying the Pythagorean Theorem. It will make a nice project as well for kids to make as a paper/pencil drawing or a computer project.

I first came across this at a math conference years ago. It came in the form of five pieces of plastic which could be rearranged to form one square by itself, with the other four pieces forming a square as well. The graphic above shows these five pieces. Then the directions said that it was possible to rearrange the five pieces to form another, larger square. This, of course, is a demonstration of the Pythagorean Theorem.

I’ve included a link to my puzzle file which gives a full-size ready-to-cut-out model of the puzzle. It’s sure to be a winner with your kids. You might consider making a transparency of this for an overhead projector model.

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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.

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We have the solution to the KenKen Conundrum posed recently. I asked my students to solve the following problem, and several of them succeeded. The original question was, “How many different arrangements (permutations) of the integers 1, 2, 3, 4 are possible in a 4×4 grid such as KenKen?

Several people commented to the post, and I also received an email from Tetsuya Miyamoto, the creator of KenKen. The problem posed for my students is a classical permutation exercise, and to make it a little more accessible for solution, I restricted the 4×4 grid to only those arrangements showing 1, 2, 3, 4 as the top row. According to an article on Latin Squares in Wikipedia, there are 576 ways to arrange 1, 2, 3, 4 in a 4×4 grid. Since 1, 2, 3, 4 is one permutation of these integers, and 24² = 576, then there must be 24 ways to complete the grid. Here is the solution for the homework exercise; the digits are arranged in numerical order from top to bottom.

My students noticed a lot of symmetry among the grids; a neat extension of this exercise is to color all the “1″ squares one color, the “2″ squares a different color, and so on. It makes a great wall display.

A simpler homework or classwork exercise is arranging all the permutations of the digits 1, 2, 3 for a 3×3 grid. There are only 12 ways to do this. It seems remarkable that going from a 3×3 to a 4×4 grid should permit so many more arrangements.

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NOTE: Stephen Shankland joined CNET News in 1998 and contributes articles to cnet.com on a wide variety of technology topics. The following are excerpts from his article about KenKen:

“Happily, just as I was getting disillusioned with Kakuro, KenKen stepped into my life. It arrived in the form of a book–a review copy of Will Shortz Presents KenKen Easiest Volume 1 from St. Martin’s Press. It is written by Shortz and Tetsuya Miyamoto, the Japanese math teacher who invented KenKen. “Ken” means wisdom in Japanese, thus KenKen means wisdom squared–or perhaps cleverness squared, depending on your translation.

It took me about 30 seconds to grasp the rules of KenKen and another 30 seconds to get sucked into the book. Shortz, The New York Times’ crossword puzzle editor since 1993 and probably the most famous puzzle guru around, praises KenKen as a puzzle that’s easy to learn yet spans a range from simple to very difficult, and he’s right.

The toughest KenKen
The hardest puzzles add a major new wrinkle. They don’t tell you which mathematical operator you need to use, so there’s an extra layer of deduction required.

KenKen’s U.S. launch came in October after Nextoy President Robert Fuhrer signed up for licensing and marketing rights outside Japan. The New York Times started publishing KenKen puzzles online in November, and The Boston Globe, The Houston Chronicle and other papers have signed up as well.

Miyamoto created KenKen to be a multifaceted challenge, and indeed I find them a great mix of logic and manageable math. They’re entertaining yet provide a slightly smug feeling that they’re good for your mind, too.

I know math puzzles aren’t for everyone, but I highly recommend KenKen. I’m giddy with the anticipation of the ones I’ll tackle in the future–or at least until my attention is grabbed by something else.

Here’s the link to read the full article by Stephen Shankland. It contains some KenKen puzzles and clickable links to solutions.

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After the recent post about How Many Possible 4×4 KenKen puzzles there are, Tetsuya Miyamoto, the creator of KenKen, gave us the following insight: (edited by Mr. L for clarity)

“Permutations of an order of number 1 to 4 can be calculated, but it is only
how many variations of placing numbers, no calculation involved.

So if they would like to know how many numbers of 4×4 puzzles, they can
limit to “addition only”, and try to create all the possible puzzles only
with one solution. (but again, they have to think of how they put freebie in
it! It will double the number of puzzles!). If they spend the time for it, it
may be possible to know the number of permutations we have for 4×4.

If we think of 9×9, with all four operations, it may be more complicated
than chess or Shogi. Like the “Four Color Theorem”, this may be someone’s lifetime study.“

After posing my original question, I realized that the question may not have been clear in its intent. I had meant to ask how many permutations of the 4 digits are possible in the 4×4 grid. This question will be the subject of a future post. Happy KenKen-ing!

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Related posts:

1. KenKen for Teachers & Students – a Harvard Professor Speaks Up
2. Games and Puzzles that Reach the Kids and Teach the Standards