Easiest KenKen Ever?

Saying that I like KenKen® would be like saying that Sigmund Freud liked cocaine. (Too soon?) ‘Twould be more proper to say that I am so thoroughly addicted to the puzzle that the length of my dog’s morning walks aren’t measured in miles or minutes but in number of 6 × 6 puzzles that I complete. (Most mornings, it’s two.) Roberto Clemente correctly predicted that he would die in a plane crash; Abraham de Moivre predicted that he would sleep to death (and the exact date on which it would occur… creepy); and I am absolutely certain that I’ll be hit by oncoming traffic as I step off the curb without looking, my nose pointed at a KenKen app on my phone and wondering, “How many five-element partitions of 13 could fill that 48× cell?”

I am forever indebted to Tetsuya Miyamoto for inventing KenKen, and I am deeply appreciative that Nextoy, LLC, brought KenKen to the United States. How else would I wile away the hours between sunrise and sunset?

I am also extremely grateful that the only thing Nextoy copyrighted was the name KenKen. This allows Tom Snyder to develop themed TomToms, and it allows the PGDevTeam to offer MathDoku Pro, which I believe to be the best Android app for playing KenKen puzzles.

The most recent release of MathDoku has improved numerical input as well as a timer. Consequently, my recent fascination is playing 4 × 4 puzzles to see how long it will take. A typical puzzle will take 20‑30 seconds; occasionally, I’ll complete a puzzle in 18‑19 seconds; and, every once in a while, I’ll hit 17 seconds… but not very often.

Today, however, was a banner day. I was in a good KenKen groove, and I was served one of the easiest 4 × 4 puzzles ever. Here’s the puzzle:

And here’s the result (spoiler):

4 × 4 KenKen solved in 15 seconds

The screenshot shows that I completed the puzzle in just 15 seconds. And it’s not even photoshopped.

This puzzle has several elements that make it easy to solve:

• The [11+] cell can only be filled with (4, 3, 4).
• The [4] in the first column dictates the order of the (1, 4) in the [4×] cell.
• The (1, 4) in the [4×] cell dictates the order of the (1, 2) in the [3+] cell.

After that, the rest of the puzzle falls easily into place, because each digit 1‑4 occurs exactly once in each row and column.

What’s the fastest you’ve ever solved a 4 × 4 KenKen puzzle? Post your time in the comments. Feel free to post your times for other size puzzles, too. (I’m currently working on a 6 × 6 puzzle that’s kicking my ass. Current time is 2:08:54 and counting.)

Think of a Number

I love to create math games almost as much as I love to play them.

My favorite professional project was leading the development of Calculation Nation. And my favorite game on the site is neXtu, though other games on the site may promote more sophisticated mathematical thinking.

I have many reasons to love my wife, not least of which is her creation of the game Dollar Nim. While I can’t take credit for the rules, I will take credit for its analysis and its popularization. (What do you call a wife who makes up a game that gets you a publication credit? A keeper!)

Recently, I’ve been frustrated by the lack of games for teaching algebra. I’ll give props to the good folks at Dragonbox, which uses a game environment to teach algebra. But I’m not yet convinced that it leads to deep algebraic understanding; even they admit “to transfer to pencil and paper, children must be explained how to rewrite equations line by line.” They also claim that “in-house preliminary tests indicate a very high level of transfer to pencil and paper,” but that’s the fox watching the henhouse.

So I’ve been thinking about games I can play with my sons that will allow them to engage in algebraic thinking. But I don’t want them to know they’re engaging in algebraic thinking. I have two criteria for all math games:

• The game mechanics depend on mathematics. The math is not tangential to the game; it is the game.
• Kids don’t realize (or at least they don’t care) that it’s a math game, because it’s fun.

It pains me to write that second criterion, because math is fun. But I know not everyone shares that opinion. So I do my best to disguise any math learning in the game and then, when they least expect it — BOOM! — I drop the bomb and show them what they’ve learned.

So here’s a game I recently devised.

• Player A chooses a number.
• Player B chooses two operations for Player A to perform on the number.
• Player A performs those operations and then tells the result to Player B.
• Player B then tries to identify Player A’s number.

