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!

Don’t Believe the HIPE

curkovicartunits.pbworks.com

Let’s get this party started with a classic word puzzle.

What English word contains four consecutive letters that appear consecutively in the alphabet?

In Mathematical Mind-Benders (AK Peters, 2007), Peter Winkler describes how the puzzle above served as inspiration for a word game.

I and three other high-school juniors at a 1963 National Science Foundation summer program began to fire letter combinations at one another, asking for a word containing that combination… the most deadly combinations were three or four letters, as in GNT, PTC, THAC and HEMU. We named the game after one of our favorite combinations, HIPE.

This seemed like a good game to play with my sons. I explained the game, and then I gave them a simple example to be sure they understood.

ER

They quickly generated a long list of solutions, including:

• tERm
• obsERver
• fishERman
• buckminstERfullerene

Since that introduction a few weeks ago, the boys and I have played quite a few games. It’s a good activity to pass the time on a long car ride. The following are some of my favorites:

WKW

RTWH
RTHW
(these two are fun in tandem)

HIPE
(the game’s namesake is a worthy adversary)

TANTAN

The practice with my sons has made me a better-than-average HIPE player, so when I recently found myself needing to keep my sons busy while I prepared dinner, I offered the following challenge:

Create a HIPE for me that you think is difficult, and I’ll give you a nickel for every second it takes me to solve it.

Never one to shy away from a challenge, Eli attacked the problem with gusto. Fifteen minutes later, he announced, “Daddy, I have a HIPE for you,” and presented me with this:

RLF

That was three days ago. Sure, I could use More Words or some other website to find the answer, but that’s cheating. Winkler wrote, “Of course, you can find solutions for any of them easily on your computer… But I suggest trying out your brain first.”

The downside to relying on my brain? This is gonna cost me a fortune.

For your reading enjoyment, I’ve created the following HIPEs. They are roughly in order from easy to hard, and as a hint, I’ll tell you that there is a common theme among the words that I used to create them.

1. MPL
2. XPR
3. YMM
4. MSCR
5. MPT
6. ITESI
7. NSV
8. RIGON
9. OEFF
10. CTAH
11. THME
12. SJU
13. TRAH (bonus points for finding more than one)

Winkler tells the story of how HIPE got him into Harvard. He wrote “The HIPE Story” as the essay on his admissions application, and four years later, he overheard a tutor who served on the admissions committee torturing a colleague with HIPEs and calling them HIPEs.

I can’t promise that HIPEs will get you into college, but hopefully you’ll have a little fun.

Periodically Crude

Old farts will know the answer to this old trivia question:

What two letters do not appear on a phone?

And if your phone still looked like this…

then it would be a reasonable question.

But phones don’t look like that anymore. They look like this…

in which case, it’s a really dumb question. (The Q is now attached to 7, and Z hangs out with 9.)

On the other hand, the periodic table looks the same today as when Mendeleev published it in 1869, so the following trivia question may be a bit better:

What two letters never appear in a chemical abbreviation on the periodic table? (I mean anywhere, bitches.)

Shouldn’t be that hard, if you’re willing to take the time to look.

Jessica Lee made headlines back in May when she placed the following quote in her yearbook:

Fluorine uranium carbon potassium bismuth technetium helium sulfur germanium thulium oxygen neon yttrium.

Seems innocuous enough, till it’s translated with the periodic table:

(A line from a Notorious B.I.G. song, for the old farts reading this.)

Are you made of nickel, cerium, arsenic and sulfur? Because you have a…

Or maybe you’re made of copper and tellurium? Because you’re…

If you got nothing better to do today, maybe you could take a ride on a ferrous wheel…

How Much Does Your Name Cost?

Here’s a contrived yet fun math problem that I shared with my sons recently:

A local hardware store sells bronze letters. However, the letters vary in price; some are more expensive than others. When I was at the store the other day, four people purchased the letters in their names. Their names and the prices they paid were:

Aiden $491  •  Ned $225  •  Dane $399  •  Ed $135

The price of a name is equal to the sum of the prices of its letters. The price for uppercase and lowercase letters is the same, and there is no additional surcharge or tax. How much would the following people pay to buy the letters in their names?

Edna  •  Ian  •  Nadine

Those of you who know a little algebra will have no trouble with that problem. Those of you who don’t shouldn’t have too much trouble, either.

But then, I realized I could extend the problem for some added fun. And who am I to keep fun things to myself? So, here ya go.

I saw this sign in a window the other day:

At first, I thought the store was engaging in human trafficking. But then I realized that $269 was the price for the bronze letters that had been used to spell the name Eli. Inside the store was a price list for other names:

AIDEN – 491	AL – 248	ART – 267	BEA – 290
EARL – 415	DANE – 399	ED – 135	ELI – 269
FAY – 220	GABI – 289	HAL – 284	IVY – 143
JACK – 234	JAY – 232	KO – 60	KAI – 283
LEXI – 272	MAVIS – 363	MAX – 215	NED – 225
PAT – 210	PERRI – 330	QI – 93	QUIN – 199
SAMMY – 338	WILL – 243	ZENO – 243

The store didn’t have a list of prices for the individual letters, but then I realized that I didn’t need one. From the table above, I could figure out how much my name  would cost.

