## Archive for September, 2016

### AWOKK, Day 8: KenKen in the Classroom

*Eight days a week…*

Yes, I know that this series is called **A Week of KenKen**, and I’m fully aware that there are only seven days in a week. But if the Beatles can love you for an extra day, then I can certainly write an extra post about KenKen. In case you’ve missed the fun we’ve had previously…

- Day 1: Introduction
- Day 2: The KENtathlon
- Day 3: KenKen Times
- Day 4: My KenKen Puzzles
- Day 5: Harold Reiter’s Puzzles
- Day 6: KenKen Glossary
- Day 7: KenKen Puzzle for 2016

Tetsuya Miyamoto created KenKen in 2004. Twelve years later, millions of KenKen puzzles are solved every day by people all over the world.

His original intent was not to create a global math sensation. Instead, he wanted to help his students improve their calculation skills, logical thinking, and persistence. Who knew that he would accomplish both?

KenKen puzzles are perfect for the classroom because they provide the same level of practice and repetition — sometimes affectionately known as *drill-and-kill* — as a worksheet full of problems, yet providing a significantly higher level of engagement.

Most students would have no more interest in answering the following questions than they would in removing their toenails with a pair of pliers:

*Use only the numbers 1-5, in how many ways can you…*

*write 300 as a product of 4 factors?**write 40 as a product of 4 factors?**write 13 as a sum of 5 numbers?**write 12 as a sum of 4 numbers?**write 5 as a quotient of 2 numbers?*

Yet wouldn’t students be willing to at least try this 5 × 5 KenKen puzzle? The cognitive demand is the same, but as any marketing guru or parent trying to get their kids to eat vegetables will tell you, it’s all about the presentation.

Because of the puzzle’s appeal and impact for students, the KenKen in the Classroom program was created. Every Friday, teachers who’ve signed up will receive free puzzles, which can be printed for distribution to students.

KenKen puzzles deal with a lot of mathematics beyond the four binary operations, including factors, parity, symmetry, modular arithmetic, congruence, isomorphism, algebraic thinking, and problem solving. Harold Reiter, John Thornton, and I wrote about these topics and how to use KenKen in a secondary classroom in the article *Using KenKen to Build Reasoning Skills*.

Even better than solving KenKen puzzles, though, is having students design their own. And to that very point… the 5 × 5 puzzle that appears above was created by my son Alex when he was 6 years old.

*I hope you’ve enjoyed reading this week of KenKen posts as much as I’ve enjoyed writing them. I’d love to hear your opinion of the series. Definitely check out the other items in this series with the links at the top of this post, and share your thoughts on all of them in the comments.
*

### AWOKK, Day 7: KenKen Puzzle for 2016

Good day, and welcome to Day 7 of the **A Week of KenKen** series. If you’ve stumbled onto this page randomly, you should definitely check out some of the fun we’ve had previously…

- Day 1: Introduction
- Day 2: The KENtathlon
- Day 3: KenKen Times
- Day 4: My KenKen Puzzles
- Day 5: Harold Reiter’s Puzzles
- Day 6: KenKen Glossary

Well, here it is, September 25, a mere 269 days into the year, and I am only now presenting you with a KenKen puzzle based on the year. I suppose I can take some solace in the fact that I’m presenting the following 2016 KenKen puzzle before the year is over. Little victories.

You might note that the puzzle has the following attributes:

- The numbers 2, 0, 1, and 6 are featured prominently as individual cells.
- Every target number uses (some combination of) the digits 2, 0, 1, and 6.
- There are two large cages — one in the shape of a 1, the other in the shape of a 6 — each with a product of 2016.

This puzzle follows the standard rules of KenKen, but there is one major exception: although it is an 8 × 8 puzzle, **instead of using the digits 1‑8, it uses 0‑7**. (Puzzles that use an atypical set of numbers are known as **KenKen Twist**.)

If you’re stuck, just fill in the squares randomly. There are only 108,776,032,459,082,956,800 different Latin squares of size 8 × 8, and your chances of guessing correctly are even better since 4 cells are already filled in.