These rules leave something to be desired, since Player B could simply ask A to “multiply by 1″ and then “add 0,” in which case finding A’s number would involve no work whatsoever. To be a stickler, an additional rule could impose that either addition or subtraction can be used exactly once and that no operation can involve either 0 or 1. In a middle school classroom, I suppose I would state such a rule explicitly; for playing this game with my seven-year-old sons, I opted not to.

We played this game three times on the car ride to school yesterday. One game went like this:

• I thought of a number (14).
• Eli asked me to add 3 to my number.
• Alex asked me to multiply by 3.
• I told them the result: 51.

Eli then guessed that my number was 16. He had subtracted 3, then divided by 3.

“No!” said Alex. “You added 3 first, so you need to subtract 9.”

“But he multiplied by 3, so if you subtract first, you have to subtract 3 × 3.”

Eli then realized that my number was 14.

He thought for a second. “Oh,” he said. “I should have divided by 3 first, then subtracted.”

Wow, I thought. This is going even better than I hoped.

Though they didn’t use the proper terminology, the boys had a great discussion about “undoing” operations by performing inverse operations in reverse order. In 10 minutes, they taught themselves how to solve a two-step equation:

3x + 3 = 51

Grace Kelemanik once said that she knew she was being effective when she didn’t have to say a word. She’d watch from the back of the room as students carried the conversation and guided one another to correct mathematical thinking.

I will never claim to be half the educator that Grace Kelemanik is. But yesterday morning, I was pretty darn effective.

I’d love to hear about math games you’ve played with kids, whether you invented them or not.

Things I Learned on a Boston Duck Tour

ConDUCKtor Dirty Waters

Dirty Waters led our Boston Duck Tour yesterday and told us a little about himself:

I’m wicked smaht. In fact, I was valedictorian of my high school. Of course, I was homeschooled… but my mom says it still counts.

Dirty was a veritable fountain of math-related trivia. For instance, he told us that the movie Good Will Hunting, in which Matt Damon roams the halls of Ford Building at MIT solving difficult math problems, wasn’t actually filmed at MIT. Rather,

The hallway scenes were filmed at Beacon Hill Community College… and let’s be honest, anyone can answer the math questions that are asked there.

Incidentally, the math problem that Damon solved involved drawing all the homeomorphically irreducible trees of degree 10. While I don’t know how well the typical BHCC student might react to this problem, I do know that my seven-year-old sons were able to solve it — once I helped them understand what a homeomorphically irreducible tree was.

Irreducible Tree

We also learned the following non-math trivia about Paul Revere:

• Paul Revere didn’t actually make it to Concord. He was captured by the Redcoats and sang like a songbird — he divulged the entirety of the colonists’ plans.
• He didn’t yell, “The British are coming! The British are coming!” That would have made no sense. At the time of his midnight ride, all of the colonists considered themselves British. Instead, he probably yelled, “The Regulars are coming!” a term used to describe British soldiers.
• That’s not Samuel Adams on the front of a Sam Adams bottle. It’s Paul Revere, who was much more handsome than Adams.

This made me realize that a lot of the things we learn(ed) in school are complete bullshit:

1. Paul Revere informed the folks in Concord that the British were coming. In fact, Samuel Prescott was the only rider to reach Concord. A third rider that night, William Dawes, accompanied Revere and Prescott, but he was thrown from his horse and walked back to Lexington.
2. Humans have five senses (sight, smell, touch, taste, hearing). Actually, no… most social scientists also include pain, hunger, thirst, pressure, balance, acceleration, and time, among others.
3. Sentences cannot end with prepositions. Not true, and sometimes you’ll sound like Yoda if you try to do otherwise (e.g., “Rained out was the baseball game”). The classic joke is, “What is a preposition? A preposition is a word one must never end a sentence with.”
4. Division by zero is impossible. It’s not impossible; it’s just a bad idea. Weird stuff happens when you divide by zero, and it’s easier to avoid it by calling the action “undefined.”
5. Chameleons change color to blend in. ‘Twould be awesome were it so, but they actually change color to communicate. While you might flip someone the bird to let them know you’re unhappy, a chameleon would just change to a darker color.
6. Columbus thought the world was flat. No, he didn’t, and neither did most educated people at the time. Columbus’s mistake was actually underestimating the size of the Earth. He was lucky to have found the West Indies, lest he and all of his crew would have died of starvation.

