## Posts tagged ‘word’

### Change the Vowel

The following puzzle contains a clue within each clue. The answer, of course, fits the clue, but each answer is also one of the words within the clue with its vowel sound changed. For instance, the clue “the distance from the top of your hat to the sole of your shoes” contains the word *hat*, and if you change the vowel sound from a short *a* to a long *i*, you get *height*, which fits the description.

As always, there’s a catch. Every answer to the clues below is a mathy word.

- The number of permutations of three different colored socks.

- This value is the same for 3 + 1 and 2 + 2.
- Do this with two odd numbers and you’ll get an even number.
- Three hours before noon.
- You may fail to correctly expand (
*a*+*b*)(*c*+*d*) if you don’t remember this mnemonic. - One represents this fractional portion of toes on the hoof of a deer.
- The last element of a data set arranged in descending order.
- The measure of central tendency made from the most common data points.
- The number of contestants ahead of third place if there’s a tie for first place.
- If the leader is forced to drop out of a race, the runner-up takes over this place.
- The square root is needed to calculate the length of the hypotenuse in this type of triangle.
- The graph of the declination of the Sun during a year can be approximated by this type of curve.

Answers

- six

- sum
- add
- nine
- FOIL
- half
- least
- mode
- two
- first
- right
- sine

### Mathy Portmanteaux

The term *portmanteau* was first used by Humpty Dumpty in Lewis Carroll’s *Through the Looking Glass*:

Well, ‘slithy’ means “lithe and slimy” and ‘mimsy’ is “flimsy and miserable.” You see, it’s like a portmanteau — there are two meanings packed up into one word.

Interestingly, the word *portmanteau* itself is also a blend of two different words: *porter* (to carry) and *manteau* (a cloak).

Portmanteaux are extremely popular in modern-day English, and new word combinations are regularly popping up. Sometimes, perhaps, there are **too many** being coined. In fact, one author refers to these newcomers as *portmonsters*, a portmanteau of, well, *portmanteau* and *monster* that attempts to capture how grotesque some of these beasts are. An abridged list of portmonsters would include *sharknado*, *arachnoquake*, *blizzaster*, *snowpocalypse*, *Brangelina*, *Bennifer*, *Kimye*, *Javanka*, *fantabulous*, and *ridonkulous*.

Portmanteaux seem to proliferate most easily in B-movie titles, weather, and celebrity couples, but the world of math and science is not free from them. Here are a few mathy portmanteaux, presented, of course, as equations.

**ginormous** = *giant* + *enormous*, really big

**guesstimate** = *guess* + *estimate*, a reasonable speculation

**three-peat** = *three* + *repeat*, to win a championship thrice

**clopen set** = *closed* + *open set*, a topological space that is both open and closed

**b****it** = *binary* + *digit*, the smallest unit of measurement used to quantify computer data

**pixel** = *picture* + *element*, a small area on a display screen; many can combine to form an image

**voxel** = *volume* + *pixel*, the 3D analog to pixel

**fortnight** = *fourteen* + *night*, a period of two weeks

**parsec** = *parallax* + *second*, an astronomy unit equal to about 3.26 light years

**alphanumeric** = *alphabetical* + *numeric*, containing both letters and numerals

**sporabola** = *spore* + *parabola*, the trajectory of a basidiospore after it is discharged from a sterigma

**gerrymandering** = *Elbridge Gerry* + *salamander*, to draw districts in such a way as to gain political advantage (In the 1800’s, Governor Elbridge Gerry redrew districts in Massachusetts to his political benefit. One of the redrawn districts looked like a salamander.)

**megamanteau** = *mega* + *portmanteau*, a portmanteau containing more than two words, such as DelMarVa, a peninsula that separates the Chesapeake Bay from the Atlantic Ocean and includes parts of Delaware, Maryland, and Virginia

**meganegabar** = *mega* + *negative* + *bar*, the line used on a check so that someone can’t add “and one million” to increase the amount

(By the way, when Rutgers University invited *Jersey Shore* cast member Snooki Polizzi to speak to students on campus in 2011, they paid her $32,000, which is $2,000 more than they paid Nobel and Pulitzer Prize winning author Toni Morrison to deliver a commencement address six weeks later.)

