## Do You Have Mathopia?

When I was young, we spent a lot of time on highways, driving to and from our summer cottage. I’d see a Pennsylvania license plate like the one below, which at the time had five digits and one letter. Most people, I suspect, would be unimpressed. But not me. I’d say to my parents, “How cool is that license plate? If *p* = 26 and the cracked bell were an equal sign, it would be 23 × 26 = 598.”

My mom would respond with, “If you say so,” or a shrug. She had failed algebra in high school and would regularly and disgustedly declare, “How the hell can *x* = 6, when *x* is a letter and 6 is a number?”

My father — who dropped out of school to join the Navy at age 15 and had never taken an algebra course — would simply grunt.

Neither of them saw the beauty in numbers. I, on the other hand, couldn’t *not* see it. I wasn’t **mad** about this. I was just **sad** that they couldn’t share my joy.

On my commute this morning, I saw a truck with the number 12448 on the tailgate. I mentally added two symbols and formed the equation 12 × 4 = 48.

When my boss told me that he was retiring on January 4, I remarked, “What a great choice! The numbers 1, 4, and 16 are all square numbers, and 1, 4, 16 forms a geometric sequence.”

The truth is, it’s not really possible for me to look at a number — whether it’s a license plate, calendar, billboard, identification card, lottery number, bar code, serial number, road sign, odometer, checking account, confirmation number, credit card, phone number, phone bill, receipt total, frequent flyer number, VIN, TIN, PIN, ISBN, or any of a million other numbers — and not try to figure out some way to give it meaning beyond just its digits.

I’m not the only one with this affliction. All mathy folks have **mathopia** — a visual disorder that causes us to see the all things through a mathematical lens.

G. H. Hardy had mathopia. He looked for a special omen in 1729, the number of the taxicab he took to visit his sick friend Srinivasa Ramanujan. Upon arriving, he mentioned that he hoped it wasn’t a bad omen to have taken a cab with such a dull number. Ramanujan had mathopia, too. He replied that 1729 was actually “an interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

Jason Padgett, whose latent mathematical powers suddenly appeared after he sustained a brain injury, has mathopia. He explained how he sees the world:

I watch the cream stirred into the brew. The perfect spiral is an important shape to me. It’s a fractal. Suddenly, it’s not just my morning cup of joe; it’s geometry speaking to me.

This is the way that math people work. We see numbers and patterns everywhere, sometimes even when they’re not really there. Or, maybe, when they’re not **meant** to be there. And while I am not trying to imply that I’m anything close to Hardy or Ramanujan or Padgett, I do think that they and I shared one characteristic — the burden, and the blessing, of seeing the world through math-colored glasses.

World Sight Day, celebrated on the second Thursday of October each year — in other words, *today* — seems like a good day to bring awareness of mathopia to the masses. It doesn’t hurt that today is 10/13/16, a date forming an arithmetic sequence, in which all three numbers are Belgian-1 numbers. (See, I can’t turn it off.)

**Do you have mathopia? What do you see when you encounter a number?**

## Dos Equis XX Math Puzzles

No, the title of this post does not refer to the beer. Though it may be the most interesting blog post in the world.

It refers to the date, 10/10, which — at least this year — is the second day of National Metric Week. It would also be written in Roman numerals as X/X, hence the title of this post.

For today, I have not one, not *two*, but **three** puzzles for you. I’m providing them to you well in advance of October 10, though, in case you’re one of those clever types who wants to use these puzzles on the actual date… this will give you time to plan.

The first is a garden-variety math problem based on the date (including the year).

Today is 10/10/16. What is the area of a triangle whose three sides measure 10 cm, 10 cm, and 16 cm?

Hint:A triangle appearing in an analogous problem exactly four years ago would have had the same area.

The next two puzzles may be a little more fun for the less mathy among us — though I’m not sure that any such people read this blog.

Create a list of words, the first with 2 letters, the second with 3 letters, and so on, continuing as long as you can, where each word ends with the letter X. Scoring is triangular: Add the number of letters in all the words that you create until your first omission. For instance, if you got words with 2, 3, 4, 5, and 8 letters, then your score would be 2 + 3 + 4 + 5 = 14; you wouldn’t get credit for the 8-letter word since you hadn’t found any 6- or 7-letter words.

2 letters: _________________________

3 letters: _________________________

4 letters: _________________________

5 letters: _________________________

6 letters: _________________________

7 letters: _________________________

8 letters: _________________________

9 letters: _________________________

10 letters: _________________________

11 letters: _________________________

12 letters: _________________________

13 letters: _________________________

14 letters: _________________________

Note: There are answer blanks above for words up to 14 letters, because — you guessed it — the longest English word that ends with an X contains 14 letters.

The third and final puzzle is a variation on the second.

How many words can you think of that contain the letter X

twice? (Zoiks!) Scoring: Ten points for the first one, and a bazillion points for each one thereafter — this is hard! Good luck!

If you’re in desperate need of help, you can access **my list of words for both puzzles** — of which I’m fairly proud, since my list of words that end in X include math words for 2 through 10 letters — or do a search at www.morewords.com.

## 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**.

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 it 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.