As is typical of all puzzles presented on this blog, I am not posting the solution. At least, not yet. Maybe someday. If you beg. Or send me money. But not today.

### AWOKK, Day 6: KenKen Glossary

KENgratulations! You’ve made it to Day 6 of MJ4MF’s **A Week of KenKen** series. If you happened to miss any of the fun we’ve had previously…

- Day 1: Introduction
- Day 2: The KENtathlon
- Day 3: KenKen Times
- Day 4: My KenKen Puzzles
- Day 5: Harold Reiter’s Puzzles

Robert F. Fuhrer is a toy inventor with a knack for coming up with creative names, including Crocodile Dentist, Gator Golf, T.H.I.N.G.S. (Totally Hilarious, Incredibly Neat Games of Skill), Rumble Bugs, Missile Toe (literally, a rocket in the shape of a toe), and many others. As the president of Nextoy, LLC, which holds the registered trademark for KenKen^{®}, Bob now uses his creative naming abilities for the appellations of KenKen-related products.

The word that started this post, KENgratulations, is just one of his many linguistic creations. I rather like the term he coined to describe the computer application that randomly generates KenKen puzzles.

**KEN·****er·****a·****tor** *n.* the “machine” (computer application) used to automatically generate KenKen puzzles

*Quick! We need more 6 × 6 puzzles. Crank up the Kenerator!*

The name conjures images of a machine from Willy Wonka.

Oompa, loompa, doom-pa-dee-do,

I’ve got a perfect puzzle for you…

Numbers and operations go in, puzzles come out.

Bob can also take credit for the following:

**KEN****·cil** *n.* a pencil used to solve KenKen puzzles

*I prefer kencils rather than pens when solving KenKen puzzles. *

**KEN****·grat·****u·****la·****tions** *n.* an expression of praise for solving a KenKen puzzle

**Kengratulations** for solving that puzzle in less than 7 years!

**KEN****·thu****·si****·ast** *n.* someone who likes to solve KenKen puzzles

*Most Kenthusiasts solve more than one KenKen puzzle a day.*

There’s no doubt, Bob is good. But as you saw in a previous post, I’ve got a knack for coining terms, too…

**KEN·tath·lon** *n.* competition involving multiple KenKen events

*I complete a Kentathlon consisting of a 4 × 4, 5 × 5, and 6 × 6 puzzle every morning.*

**e·go·KEN·tric** *adj.* a person who thinks that they are better than others at solving KenKen puzzles

*He’s completely egokentric, even though he’s never won a KenKen tournament.*

**KEN·tral of·fice** *n.* where KenKen puzzles are made

*The Kenerator resides in the kentral office.*

**KEN·te·nar·i·an** *n.* person who has solved 100+ KenKen puzzles

*He became instantly addicted to KenKen puzzles; he became a kentenarian in less than 3 weeks.*

**su·per·KEN·te·nar·i·an** *n.* person who has solved 100+ KenKen puzzles *in one day*

*She became a superkentenarian by completing all the puzzles in Ferocious KenKen on Saturday.
*

**KEN·ta·gon** *n.* the arena in which KenKen tournaments take place (analogous to the Octagon, the eight-sided chain-link enclosure used for Ultimate Fighting Championship matches, though usually less violent)

*Enter the kentagon, prepare to solve!*

**KEN·o·pause** *n.* the period in a puzzle solver’s life when KenKen ceases to be fun

*Kenthusiasts who have entered kenopause usually solve fewer than one puzzle a day, on average.*

**KEN·i·ten·tia·ry** *n.* where KenKen solvers are locked up if they’re caught cheating (*syn* **prism**)

*If you copy off your neighbor at a KenKen tournament, you’ll be sent to the kenitentiary.*

**KEN·al·ty** *n.* a disadvantage imposed on a puzzle solver at a tournament for an infringement of the rules

*He was given a 15-second kenalty for “using a kencil in an unsafe manner.” *

**KEN·ais·sance** *n.* the period from roughly 2008 to 2010 when KenKen puzzles experienced tremendous growth in popularity, likely the result of publication in *The* *Times* (London), the *NY Times*, and other newspapers