Can you figure out how much your name would cost?

You can download both of these problems for use in a classroom (or at a mathy party) from the following link:

Name Letters (PDF)

And while I don’t believe in answer keys, you can check your work by using the form found on this page.

Name Letter Form

For what it’s worth, the longest name ever — according to Wolfe + 585, Senior, who has a pretty long name himself — is Rhoshandiatellyneshiaunneveshenk Koyaanisquatsiuth Williams. Her entire name name would have cost $4,073 at this store — an astounding $2,359 for her first name, $1,119 for her middle name, and a veritable bargain at $595 for her tame-by-comparison last name. (Incidentally, this is the name that appeared on her birth certificate. As the story goes, her father later increased her first name to 1,019 letters and added an additional 36 letters to her middle name. You know… just in case the name wasn’t long or unique enough already.)

Sentences Are Commutative, Words Are Not

While playing Scrabble^® on my phone today, I had a rack with following letters:

AABEILN

Near the top of the board was TAVERNA, and it was possible to hook above the first six letters or below the first two letters. There were other spaces on the board to place words, but this was clearly the most fertile. The full board looked like this:

On my rack, the letters weren’t in alphabetical order (as above), so I missed a seven-letter word that would have garnered 78 points. Instead, I played ABLE for a paltry 13 points.

After my turn, the Teacher feature showed me the word I should have played:

ABELIAN

Kickin’ myself. I’ll get over not seeing BANAL, LANAI, or even LEV. But how does a math guy miss ABELIAN? I would not put up a fight if someone wanted to rescind my Math Dorkdom membership card.

What loves letters and commutes?
An abelian Scrabble player.

(That’s a joke. Please don’t play Scrabble while driving.)

Unique Words

Using Scrabble^® tiles, my sons were making anagrams. One would select four tiles, and the other would have to rearrange them to form a word.

This struck me as interesting, so I posed the following question to them:

Take four consecutive letters from the alphabet, and rearrange them to form a common English word.

How many solutions do you think there are? Before you solve the problem, take a guess. Can five words be formed from four consecutive letters? Maybe ten words? Or fifteen?

Okay, now solve the problem. Take your time. We’ll wait for you.

There are 23 ways to select four consecutive letters, and each set of four letters can be arranged in 4! = 24 ways. With 23 × 24 = 552 possibilities, it seems like there ought to be several solutions.

Were you as surprised as I was to find that there was only one?

But maybe I shouldn’t be too surprised. Lots of things in life are unique…

Always remember that you’re unique, just like everybody else.

Student: Do you believe in God?
Professor: Yes — up to isomorphism!

Then again, lots of things aren’t unique…

Don’t think you’re special. Even if you’re 1 in a million, there are still 7,000 people in the world just like you.

Here are two unique, non-math jokes…

How do you catch a unique rabbit?
Unique up on it.

How do you catch a tame rabbit?
The tame way!

What (Math) is in a Name?

One of my favorite online tools is the Mean and Median app from Illuminations. This tool allows you to create a data set with up to 15 elements, plot them on a number line, investigate the mean and median, and consider a box-and-whisker plot based on the data. Perhaps the coolest feature is that you can copy an entire set of data, make some changes, and compare the modified set to the original set. For example, the box-and-whisker plots below look very different, even though the mean and median of the two sets are the same.

It’s a neat tool for learning about mean and median, and I plan to use this tool in an upcoming presentation.

• Exceptional, Free Online Resources for the Middle Grades Classroom
  G. Patrick Vennebush
  Thursday, October 20, 12:30-2:00pm
  Room 401 (Atlantic City Convention Center)

For classroom use, I like to use this app with real sets of data. However, the app requires all elements of a data set to be integers from 1-100. Can you think of a data set with a reasonable spread that has no (or at least few) elements greater than 100? If so, leave a comment.

Recently, and rather accidentally, I found a data set that works well. Do the following:

Assign each letter of the alphabet a value as follows: A = 1, B = 2, C = 3, and so on. Find the sum of the letters in your name; e.g., BOB → 2 + 15 + 2 = 19.

Now imagine that every student in a class finds the sum of the letters in their first name. For a typical class, what is the range of the data? What is the mean and median?

The name with the smallest sum that I could find?

ABE → 1 + 2 + 5 = 8

The name with the largest sum?

CHRISTOPHER → 3 + 8 + 18 + 9 + 19 + 20 + 15 + 16 + 8 + 5 + 18 = 139

The Social Security Administration provides a nice resource for investigation, Popular Baby Names. Using a randomly selected set of 2,000 names and an Excel spreadsheet, I found the mean name sum to be 62.49, and 96% of the names had sums less than 100. Of the 80 names with sums greater than 100, many (such as Christopher, Timothy, Gwendolyn, Jacquelyn) have shortened forms (Chris, Tim, Gwen, Jackie) for which the sum is less than 100.

