## Posts tagged ‘letter’

### Math Puzzles with Letters

This week on the NPR Sunday Puzzle, host Will Shortz offered the following challenge:

Name a famous city in ten letters that contains an

s. Drop thes. Then assign the remaining nine letters their standard value in the alphabet — A = 1, B = 2, C = 3, etc. The total value of the nine letters is only 25. What city is it?

It’s not much of a spoiler to note that the average value of those nine letters must be less than three, since their sum “is only 25.” Consequently, a lot of those letters must occur at the beginning of the alphabet and — if eight of them were *a*‘s — there would be no letters later than *q* in the name of the city. But that’s as much as I’ll say; you can solve the puzzle on your own. (When you do, you can submit your answer for a chance to play next week’s on-air puzzle live with Will Shortz.)

Mathematician Harold Reiter uses a similar problem with elementary school students. Using the same idea — that each letter has a value (in cents) equal to its position in the alphabet — he asks students to find a dollar word, that is, a word whose letters have a sum of 100. As it turns out, there are many. Based on a nonexhaustive search, there are at least 3,500 dollar words, and likely a whole lot more. In a quick perusal of the list, one word jumped out: **oxygon**. Nope, that’s not a typo. It’s an archaic term meaning “a triangle with three acute angles.”

All of this talk of letters reminds me of my favorite puzzle, which I call Product Values. Using the same scheme — that is, A = 1, B = 2, C = 3, etc. — find the product value of a word by *multiplying* the values of the letters. So, for instance, *cat* has a product value of 3 × 1 × 20 = 60. How many words can you find that have a product value of 100? Based on the ENABLE word list, there are nine. (If you need some help, you can use the Product Value Calculator at www.mathjokes4mathyfolks.com.)

To end this post, a few math jokes that involve letters:

And Satan sayeth, “Let’s put the alphabet in math.” Bwa-ha-ha-ha-ha.

Romans had no trouble with algebra, because X was always equal to 10.

### 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.

### One-Letter Quiz

The answer to each question below is a letter of the alphabet. Each letter is used exactly once. (Thanks for the idea, *Ask Me Another*.) Good luck!

**Want to amuse your friends, irritate your students, or annoy people you’ve just met? Download a ****PDF version of the One-Letter Quiz**** (without answers).**

- The letter used to represent the square root of -1.
- This letter is often added to indefinite integrals to show that any function with at least one antiderivative has an infinite number of them.
- The most frequently occurring letter in English words.
- The letter most recently added to the modern, 26-letter English alphabet.
- The letter represented by four dots in Morse Code.
- A type of road intersection with three arms.
- Although long out of use, this letter was used in the middle ages as the Roman numeral to represent 90.
- This letter is used for the temperature scale in which the boiling point is 212 degrees and the freezing point is 32 degrees.
- The most common blood type.
- The rating from the Motion Picture Association of America that requires children under 17 to be accompanied by an adult.
- The 43rd President of the United States.
- The only vowel that does not appear in the spelling of any single-, double-, or triple-digit numbers.
- Between
*s*and*c*, the second most common letter with which English words begin. - With
*plan*, the letter used to refer to a typically less desirable alternative. - The Roman numeral for 500.
- The symbol for potassium on the periodic table.
- The most common variable in algebra.
- The Roman numeral for 5.
- The “score” used to indicate the number of standard deviations a data point is from the mean.
- The letter commonly used to refer to the vertical axis on a coordinate graph.
- Although every adult can recognize the loop-tail version of this lowercase letter in print, less than one-third of participants in a Johns Hopkins study could correctly pick it out of a four-option lineup.
- The clothing size that increases when preceded by an X.
- The shape of the “happiness curve,” which implies that most people are least happy in their 50’s.
- The shape of a logistic growth curve, which increases gradually at first, more rapidly in the middle, and slowly at the end, leveling off at a maximum value after some period of time.
- The only letter that does not appear in the name of any US state.
- The answer to the riddle, “It occurs once in a minute, twice in a moment, but never in a thousand years.”

**Answers (and Notes of Interest)**

- I
- C
- E
- J : in 1524, Gian Giorgio Trissino made a clear distinction between the sounds for
*i*and*j*, which were previously the same letter - H
- T
- N : see Wikipedia for a list of other Roman numerals used in medieval times
- F
- O
- R
- W : should probably be “Dubya” instead of “Double U,” but whatever
- A
- P : as you might expect, more English words start with S than any other letter; based on the ENABLE word list, P is the second most common initial letter, followed by C
- B
- D
- K : the symbol K comes from
*kalium*, the Medieval Latin for*potash*, from which the name*potassium*was derived - X
- V
- Z
- Y
- G : a lowercase
*g*can be written in two different ways, and the more common version in typesetting (known as the “loop-tail*g*“) can be recognized but not written by most adults, as recounted on the D-Brief blog - L
- U : see this article from
*The Economist*, especially this image - S
- Q
- M

### A Mathematical Thank-You Note

I recently sent a copy of **More Jokes 4 Mathy Folks** to a friend of an acquaintance, and today I received a thank-you letter from the recipient. I think the letter is worth sharing, so here you go:

Patrick —

Thanks so much for sending me

… I’ve already annoyed my entire family, and I’m only 46/111 through the book. (I think your work here is done.)More JokesI can’t provide a formal proof, but empirically speaking, this is the best collection of math jokes known to exist. Honestly, I enjoy the blend of quips and jokes; almost everything I read translates well into a groaner I can splice into my physics classroom repertoire!