### More HIPE

Nearly five years ago, I wrote about HIPE, a parlor game in which one person gives a particular string of letters, and the other people in the parlor try to guess a word with that same string of letters (consecutively, and in the same order).

Well, I recently rediscovered *Can You Solve My Problems?* by Alex Bellos, and I was pleasantly surprised to find that he included four HIPEs in that book:

- ONIG
- HQ
- RAOR
- TANTAN

The fourth is one that I had included in my previous post, Don’t Believe the HIPE, and all are good enough that they deserve wide distribution.

Just for fun, here’s a new list of HIPEs that might prove interesting.

- SSP
- LWE
- NUSCU
- CUU
- CTW
- KGA
- UIU
- XII

In an effort to collect a bunch of excellent HIPEs, I’m asking for your help. If you play the game with friends and discover a particularly delectable combination of letters, please share below or at https://forms.gle/otddCw1uLeDALrMo7.

### Math Words for National Dictionary Day

Want to start today the right way? Say, “Good morning!” to Alexa today, and she’ll respond:

Good morning! It’s National Dictionary Day. Ever wonder what the shortest word is? Technically, it’s a toss-up between the single letter words

andI, but sinceais always capitalized, I’d sayIis just a little shorter.a

Is there anything more powerful than a language arts joke to get the day off to a good start?

I have no words to describe today. I do, however, have a ton of obscene gestures.

So, what’s the shortest **math** word? Technically, *e* and *i*, but if you don’t like constants, then you’ll have to settle for the three-letter words *set* and *box*.

And what’s the longest math word — at least based on the list at Math Words? It has 17 letters, and you’ll get a big hint if you check the time.

What two math words, both having the same number of letters, are equally appropriate to describe a triangle whose sides are congruent?

And what’s the funniest math word? Personally, I think it’s *syzygy*, but according to Tomas Engelthaler, it’s *logic*. In Humor Norms for 4,997 English Words, Engelthaler and Hills (2017) describe a method for determining which words are funniest. I emailed Engelthaler to ask which math word is funniest, and he responded as if it were a completely reasonable question. Without hesitation, he shared a list of math words and their humor rankings, and these five were at the top of the list:

- logic
- math
- theory
- science
- graph

The overall funniest English word, according to Engelthaler’s research? *Booty*. Go figure.

While you may not think that any of those words, mathy or otherwise, are laugh-out-loud funny, this isn’t debatable; it’s based on science.

If you take issue with this research, you’ll need to discuss it with Engelthaler and his colleagues. Please write to him directly to say that you’re bumfuzzled, that his research is malarkey, or that you think he’s a nincompoop.

### Friday Word Puzzle

Sometimes, a small word is contained in a longer word. For example, you can see the three-letter word *rid* tucked nicely inside *F riday* in the title for this post, and

*zip*can be found in the middle of

*mar*.

**zip**anSome folks have told me that the following word-in-a-word is particularly appropriate for this blog…

…since my puns put the UGH in LAUGHTER.

Words within words are the basis of today’s puzzle.

Complete each of the nine words below by placing a three-letter word in the blank. The three-letter words that you use all belong to the same category. But there is a tenth three-letter word from the same category that is not used below. What is the category, and what is the missing word?

- OB _ _ _ D
- C _ _ _ PY
- M _ _ _ OT
- PH _ _ _ M
- H _ _ _ SE
- VE _ _ _ D
- CA _ _ _ OU
- EL _ _ _ SE
- LE _ _ _ E

When I started to create this puzzle, I was hoping to give you a similar list in which a short math word was found in a longer word. I found several, but they seem pretty darn hard, and the missing words aren’t always obviously mathy. But for fun, you can try your hand at these, too…

