NPR Puzzle Combinations

During yesterday’s NPR Sunday Puzzle, puzzlemaster Will Shortz presented the following challenge:

I’m going to give you some five-letter words. For each one, change the middle letter to two new letters to get a familiar six-letter word. For example, if I said FROND, F-R-O-N-D, you’d say FRIEND, because you’d change the O in the middle to I-E.

He then presented these nine words:

• EARLY
• TULIP
• MOURN
• JUROR
• FUTON
• DEITY
• PANDA
• SLOTH
• VISOR

You can figure out the answers for yourself. For those that give you real trouble, you can either listen to the broadcast or search for the answer at More Words.

For those of you who don’t know who Will Shortz is, you have something in common with detective Jake Peralta from Brooklyn Nine-Nine:

The puzzle was fun. But what was more fun was the conversation that our family had about it. After the third word, Alex announced, “This shouldn’t be that hard. There are only 676 possible combinations.”

What he meant is that there are 26 × 26 = 676 possible two-letter combinations, which is true.

He continued, “But you can probably stop at 675, because Z-Z is pretty unlikely.”

I smiled. He had chosen to exclude Z-Z but not Q-K or J-X or V-P.

Yet his statement struck me as a challenge. Is there a five-letter word where the middle letter could be replaced by Z-Z to make a six-letter word? Indeed, there are several:

• BUSED or BUSES
• CONED or CONES
• FETED
• FUMED or FUMES
• GUILE
• MEMOS
• NOBLE
• PITAS
• RAVED or RAVES
• ROVED or ROVER or ROVES
• TAPAS
• WIDEN
• WINED or WINES

None of them are perfect, though, because Z-Z is not a unique answer. For instance, ROVER could become ROBBER, ROCKER, ROMPER, ROSTER, or ROUTER, and most puzzle solvers would surely think of one of those words before arriving at ROZZER (British slang for a police officer).

From the list above, the best option is probably GUILE, for two reasons. First, stumbling upon GUZZLE as the answer seems at least as likely as the alternatives GUGGLE, GURGLE, and GUTTLE. Second, the five-letter hint has only one syllable, but the answer has two, and such a shift makes the puzzle just a little more difficult.

But while Alex had reduced the field of possibilities to 675, the truth is that the number was even lower. The puzzle states that one letter should be “changed to two new letters,” which implies that there are only 25 × 25 = 625 possibilities. Although that cuts the number by 7.5%, it doesn’t help much… no one wants to check all of them one-by-one to find the answer.

When Will Shortz presented DEITY, the on-air contestant was stumped. So Will provided some help:

I’ll give you a tiny, tiny hint. The two letters are consonant, vowel.

Alex scrunched up his brow. “That’s not much of a hint,” he declared.

Ah, but it is — if you’re using brute force. To check every possibility, this reduces the number from 625 to just 21 × 5 = 105, which is an 80% reduction.

Still, Alex is correct. The heuristic for solving this type of puzzle is not to check every possibility. Rather, it’s to think of the word as DE _ _ TY, and then check your mental dictionary for words that fit the pattern. It may help to know that the answer isn’t two consonants, but most puzzle solvers would have suspected as much from the outset. In the English language, only SOVEREIGNTY, THIRSTY, and BLOODTHIRSTY end with two consonants followed by TY.

Below are five-letter math words for which the middle letter can be changed to two new letters to form a six-letter word. (Note that the answers aren’t necessarily mathy.)

DIGIT :: DI _ _ IT (unique)

POINT :: PO _ _ NT

FOCUS :: FO _ _ US

MODEL :: MO _ _ EL (unique)

POWER :: PO _ _ ER

RANGE :: RA _ _ GE (unique)

SOLID :: SO _ _ ID (unique)

SPEED :: SP _ _ ED

And below, your challenge is reversed: Find the five-letter word that was changed to form a six-letter math word.

CO _ EX :: CONVEX (unique)

LI _ AR :: LINEAR (unique)

OR _ IN :: ORIGIN

RA _ AN :: RADIAN (unique)

SE _ ES :: SERIES

SP _ RE :: SPHERE

Enjoy!