*Harold Reiter’s interest in KenKen started long before the Kenaissance.*

**KEN·der·foot, ***n.* an amateur; someone who has solved only a few KenKen puzzles

*He’s such a kenderfoot; he doesn’t even know the X-wing strategy!*

**KEN·den·cies** n. the habits of a KenKen solver; analogous to a “tell” in poker

*He has a kendency to complete all of the addition cages before attempting any subtraction cage.*

**hy·per·KEN·ti·late** *n.* to breathe heavily while solving a puzzle (usu., a result of having difficulty)

*At the 2013 KenKen International Championship, she started to hyperkentilate when she had trouble with a difficult 6 × 6 puzzle.*

### AWOKK, Day 5: Harold Reiter’s Puzzles

You’ve made it! Today is Day 5 in MJ4MF’s **A Week of KenKen** series. In case you missed the fun we’ve had previously…

- Day 1: Introduction
- Day 2: The KENtathlon
- Day 3: KenKen Times
- Day 4: My KenKen Puzzles

Like Thomas Snyder, Harold Reiter believes that human-generated KenKen puzzles are superior to those created by computer. But Harold pushes the envelope by developing variations on KenKen that include Primal KenKen, 3-D KenKen, *n* × *n* KenKen puzzles that use numbers other than 1‑*n*, and Clueless KenKen — which contain no clues and, contrary to popular belief, were not invented by Cher Horowitz. Recently, in fact, he sent me a No-Op KenKen puzzle that used the floor function.

The following standard KenKen puzzle is one of Harold’s “Big Letter” puzzles. One of the cages forms a capital E, and he made this puzzle for a friend whose name began with — wait for it — the letter E.

No-Op KenKen Puzzles have target numbers in the cages but no operations. All the typical rules of KenKen apply, but it’ll take a little more thought to determine which operation is needed in each cage. The following is Harold’s “No-Op 12” puzzle.

And the piece de resistance of Harold’s creations — though he says the idea for this one actually came from his daughter, Ashley — is the following 3‑D KenKen. Here are Harold’s instructions for this puzzle:

In the puzzle below, there are 48 lines. Each must contain exactly one of the digits 1, 2, 3, and 4. The 16 cages are identified by letters. Each cage is either additive or multiplicative, and the target number for every cage is 12 — meaning that the numbers in the cells of every cage either add or multiply to give 12. Ken you solve it?

This one might need a little more explanation. The four layers of the puzzle work together; for instance, the two A’s on the top layer and the two A’s on the second layer form a four-number cage with a target number of 12. Similarly, the four G’s on the second layer and the lone G in the third layer form a five-number cage with a target number of 12. All standard KenKen rules apply and are extended to three dimensions — no repeat of a digit in any row, column, or tube vector.

I prefer this presentation of the puzzle, which looks like a 3‑D tic-tac-toe board:

But Harold’s presentation of it gives you a blank version side-by-side with the original, so you have a place to record your work:

The KenKen world is indebted to Harold for creating interesting puzzles *and* for sharing them with a new generation of students at his local math events.

For more of Harold’s puzzles, and for suggestions on how to create your own, check out KenKen, Thinking Globally.

### AWOKK, Day 4: My Puzzles

It’s Day 4 of the MJ4MF **A Week of KenKen** series, and I’m very excited about today’s offering. But first, in case you missed the fun we’ve had previously…

- Day 1: Introduction
- Day 2: The KENtathlon
- Day 3: KenKen Times

The puzzles that appear at KenKen.com and within the KenKen app are automatically generated by something that those folks call the **KENerator**. (Clever, no?) Likewise, the puzzles that appear in the MathDoku Pro app are also generated by a computer algorithm. The benefit is that both of these sources will provide a nearly infinite supply of puzzles. The downside is that computers don’t think as well as humans, so the puzzles range from mundane to amusing, and, in the words of Thomas Snyder, are “too easy and too computer-generated.” Rarely do they fall into the category of truly interesting.

That’s why Thomas Snyder attempted to outdo *The* *New York Times* KenKen puzzle back in 2009, when he presented a new KenKen puzzle every day. His themed puzzles were meant so show “what the puzzle should be” and how to make them interesting. Of the 20+ puzzles he presented, this is my favorite, which he created for March 3 (3/3).