Why do these inaccuracies persist? I suspect most of the errors are legacy content from hundred-year-old curriculum; the alternative is that it’s willful deceit on the part of educators, and that’s hard to swallow.

What other complete bullshit is still perpetuated in American classrooms?

Math for International Beer Day

I don’t need an excuse to do math. Nor do I need one to drink beer.

But if I did, today is IBD.

I’m most happy when I can pursue my two passions simultaneously…

That said, I never attempt calculus while imbibing. It’s not safe to drink and derive.

And like most of the American populace, I almost never attempt to think logically while drinking beer…

A bartender asks three logicians, “Would all of you like beer?”
The first answers, “I don’t know.”
The second answers, “I don’t know.”
The third answers, “Yes!”

Judge This Joke By Its Size, Do You?

Most everyone knows the classic 7-8-9 joke:

What is 6 afraid of 7?
Because 7 8 9.

I recently heard a Star Wars variation:

According to Yoda, why is 5 afraid of 7?
Because 6 7 8.

This joke isn’t funny unless you understand the syntax often used by Yoda, which involves inverting the word order. See www.yodaquotes.net for some examples.

There are two other variations that have long been part of my arsenal. My favorite is:

Why don’t jokes work in base 8?
Because 7 10 11.

When I told this joke to my seven-year-old son, he said, “I don’t get it.” I asked him how 7, 10, and 11 would be represented in base 8. He thought for a second then said, “7… 8… oh, yeah… yeah, that works.”

That’s why I call this version a joke grenade. You pull the pin, and five seconds later, people laugh. Well, some people will laugh. Not everyone. I estimate that 5% of the population would understand this joke, and only about 1% would find it funny.

The last variation is multicultural:

What is ε afraid of ζ?
Because ζ η θ.

If you’re thinking, “That’s all Greek to me,” you’re right. The translation is, “Why is epsilon afraid of zeta? Because zeta eta theta.” The Greek alphabet proceeds, in part, as, “…δ (delta), ε (epsilon) ζ (zeta), η (eta), θ (theta), ι (iota)….” But as with all jokes, if it has to be explained to you, then you’re probably not going to find it funny.

A Father’s Day Gift Worth Waiting For

Alex made a Father’s Day Book for me. Because the book didn’t make it on our trip to France, however, I didn’t receive it until this past weekend. It was worth the wait.

The book was laudatory in praising my handling of routine fatherly duties:

I loved when you took me to Smashburger.

I appreciated when you helped me find a worm.

I love when you read to me at night.

I love when I see you at the sign-out sheet [at after-school care]. It means I can spend time with you.

But my favorite accolade — surprise! — was mathematical:

I liked the multiplication trick you taught me. Take two numbers, find the middle [average], square it. Find the difference [from one number to the average], square it, subtract it. (BOOM! Done!)

Priceless.

The trick that I taught him was how to use the difference of squares to quickly find a product. For instance, if you want to multiply 23 × 17, then…

• The average of 23 an 17 is 20, and 202 = 400.
• The difference between 23 and 20 is 3, and 32 = 9.
• Subtract 400 – 9 = 391.
• So, 23 × 17 = 391.
• BOOM! Done!

This works because

$(a + b)(a - b) = a^2 - b^2$,

and if you let a = 20 and b = 3, then you have

$23 \times 17 = (20 + 3)(20 - 3) = 20^2 - 3^2$.

In particular, I suggested this method if (1) the numbers are relatively small and (2) either both are odd or both are even. I would not recommend this method for finding the product 6,433 × 58:

• The average is 3,245.5, and (3,245.5)2 = 10,533,270.25.
• The difference between 6,433 and 3,245.5 is 3,187.5, and (3,187.5)2 = 10,160,156.25.
• Subtract 10,533,270.25 – 10,160,156.25 = 373,114.
• So, 6,433 × 58 = 373,114.