Dreidel is Not Fair

play an online version of dreidel at the Jewish Outreach Institute (but turn the sound off)

Is dreidel fair?

The rules of dreidel are straightforward. At the beginning of each round, players put one coin into the pot. (For young kids, the “coins” are actually chocolate pieces in the shape of a coin and wrapped in gold foil. This is known as geld, and as far as I’m concerned, chocolate is a currency to kids.) Players then take turns spinning the dreidel, and a reward is earned based on which of the four Hebrew letters appears on top when the dreidel stops spinning:

• Nun: nothing.
• Hey: half the pot.
• Gimel: all of the pot.
• Shin: put one in.

Play continues clockwise, with each person spinning the dreidel until Gimel occurs and all coins are removed from the pot. At that point, everyone antes another coin, and a new round starts with the next player.

Officially, a player is out of the game when she or he has no coins left to contribute to the pot, and the game ends when one person has all the coins. But practically speaking, the game often ends much earlier, because players get bored and quit or, in the case of very young kids, the game lasts beyond bedtime and the children are pulled away by their parents.

No matter how the game ends, though, it’s not fair.

The following table is courtesy of Paul J. Nahin (Will You Be Alive 10 Years from Now?, Princeton University Press, 2014, p. 81). It shows the amount, over the long run, that each player will win during a dreidel game.

 Player 1 2 3 4 5 Number of Players 2 1.143 0.857 3 1.361 0.956 0.680 4 1.617 1.102 0.757 0.524 5 1.900 1.267 0.855 0.580 0.398

In other words, the first player has a significant advantage over the others. In a game of five players who start with 10 coins each, the first player will finish the game with 19 coins, on average, whereas the fifth player will finish with just 4 coins. That’s if the game ends early. If played until one person gets all the coins, then the first player is five times more likely to win than the fifth player.

This disparity in odds is likely the reason that an unofficial rule of dreidel is that the youngest player goes first, the second-youngest player goes second, and so on.

The word dreidel is Yiddish and means “to turn around.” Because the dreidel is, after all, a top.

This fact is not lost on comedian Lewis Black, who has some thoughts on the matter.

Happy Chanukah!

The Mathematization of Commercials

People have been commercializing mathematics for years. There are numerous examples on Zazzle or ThinkFun or, heck, even Math Jokes 4 Mathy Folks. My first experience with this phenomenon occurred in eighth grade, when I learned that Russell Hardy would do your algebra homework for the price of a school lunch. There was demand for this service, so this was a simple market economy.

But recently, a different phenomenon has arisen: the mathematization of commercials.

My first exposure to this phenomenon was a commercial for TireRack.com, which shows a female scientist on her way to a conference, and there is mathematical graffiti all over her window — a derivative, a summation, a triple integral, a logarithmic spiral, presumably a totient function, and a few randomly placed parentheses ostensibly for aesthetic balance only. The narrator says, “Phyllis isn’t thinking about tires. Phyllis is thinking… uh… well, she’s thinking, eh… I’m not really sure, but it’s probably important.”

Similarly, Planet Fitness mathematized their commercial “Upsell” in a humorous way when a smooth-talking salesman makes three offers that could hardly be refused.

Hold on! What did he say?

For you? Bronze package. Triple the price doubles the package.

Or the platinum package — 100% of the fee goes toward 75% of the total cost.

Okay, onyx package. Three percent, divided by 7, minus your budget.

The nonsensicality of the statements is why they’re funny. But the would-be gym member is both confused and slightly fearful, making the look on his face similar to those worn by many high school math students, especially those who are subjected to Saxon textbooks. (Zing!)

But the pièce de résistance of the commercial mathematization movement appears in Chevrolet’s “Equinox Forward Collission Alert” advertisement, in which a number of unsuspecting citizens are presented the following problem:

This is a traditional algebra problem, one that could have been pulled from any number of textbooks currently on the market. I suspect that most high school algebra students would fare better than the engineers, educators, and analysts in the commercial.