You can see the large numeral “3” formed by the cage along the right edge of the puzzle. Further, every cage includes at least one digit 3 as part of the target number.

I also like his “Perfect Ten” puzzle, where every cage has 10 as the target number.

Like Snyder, I agree that KenKen puzzles are generally more interesting — and, usually, more difficult — when they’re created by a human instead of a computer. The following are several themed puzzles that I’ve created.

This puzzle is relatively easy, but I like that it uses only 8’s and 4’s. In honor of the small town of Eighty-Four, PA — which apocryphally is said to be named after the town’s mile marker on the Baltimore & Ohio Railroad — I call this the “B&O” puzzle.

You may know that 1! = 1, 2! = 2, 3! = 6, …, 6! = 720, and that these numbers are known as *factorials*. Consequently, I call this the “Factorial” puzzle, since the target numbers are the first six factorials.

Of all the puzzles that I’ve created, my all-time favorite is a special 8 × 8 puzzle that I created in honor of the current year. But if you want to see it, you’ll need to check back on Sunday, September 25, for AWOKK, Day 7.

### AWOKK, Day 3: KenKen Times

Today is Day 3 in MJ4MF’s **A Week of KenKen** series. In case you missed the fun we’ve had previously…

- Day 1: Introduction
- Day 2: The KENtathlon

Yesterday, I introduced you to the **KENtathlon**.

While completing a KENtathlon, my goal is to complete a 6 × 6 puzzle in less than 2 minutes; a 5 × 5 puzzle in less than 1 minute; and a 4 × 4 puzzle in less than 20 seconds. Even though the sum of those times for all three puzzles is 3 minutes, 20 seconds, my goal is a combined time of 3 minutes. It’s good to have goals.

Puzzle Size |
Goal Time |
Personal Best |

4 × 4 | 0:20 | 0:12 |

5 × 5 | 1:00 | 0:27 |

6 × 6 | 2:00 | 1:29 |

KENtathlon | 3:00 | 2:32 |

I don’t always perform well enough to meet those goals. And when I don’t, I repeat the same size puzzle again… and again… and again… for as many attempts as it takes to complete each puzzle in the allotted time. And when I’ve met the time goal for each puzzle individually, if the combined time isn’t satisfactory, then I start the whole thing over.

To say that I’m slightly obsessive would be like saying that the Pope is a little bit Catholic.

As you may have noticed in the table, I once finished a 4 × 4 puzzle in 12 seconds. The key word there is **once**. The stars were in alignment that day — it was an easy puzzle, and the dexterity of my thumbs and fingers was at an all-time high. Though I’ve attained 13 a handful of times, I’ve never replicated that 12-second feat.

That said, I regularly complete 4 × 4 puzzles in 14 or 15 seconds. With that being the case, you have to wonder if the 20-second goal is really a challenge. And what about the goal times for 5 × 5 and 6 × 6 puzzles?

Admittedly, my time goals are arbitrary, though not random. When I chose those goals, I had completed enough KenKen puzzles that I intuitively knew what felt right. Still, it wasn’t based on hard data… and if you’ve read this blog long enough, you know that that bothered me. A lot.

But what’s a boy to do?

I suppose a well-adjusted human might do nothing, think it’s not worth the trouble, and just let the whole thing go. But an obsessive numbers guy? Well, he’d painstakingly solve 132 KenKen puzzles, collect data on the amount of time each one took to complete, meticulously record the data in an Excel spreadsheet, and perform a thorough analysis. You may think that undertaking such a project is ludicrous; but to me, it was absolutely essential.

The graph below shows the results. The circular dots represent my median time for each puzzle size, and the square dots represent the upper and lower quartiles. For instance, the median time for 6 × 6 puzzles was 217 seconds, while the interquartile range for 6 × 6 puzzles extended from 163 to 284 seconds.

What this reveals is that my intuition wasn’t perfect, but not bad.