- D _ _ _ Y
- AS _ _ _ E
- SE _ _ _ H
- BU _ _ _ _ SS
- EL _ _ _ _ TH
- C _ _ _ _ RA
- RU _ _ _ _ NT
- SH _ _ _ _ BLE
- BRA _ _ _ _ ILD
- HU _ _ _ _ D
- PR _ _ _ _ _ IFY
- RE _ _ _ _ NT
- WA _ _ _ _ ELON
- DE _ _ _ _ OR
- HO _ _ _ _ SS
- H _ _ _ _ HOG
- IM _ _ _ _ ST

ANSWERS

- OB eye D
- C hip PY
- M arm OT
- PH leg M
- H ear SE
- VE toe D
- CA rib OU
- EL lip SE
- LE gum E

The three-letter words are all parts of the body. The tenth word in that category is *jaw*, which never appears in the interior of a longer word (only at the beginning or end, such as * jawbone* or

*lock*).

**jaw**- D add Y
- AS sum E
- SE arc H
- BU sine SS
- EL even TH
- C hole RA
- RU dime NT
- SH area BLE
- BRA inch ILD
- HU more D
- PR equal IFY
- RE side NT
- WA term ELON
- DE mean OR
- HO line SS
- H edge HOG
- IM mode ST

### 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.

### Sound Smart with Math Words

When law professor Richard D. Friedman appeared in front of the Supreme Court, he stated that an issue was “entirely orthogonal” to the discussion. Chief Justice John G. Roberts Jr. stopped him, saying, “I’m sorry. Entirely *what*?”

“Orthogonal,” Friedman replied, and then explained that it meant *unrelated* or *irrelevant*.

Justice Antonin Scalia was so taken by the word that he let out an **ooh** and suggested that the word be used in the opinion.

In math class,* orthogonal* means “at a right angle,” but in common English, it means that two things are unrelated. Many mathematical terms have taken a similar path; moreover, there are many terms that had extracurricular meanings long before we ever used them in a math classroom. *Average* is used to mean “typical.” *Odd* is used to mean “strange” or “abnormal.” And *base* is used to mean “foundation.” To name a few.

The stats teacher said that I was average, but he was just being mean.

You know what’s odd to me? Numbers that aren’t divisible by 2.

An exponent’s favorite song is, “All About the Base.”

Even words for quantities can have multiple meanings. Plato used *number* to mean any quantity more than 2. And *forty* used to refer to any large quantity, which is why Ali Baba had forty thieves, and why the Bible says that it rained for forty days and forty nights. Nowadays, we use *thousands* or *millions* or *billions* or *gazillions* to refer to a large, unknown quantity. (That’s just grammatical inflation, I suspect. In a future millennium, we’ll talk of *sextillion* tourists waiting in line at Disneyland or of *googol* icicles hanging from the gutters.)

Zevenbergen (2001) provided a list of 36 such terms that have both math and non-math meanings, including:

- angle
- improper
- point
- rational
- table
- volume

The alternate meanings can lead to a significant amount of confusion. Ask a mathematician, “What’s your point?” and she may respond, “(2, 4).” Likewise, if you ask a student to determine the volume of a soup can, he may answer, “Uh… quiet?”

It can all be quite perplexing. But don’t be overwhelmed. Sarah Cooper has some suggestions for working mathy terms into business meetings and everyday speech. Like this…

For more suggestions, check out her blog post How to Use Math Words to Sound Smart.

If you really want to sound smart, though, be sure to heed the advice of columnist Dave Barry:

Don’t say:“I think Peruvians are underpaid.”

Say instead:“The average Peruvian’s salary in 1981 dollars adjusted for the revised tax base is $1452.81 per annum, which is $836.07 below the mean gross poverty level.”

NOTE: Always make up exact figures. If an opponent asks you where you got your information, makethatup, too.

This reminds me of several stats jokes:

- More than 83% of all statistics are made up on the spot.
- As many as one in four eggs contains salmonella, so you should only make three-egg omelettes, just to be safe.
- Even some failing students are in the top 90% of their class.
- An unprecedented 69.846743% of all statistics reflect an unjustified level of precision.