What do we make of the fact that none of the participants were able to solve this problem? We could surmise, perhaps, that they’re just plain dumb. Rather, I believe the conclusion to be drawn is that the problem is hardly worth solving. As revealed at the end of the commercial, there is no need to solve the problem, because the Chevy Equinox will solve it for you and, if necessary, alert you when there’s something to worry about.

Spoiler: Car A is traveling 25 mph faster than Car B, and 25 mph = 36⅔ feet per second. That means that Car A would collide with Car B in 170.2 ÷ 36⅔ ≈ 4.64 seconds… unless, of course, the driver of Car A isn’t a complete idiot and isn’t texting his best friend and — instead of driving up the tailpipe of Car B — hits the brakes.

Never Ask a Mathematician for Directions

I spent my formative years in a very rural county. Roads didn’t have names, or at least they didn’t have road name signs.

In college, my urban friends used to claim that if you asked someone in my hometown for directions, they’d say:

Go about a half mile, and turn left at Old Man Johnson’s farm. Then take a right at the huckleberry tree.

That very well may be true, but it’s no worse than the directions you might be given by a mathematician:

Well, you’re facing the wrong way, so do a reflection about this cross street. Go about 0.628397154 miles, then rotate π/2 radians and travel orthogonally to your previous vector for 600 light-nanoseconds. But they’ve got radar on that road, so keep your speed between 72 and 43 miles per hour. Turn left, and you should reach your destination in 2.4 minutes ± 0.3%, if you maintain an average speed of 46.8 mph.

Now, that’s bad. But at least I could understand all of it. Which is more than I can say for the directions that Google Maps provided yesterday. Traveling through the suburbs near Washington, DC, I had just crossed MacArthur Boulevard, and I was heading southwest on Arizona Avenue. As you can see from the screenshot below, Google Maps was suggesting that I turn right on Carolina Place, right on Galena Place, right on Dorsett Place, and then left on Arizona. In essence, it suggested that I reverse direction to take a 10-mile, 45-minute route.

I ignored that suggestion. Instead, I stayed on Arizona Avenue with the intention of turning right onto Canal Road in a quarter-mile. Just as I passed Carolina Place, Google Maps said that it was “Rerouting…,” and within 15 seconds, it confirmed that I had made the correct choice:

By ignoring Google Maps, I shaved 3.8 miles and 23 minutes from my commute.

WTF?

My speculation is that Google Maps attempts to avoid my chosen route because it follows Canal Road, which parallels the C&O Canal National Historic Park; it requires me to cross Chain Bridge, which offers a beautiful view of the Potomac River; and it then winds through an affluent neighborhood, where I can feel safe on tree-lined streets with elegant homes. Honestly, who would choose that when Google Maps is offering double the travel time and an opportunity to drive on the beltway?

I once asked Google Maps which highway I should take to California. It replied…

$\sqrt{1^2 + 4^2 + 7^2}$

Oh, yeah. Root 66.

The logic employed by Google Maps reminds me of a college friend…

He would always accelerate when coming to an intersection, race through it, and then brake on the other side. I asked him why he went so fast through intersections. He replied, “Well, statistics show that more accidents happen at intersections, so I try to spend less time there.”

Don’t Drink the Flavor Aid

Okay, boys and girls. Time for a pop quiz.

Question 1. What cyanide-laced drink did 913 members of the Peoples Temple consume on November 18, 1978, as part of the largest mass murder-suicide in modern history?

If you said Kool-Aid, well, you’d be in good company. Most Americans think that that was the drink of choice in Jonestown, and it’s the reason for the idiom, “Don’t drink the Kool-Aid.” (More on that in a moment.) In fact, members of the cult consumed Flavor Aid, but media outlets at the time and revisionist history since have used Kool-Aid because of its status as a more genericized trademark.

Question 2. Why does Kool-Aid have a hyphen but Flavor Aid does not?

That question is rhetorical. Damned if I know. Ostensibly, the unnecessary hyphen in Kool-Aid is a marketing gimmick, like the removal of the period in Dr Pepper, the inclusion of the backwards R in Toys Я Us, or the use of the numeral 4 in Math Jokes 4 Mathy Folks.