- I completed
**49% of 4 × 4 puzzles**in less than the goal time of 20 seconds. - I completed
**58% of 5 × 5 puzzles**in less than the goal time of 1 minute. - But, I completed
**only 14% of 6 × 6 puzzles**in less than the goal time of 2 minutes.

Further analysis revealed that I completed 40% of the 6×6 puzzles under 3 minutes, and that seems a bit more reasonable, so **my new goal time for 6 × 6 puzzles is 3 minutes**.

Now, I know you thought this analysis was completely unnecessary, but the proof is in the pudding. The results were invaluable. By considering the data, interpreting the results, and revising my goal time for 6 × 6 puzzles, the probability that I can now complete each size puzzle in the allotted time on the first or second try has increased from 16% to 39%. Or said another way, Remy’s morning walks now last an average of just 15 minutes, whereas some of them used to take an hour-and-a-half.

### AWOKK, Day 2: The KENtathlon

You’ve come back! Great! Welcome to Day 2 of MJ4MF’s **A Week of KenKen** series. In case you missed the fun we had yesterday…

- Day 1: Introduction

As the early riser in our family, I’m responsible for taking our dog Remy on his morning walk. Given his advanced age (13) and breed (golden retriever), our veterinarian says that I should walk him 15-20 minutes every morning.

But the length of Remy’s morning walk is completely independent of any professional recommendations. Instead, it depends entirely on the amount of time it takes me to solve a 6 × 6, a 5 × 5, and a 4 × 4 KenKen puzzle.

I call it the **KENtathlon**.

Here are three puzzles for you to solve as a KENtathlon. The order in which you solve them doesn’t matter. I don’t even care if you take a break between puzzles. The only thing I would recommend is that you complete each puzzle in one sitting with no break — but, hell, this is for fun, so I don’t much care if you ignore that suggestion, too.

**Post your combined time for all three puzzles in the comments.**

If you’d like to compete against other KENthusiasts, you can register for the **KenKen International Championship**, to be held in New York City on December 18. (Unfortunately, registration isn’t open yet… check www.kenken.com in the coming months for details.)

### A Week of KenKen, Day 1: Introduction

Welcome to **A Week of KenKen** (AWOKK). Every day this week, the MJ4MF blog will feature a new post about the number puzzle that Sudoku wishes it could be. That’s right — seven days, nothing but KenKen.

Here’s a list of the posts that you’ll see in coming days:

- Day 1: Intro (that’s
*today*!) - Day 2: The KENtathlon
- Day 3: KenKen Times
- Day 4: My KenKen Puzzles
- Day 5: Harold Reiter’s Puzzles
- Day 6: KenKen Glossary
- Day 7: KenKen Puzzle for 2016
- Day 8: KenKen in the Classroom

If the Beatles got nothin’ but love, babe, eight days a week, then I can certainly have a week with eight days of KenKen. Deal with it.

Today is an introduction, for those of you unfamiliar with KenKen. Here are the rules of the puzzle:

- For an
*n*×*n*grid, fill each row and column with the numbers 1 through*n*. A number may not be repeated in any row or column. - Each heavily outlined set of cells, called a
*cage*, contains a mathematical clue that consists of a number and an arithmetic operation: +, –, ×, or ÷. The numbers in that cage must combine (in any order) to produce the target number using the mathematical operation indicated. - Cages with just one cell should be filled with the target number.
- A number may be repeated within a cage, provided it’s not in the same row or column.

*The New York Times* crossword puzzle editor and *Weekend Edition* puzzlemaster Will Shortz explains KenKen in this short video:

Ready to try for yourself? Here’s a simple puzzle, which is dubbed an “easy” puzzle from the KenKen website:

Too easy? Here’s a slightly more interesting one that I created:

Did that whet your appetite for more? If you were a kid who could’ve held out for several minutes to get two marshmallows, then check back tomorrow for the next installment. But if you were a kid who just couldn’t wait and would’ve gobbled that single marshmallow immediately, then here’s your instant KenKen gratification:

- kenken.com
- Illuminations KeKen app (provided by NCTM, iPhone only)
- MathDoku Pro (my personal favorite, but Android only)

Till tomorrow, happy solving!