You can see the original version of “How to Win an Argument” at Dave Barry’s website, or you can check out a more readable version from the Cognitive Science Dept at Rensselaer.

Zevenbergen, R. (2001). Mathematical literacy in the middle years. *Literacy Learning: the Middle Years*, *9*(2), 21-28.

### 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

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.

- MPL
- XPR
- YMM
- MSCR
- MPT
- ITESI
- NSV
- RIGON
- OEFF
- CTAH
- THME
- SJU
- 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.

### Number Words and Learned Helplessness

How about some number word puzzles? Here’s a well-known puzzle that you’ve likely seen before:

What is the first positive integer that, when spelled out, contains the letter

a?

And here’s a modification of that puzzle that you may find a little more difficult:

What is the first positive integer that, when spelled out, contains the letter

c?

And taking it one step further:

What letters are

neverused in the spelling of any positive integer?

Who says that math isn’t useful in English class?

One more problem in a similar vein:

Pick any positive integer you like, and count the letters when that number is spelled out. Now count the letters when the resulting number is spelled out. Continue ad infinitum. What do you get?

Maybe those weren’t your cup of tea. Perhaps anagrams are more to your liking, so here are two (related) puzzles for you.

Try to make an anagram for each of the following three words.

- whirl
- slapstick
- cinerama
Too tough? Then try these three words instead.

- bat
- lemon
- cinerama

If you had trouble with the first set, you’re in good company. There are **no** anagrams for the words *whirl* or *slapstick*.

These two sets of words were used by Charisse Nixon, a pyschologist at Penn State–Erie, who gave the first set of words to half her class and the second set of words to the other half. She instructed them to find an anagram of the first word on their list; those students who had received the second set were successful. Nixon then instructed them to find an anagram of the second word on their list; again, those students who had received the second set were successful. When she then instructed them to find an anagram of the third word on their list — of which there is exactly one, **American** — those who hadn’t found anagrams for the first two words were less successful than their peers, even though the final challenge was identical.

Afterwards, students who received the first set of words admitted to feeling confused, rushed, frustrated, and stupid.

Nixon was studying **learned helplessness**, a condition in which a person suffers from a sense of powerlessness, often arising from persistent failure.

This has implications the math classroom. Students who perform at a fourth-grade level but are asked to participate in an eighth-grade class are surely as confused and frustrated as the subjects in Nixon’s experiment. Students need to occasionally feel success, or else they’ll shut down. If you’re a teacher, you don’t need me or a psychological research study to tell you that. So the question is, how can you get students to feel success? That is, what can you do to prevent learned helplessness?

My suggestion is to look for acceptable and accessible entry points.

Consider the following problem, which might be seen in a middle school classroom:

What is the maximum possible product of a set of positive integers whose sum is 20?

As written, that problem contains three words — *maximum*, *product*, and *integers* — that may confound some students. For middle school students who do understand the terminology, finding an appropriate strategy might be daunting.

In my opinion, the following is a better way to present this problem so that all students have an entry point:

Find some numbers with a sum of 20. Now, multiply those numbers together. Compare your result with a partner. Whose result was greater? Can the two of you work together to find a product that’s greater still?

Even a struggling middle school student could start this activity. Surely he could find some numbers with a sum of 20. Certainly, he could multiply them without a problem.

Why is this a better presentation? The wording is simplified. There is encouragement to work with a partner. It feels more like a collaborative game than a traditional math problem. It sounds — dare I say it? — like **fun**.

When a struggling student is able to get into a problem, and they’re able to make some strides in the right direction, and they’re rewarded by your positive encouragement, they attain some level of success. Maybe they won’t solve the problem entirely, but who cares? For many students, trying is progress.

And for students who are having trouble finding any success, perhaps the following words of encouragement will help.

If at first you don’t succeed, call it version 1.0.

If at first you don’t succeed, destroy all evidence that you ever tried.

If at first you don’t succeed, blame someone else and seek counseling.

If at first you don’t succeed, then skydiving is not for you.

If at first you don’t succeed, get new batteries.

If at first you don’t succeed, try two more times so your failure is statistically significant.