Posts tagged ‘word’

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.

dos-equis-logo-web

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.

10-10-10 Triangle

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.

October 6, 2016 at 10:10 am Leave a comment

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.

Orthogonal Definition

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…

deltaFor 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, make that up, 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.

May 4, 2016 at 7:47 am Leave a comment

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!

December 8, 2015 at 6:20 am Leave a comment

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.

  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.

April 9, 2015 at 10:17 pm 2 comments

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 never used 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.

 

January 8, 2015 at 7:42 am Leave a comment

Word + Letter = Math Term

AnagramOn a recent Sunday Puzzle on NPR, Will Shortz gave a letter and a word, and the contestant was to guess the name of a popular TV show using an anagram of the letters (“Coming to TV This Fall: Anagrams,” Oct 12, 2014). For instance,

M + NAMED

gave the answer

MAD MEN.

This struck me as an interesting puzzle format. My only criticism is that it just wasn’t mathy enough.

But I’m not a problem maker, I’m a problem solver… so rather than cast aspersions at the puzzle, I’ll instead use the format to offer my own version.

Each of the 26 letters of the alphabet has been paired with a common English word. An anagram of the pair will yield a common math word. How many can you find?

  1. A + ERA
  2. B + AGLARE
  3. C + BITES
  4. D + NOTICER
  5. E + EDGERS
  6. F + SAUCER
  7. G + LEAN
  8. H + OPERABLY
  9. I + TANGLER
  10. J + INDUCTIONS
  11. K + SEW
  12. L + POSE
  13. M + RIPS
  14. N + AIMED
  15. O + PINT
  16. P + MYRIAD
  17. Q + AURES
  18. R + ENVIES
  19. S + RECITED
  20. T + HAM
  21. U + RAIDS
  22. V + EXERT
  23. W + ROPE
  24. X + SEA
  25. Y + PENTHOUSE
  26. Z + ORE

I don’t believe in providing an answer key, but you can find some help at Math Words, and you can click over to More Words if you run into real trouble. But give it the old college try before seeking assistance. Honestly, you’ll feel better about yourself if you solve these on your own.

October 31, 2014 at 7:10 am Leave a comment

Science Word Game for Road Trips

For your next road trip, a fun, family-friendly game… especially if your family tree has a lot of geeky branches.

Couldn’t be simpler:
Combine abbreviations from the periodic table to create a word.

For instance, you could combine the abbreviations for barium and rutherfordium to make BaRf.

Or use chromium, iodine, titanium, carbon, and aluminum to make CrITiCAl.

Or my current favorite — combine titanium, platinum, oxygen, and einsteinium to form TiPtOEs. How fun!

Post your longest words in the comments. (If you choose to write a computer program to find the longest possible word, please don’t spoil everyone else’s fun.)

Uber-geeks will want to use a formula such as

Points = 3 × No. of Letters + 5 × No. of Elements Used

to score the game. And you can if you like; play ten rounds, and highest score wins. But I say just have fun, make words, and impress everyone riding in the car with you!

You might find the following lists helpful.

Single-Letter Abbreviations:
B, C, F, H, I, K, N, O, P, S, U, V, W, Y

Two-Letter Abbreviations, With A Vowel:
Ac, Ag, Al, Am, Ar, As, At, Au, Ba, Be, Bh, Bi, Bk, Br, Ca, Ce, Co, Cu, Er, Es, Eu, Fe, Ga, Ge, He, Ho, In, Ir, La, Li, Lu, Mo, Na, Ne, Ni, No, Os, Pa, Po, Pu, Ra, Re, Ru, Se, Si, Ta, Te, Ti, Xe

Two-Letter Abbreviations, No Vowel:
Cd, , Cf, Cl, Cm, , Cp, Cr, Cs, Db, Ds, Dy, Fm, Fr, Gd, Hf, Hg, Hs, Kr, Lr, Md, Mg, Mn, Mt, Nb, Nd, Np, Pb, Pd, Pm, Pr, Pt, Rb, Rf, Rg, Rh, Rn, Sb, Sc, Sg, Sm, Sn, Sr, Tb, Tc, Th, Tl, Tm, Yb, Zn, Zr

Three-Letter Abbreviations:
Uuh, Uuo, Uup, Uuq, Uus, Uut

Periodic Table

Click Image for Larger Version

October 17, 2012 at 6:45 pm 2 comments

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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.

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