Sure, it works, but that problem screams for a calculator. The trick only has utility when the numbers are small and nice enough that finding the square of the average and difference is reasonable.

Then again, it’s not atypical for sons to do unreasonable things…

Son: Would you do my homework?

Dad: Sorry, son, it wouldn’t be right.

Son: That’s okay. Can you give it a try, anyway?

I’m just glad that my sons understand math at an abstract level…

A young boy asks his mother for some help with math. “There are four ducks on a pond. Two more ducks join them on the pond. How many ducks are there?”

The mother is surprised. She asks, “You don’t know what 4 + 2 is?”

“Sure, I do,” says the boy. “It’s 6. But what does that have to do with ducks?”

Math in France

Driving through the French countryside using smartphone GPS for navigation is a lot like driving through rural Pennsylvania with my redneck cousin riding shotgun — there is a significant lack of sophistication, an ample amount of mispronunciation, and myriad grammatical errors.

In Pennsylvania:

Take that there right onto See-Quo-Eye-Ay (Sequoia) Drive.

In France:

At the roundabout, take the second right toward Ow-Bag-Nee (Aubagne).

Take D51 to Mar-Sigh-Less (Marseilles).

And of course, the GPS pronounced the coastal town of Nice like the adjective you’d use to describe your grandmother’s sweater, though it should sound more like the term you’d use to describe your brother’s daughter.

I was half expecting the computer voice to exclaim,

Hey, cuz, watch this!

Otherwise, the rest of my recent week-long trip to the south of France was intellectually and often mathematically stimulating. The image below shows a -1 used to describe an underground floor (parking) in a hotel:

And though I didn’t get a picture, the retail floor of the parking garage at the Palais de Papes in Avignon was labeled 0, with the three floors below for parking labeled -1, -2, and -3.

This is a country that does not fear negative integers.

I also noticed that the nuts on fire hydrants in Aix-en-Provence were squares.

Hydrant with Square Nut

The nuts on American hydrants used to be squares, until hoodlums realized that two pieces of strong wood could be used to remove them, release water into the streets, and create an impromptu pool party for the neighborhood. As a result, pentagonal nuts are now used on most hydrants.

Alas, an adept hoodlum can even remove pentagonal nuts, so some localities have replaced them with Reuleaux triangle nuts, like the ones on hydrants outside the Philadelphia convention center, which can only be removed with a specially forged wrench.

with Reuleaux Triangle Nut

But perhaps the most mathematical fun that France has to offer is the Celsius scale. While there, our cousins taught my sons a poem for intuitively understanding the Celsius scale:

30 is hot,
20 is nice,
10 is cold,
and 0 is ice.

And I was able to teach them a formula for estimating Fahrenheit temperatures, which is easy to calculate and provides a reasonable approximation:

Double the (Celsius) temperature, then add 30.

Or algebraically,

F = 2C + 30

The actual rule for converting from Fahrenheit to Celsius is more familiar to most students:

F = 1.8C + 32

This rule, however, sucks. It’s not easy to mentally multiply by 1.8.

My sons were not convinced that the rule for estimating would give a close enough approximation. I showed them a table of values from Excel:

I also showed them a graph with the lines y = 1.8x + 32 and y = 2x + 30:

With both representations, it’s fairly clear that the estimate is reasonably close to the actual. For the normal range of values that humans experience, the estimate is typically within 5°. Even for the most extreme conditions — the coldest recorded temperature on Earth was -89°C in Antartica, and the hottest recorded temperature was 54°C in Death Valley, CA — the Fahrenheit estimates are only off by 9° and 20°, respectively. That’s good enough for government work.

And here’s a puzzle problem for an Algebra classroom, using this information.

The Fahrenheit and Celsius scales are related by the formula F = 1.8C + 32. But a reasonable estimate of the Fahrenheit temperature can be found by doubling the Celsius temperature and adding 30. For what Celsius temperature in degrees will the actual Fahrenheit temperature equal the estimated Fahrenheit temperature?

It’s not a terribly hard problem… especially if you look at the table of values above.

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.