As it turns out, the frequency with which letters occur in first names differs from their frequency in common English words. The most common letter in English words is e, but the most common letter in names is a. The chart below shows the frequency with which letters occur in first names.

Because of this distribution, the average value of a letter within a first name is 10.54, which is slightly less than the 13.50 you might expect. This is because letters at the beginning of the alphabet, which contribute smaller values to the name sum, occur more often in names than letters at the end of the alphabet.

The chart below shows the distribution for the number of letters within first names. The mean number of letters within first names is 5.92 letters, and the median is 6. (In the data set of 2,000 names from which this chart is derived, no name contained more than 11 letters.)

Do you know a name that has more than 11 letters or has a name sum greater than 139 or less than 8? Let me know in the comments.

Plainly Stated

One of my favorite applets at Illuminations is the State Data Map, which allowed me to create the following map depicting the number of U.S. Presidents born in each state:

Note that the states are color‑coded. Those states in which the greatest number of Presidents were born are the darkest shade of red; those in which no Presidents were born are white. In addition to allowing you to enter data, there are also pre‑loaded data sets. My favorite is the “Letters in State Name” set, from which I concocted the following trivia questions:

• Which state names have the most letters?
• Which state names have the fewest letters?

Feel free to think about it a few seconds before reading the next paragraph.

As it turns out, there are three states whose names contain 13 letters, and there are three states whose names have 4 letters. For what it’s worth, the mean number of letters is 8.24, and the median is 8.

My sons have a collection of foam letters for the bath tub. When the letters get wet, they stick to the side of the tub, and Alex and Eli love to use the letters to spell the names of states. Tonight, Eli spelled WYOMING. We then played a game where I’d give them the name of a state, and they’d try to spell it — but they couldn’t spell many of the state names because the set contains only one copy of each letter of the alphabet. This led to the following trivia question: 

• Which states have names that can be spelled with bath tub letters, i.e., the state name contains no repeated letters?

Feel free to cogitate on that a while, too, then read on.

There are nine states with no repeated letters in their names. (Don’t feel bad if you weren’t able to identify all of them. I had to look at a map.)

Finally, here is a state trivia question a pro pos of absolutely nothing. For each pair of states below, identify the only state that borders both of them. (Each question has a unique answer.)

1. North Carolina, South Carolina
2. South Dakota, Illinois
3. New Mexico, Missouri
4. Oregon, Wyoming
5. Missouri, West Virginia
6. Wisconsin, Ohio

For the answers to all questions, check a map.

What’s in a Name?

The product value of a word can be calculated as follows:

Assign each letter of the alphabet a value as follows: A = 1, B = 2, C = 3, and so on. The product value of a word is the product of its letters. For instance, the word CAT has a product value of 60 because C = 3, A = 1, T = 20, and 3 × 1 × 20 = 60.

During a recent webinar, I introduced participants to my collection of Product Value Puzzles. The following product value puzzle is credited to John Horton Conway:

Find an English word with a product value of 3,000,000.

Finding the solution is up to you. But I will give you some good news — there’s not a unique answer. In fact, there are two English words that satisfy the conditions of the problem.

What most folks found interesting, though, are the Product Value Calculators on my web site. With these two tools, you can:

1. Enter an integer value, and the first calculator will return all words in the English language whose product value equals the number you enter.
2. Enter a word, and the second calculator will return the product value.

One of the participants during the webinar said that her middle school students, when confronted with any type of math puzzle involving words, will first apply the rules of the puzzle to their name. Apparently, I’m not much different from a middle school kid, because that’s what I did, too. Turns out, my name has a product value of 1,710,720:

Patrick = 16 × 1 × 20 × 18 × 9 × 3 × 11 = 1,710,720

So, then I wondered, “Are there any other words that have a product value of 1,710,720?” Of course, I could have used the Product Value Calculators to find the answer, but that would have been unsatisfying. With a little trial-and-error, I found that blackboard also has a product value of 1,710,720:

blackboard = 2 × 12 × 1 × 3 × 11 × 2 × 15 × 1 × 18 × 4 = 1,710,720

There were three things about solving this problem that I really enjoyed:

1. My strategy involved substitutions: I replaced a letter or a pairs of letters by other pairs of letters that have the same product value. For instance, the t and c in Patrick could be replaced by o and d, because both pairs have a product value of 60.
2. Calculating the product values for Patrick and blackboard reveal two distinct factorizations for 1,710,720.
3. How cool is it that I’m a mathy folk, and my name and blackboard have the same product value?

(Incidentally, my boss David found that his name and the word chalk have the same product value. Some would argue that its numerological destiny that we work together and are friends.)

So now I’ll offer  the challenge to you. Can you find a word that has the same product value as your name? Good luck!

Of course, if that’s more thinking than you care to do right now, you could just access the product value calculator. But what fun would that be?