Question 3. Why do educators, policymakers, and other adults continually offer Kool-Aid to math students?

Not literally. Let me explain what I mean.

“Drinking the Kool-Aid” is a figure of speech that refers to a person or group holding an unquestioned belief, argument, or philosophy without critical examination. Wikipedia says, “It could also refer to knowingly going along with a doomed or dangerous idea because of peer pressure.”

In math class, we offer Kool-Aid every time we tell students how important some mathematical topic is, how useful it will be to them in the future, or how knowing it will prepare them for a fantastic career.

C’mon!

When was the last time you were asked to add fractions at a job interview?

When I was a student teacher at West Mifflin High School, one of the teachers made the following statement to a student:

Unless you become a math teacher, you’re never gonna use most of what we teach you.

I hated that statement. Still do. But I’ve come to appreciate its honesty.

At the time, I was an optimistic 23-year-old. All I could think was, “Then why are you teaching it?” Now, I’m a cynical 44-year-old, and I know that he had little choice… students were going to be tested on it, whether he thought it was relevant or not.

How do we offer the Kool-Aid? Lots of ways.

An algebra student asks, “When will I ever need to factor a trinomial in real life?” Teachers often respond with answers like, “You’ll use it when you get to chemistry or physics,” or perhaps they’ll offer, “It can be used in area and construction problems. For instance, what if you know that the floor of a rectangular building is 1,344 square feet and its length is 20 feet greater than its width? Then you could use the equation w2 + 20w – 1344 = 0 to find the length and width.”

C’mon!

The simplistic trinomials presented in a typical Algebra I curriculum are not what you’ll find when you get to physics. There, you’ll be confronted with -16x2 + 25x + 42 = 0, which can’t be factored easily.

And if you’re ever on a construction site and someone gives you the math problem about length and width that’s described above, you should do two things. First, punch that person in the face. Second, while they’re tending to their bloody nose, ask them how they know that the length is 20 feet greater than the width and that the area is 1,344 square feet if they didn’t actually measure the damned building.

But it’s not just teachers who offer mathematical Kool-Aid to students. Policymakers do it all the time, too.

The Common Core State Standards for Mathematics (CCSSM) contain topics that should have been removed from the curriculum a long time ago. For instance, students are still expected to learn long division (6.NS.B.2), even though it’s an antiquated paper-and-pencil skill that — let’s be honest — no one uses anymore. Your smartphone can do the job just fine, yet CCSSM still gives it a place in the curriculum.

C’mon!

In defense of Common Core, the standard says that students should divide by “using the standard [long division] algorithm,” though it is referenced only one time in Grade 6 and then never mentioned again. Perhaps they’re not that serious about it being a staple in students’ mathematical diet. (Based on my experience, I’d bet that someone on the standards writing committee vehemently argued for its inclusion despite repeated objections, and it was included just to quell the discussion.)

And lest we be too quick to blame educators and politicians, your typical, run-of-the-mill, do-right-by-your-kids parents are equally at fault. What do they do?

• They buy books like Skippyjon Jones Shape Up, which does nothing more than teach shape names without understanding (a better option is this free shapes book from Chrisopher Danielson, Which One Doesn’t Belong?).
• They use resources like Bedtime Math to perpetuate the myth that story problems with keywords and no extraneous information is what math looks like in the real world.
• They tell their kids that fractions are important because you’ll use them in baking, yet they fail to realize that there are over 400,000 different recipes for German Chocolate Cake alone that can be found online, thereby proving that fractions aren’t that big a deal, since lots of different fractions result in yummy desserts and, if you need to double a recipe, you don’t have to know how to add or multiply fractions, you just need to use each measuring cup twice.

C’mon!

The counterargument isn’t that the specific skill will be necessary, but the way of thinking associated with that topic is an important life skill. I’m as much a fan of the phrase “habits of mind” as the next guy, but does anyone really believe that the thinking associated with finding common denominators will increase a student’s quality of life?

More Kool-Aid, as far as I’m concerned.

