## Archive for August, 2014

### 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):

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.”

“Why 9?” Eli asked. “Daddy only added 3.”

“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:

3*x* + 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

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.

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:

**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.**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.**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.”**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.”**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.**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?**

Leave a comment.

### 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!”