So, what can we do? First of all, use authentic problems in your classroom, and show students the value of creating a mathematical model (and then improving it) instead of just providing models into which they substitute values or assign measurements. Conversely, stop using fabrications that include a real-world context only to make students do math that no one in their right mind would ever do.

Next, stop teaching things that don’t need to be taught. Dr. Henry Giroux once told me, “To thine own self be true, even if it means being a troublemaker.” Be a troublemaker! Tell your principal that you refuse to teach trinomial factoring on the grounds that it violates the fourth amendment. Tell her that forcing kids to use substitution or elimination to solve a system of equations — just because that’s what creators of the state test are trying to assess — is lunacy if guess-and-check is a more reasonable strategy. And for Pete’s sake, don’t even think about pawning off Ceva’s theorem on unsuspecting geometry students as a potentially useful piece of information. (It’s not, and you know it.)

Finally, be honest to yourself and your students. When a freshman in your algebra class asks, “When are we ever going to use this?” look him in the eye and say, “Probably never, but it’ll be on the state test.”

Yeah, that answer will leave a bad taste in your mouth and in your student’s mouth, too. But you can wash it away with a nice, tall glass of Kool-Aid.

Glug, glug.

Prime Time at NCTM Minneapolis

This afternoon, I’ll be presenting “Experience the Math Practices with Games and Online Tools” at the NCTM Regional Conference in Minneapolis. So if you unwittingly find yourself in the Minneapolis Convention Center at 1:30 p.m. today, please stop by.

But how cool is this? My session is #210, and today is Friday, November 13. Awesome, huh?

Wait, maybe you don’t see it:

210 = 2 × 3 × 5 × 7

Friday, November 13 = 11/13

Yeah, that’s right! The factors of my session number combined with today’s date are the first six prime numbers. You don’t have to be a math dork to appreciate that! (Though it doesn’t hurt.)

Why is 6 afraid of 7?

I assume it’s because 7 is a prime number, and prime numbers can be intimidating.

Thanks to Castiel from Supernatural for that new twist on an old classic.

Wait, Wait… I’ve Got a Math Question

“Not My Job” is a segment on the NPR game show Wait, Wait… Don’t Tell Me! During the segment, host Peter Sagal asks a celebrity three questions, on a topic about which they likely have no clue. For instance,

• Rob Lowe was asked questions about bratwurst, not the Brat Pack;
• Leonard Nimoy was asked questions about the other Dr. Spock (you know, the celebrity pediatrician).

My favorite of these segments, however, was with singer-songwriter Will Oldham, better known by his stage name Bonnie “Prince” Billy. Sagal explained, “You sing mostly sad or mournful songs, interspersed with the occasional tragic one. And we were thinking, who’s the singer least like you? […] So, we’re going to ask you three questions about Ms. Doris Day, the sweet-faced, sweet-voiced singer most famous in the 50s and 60s.”

As Sagal congratulated him, Oldham pretended that Doris Day was in the room with him. “Hey, Doris, you were right!” he said. “All the questions were about you!”

Now, that’s funny!

(As an aside, let me share with you a slide that I sometimes show during presentations about classroom technology:

Yep, that’s Doris Day in the 1958 movie Teacher’s Pet. Hopefully that image looks a little odd to you in 2015. Honestly, if you’re teaching math to adolescents and still using chalk and a wooden pointer, I’ll kindly ask you to consider a different career. There are other options for you. For instance, you could become a dentist and specialize in square root canals. But, I digress.)

So, back to the point.

The celebrity quiz contains three multiple-choice questions, each with three choices. If the guest answers at least two questions correctly, he or she wins. Which got me to thinking…

What is the probability that a celebrity guest will get at least two questions correct if she guesses randomly?

Brute force is definitely an option here. Write down all possible answer choice combinations, randomly decide which configuration will be correct, and then figure out how many of the possible combinations would yield a winner. Not pretty, but it works.

Speaking of combinations, here’s a joke that just has to be shared:

Courtney Gibbons’s comics used to appear at Brown Sharpie, but then she got a job.

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.