Hoping you don’t object to the informality of my “stationery” — thought you’d appreciate the utility of engineering paper.

^{1}Thanks again! Best,

RobP.S. I noticed — when I looked up your address (ed. note:

STALKER!) — that we only live about seven root two^{2}minutes apart. How would you feel about raising a pint some evening? I’d love to hear about your work and exchange a few quips!

This is one of the nicer letters I’ve received, and it was fun to be reminded that not all mail needs to be electronic. And you can bet that some of those compliments will be used on a promotional flyer soon!

^{1} Quarter-inch quadrille paper, if you must know.

^{2} He had correctly written the irrational number as

but since LaTeX looks like shit as inline text, I converted it to straight English.

### 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.

### Word + Letter = Math Term

On 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?

- A + ERA
- B + AGLARE
- C + BITES
- D + NOTICER
- E + EDGERS
- F + SAUCER
- G + LEAN
- H + OPERABLY
- I + TANGLER
- J + INDUCTIONS
- K + SEW
- L + POSE
- M + RIPS
- N + AIMED
- O + PINT
- P + MYRIAD
- Q + AURES
- R + ENVIES
- S + RECITED
- T + HAM
- U + RAIDS
- V + EXERT
- W + ROPE
- X + SEA
- Y + PENTHOUSE
- 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.

### Book Review: *365 Things To Make You Go Hmmm…*

Before reading *365 Things That Make You Go Hmmm…*, I hadn’t realized that I’d been on Earth for 1.3 billion seconds, and I never thought about what someone would feel like after spending a day in my mind. That’s the beauty of this incredible book — it asks you to think about things that you’ve probably never thought about before. The questions are great for starting classroom discussions, but they also work well for sparking a conversation between a parent and child, or as an icebreaker at your next social event.

The book contains introspective questions (“What makes you irreplaceable?”), but it also contains math and logic puzzles like the following:

Before this piece of paper was folded over once, it was a capital letter. It wasn’t the letter L — that would be too easy. Which letter was it?

I’m also a big fan of puzzle #110, which starts:

An

antigramis word [or phrase] that when you rearrange the letters you can make a new word or phrase that means something very different — in fact, almost the opposite! For example:.earliest→rise late

It then provides a list of antigrams and asks for the opposite word or phrase. One of the antigrams is:

within earshot

Flummoxed, I looked at the answer in the back of the book, which read:

I won’t hear

I realized immediately that something was wrong. The given answer did not contain enough letters. And then I gasped, because I realized which letters had been omitted:

Wow! I emailed Paul Wrangles (the author) immediately and asked if the answer was given as “I won’t hear” so as to avoid writing “I won’t hear shit,” or if this was simply a typo. He assured me that it was only a typo, and the correct answer is supposed to be:

I won’t hear THIS

Whew!

With that mystery solved, I viewed the other 360 things and thoroughly enjoyed them. My sons and I have been working our way through them, though they’re so addictive, we rarely stop at answering just one a day. We’re hoping for a second volume — we need more questions to last an entire year!

*365 Things That Make You Go Hmmm…* is an amazing resource. Chock full of questions from ordinary to extraordinary, it made my head hurt — but in a good way!

I highly recommend this book for any teacher, parent, or curious individual.

### Permutations

Tonight at dinner, my wife told Alex that he had to eat his carrots. “How many?” he asked. There were three on his plate. “Two,” she responded. (For what it’s worth, I deplore these dinnertime negotiations. I put three carrots on his plate — I expect him to eat three carrots. If I only wanted him to eat two, I would have given him only two. On the other hand, my kids are 3½ and they actually ask for broccoli, edamame, carrots, kale, and a host of other veggies, so I can’t really complain.)

Alex then turned to his brother and asked, “Which two should I eat, Eli?” He picked up Carrot A and Carrot B and asked, “These two?”; he then returned Carrot B to his plate and picked up Carrot C and asked, “These two?”; and, finally he picked up Carrot B and Carrot C and asked, “Or these two?”

I was pretty psyched about Alex’s “proof without words” that _{3}*C*_{2} = 3.

Of course, we all know how cool permutations are, since all of our inboxes have been filled by the text of a letter written by Graham Rawlinson to *New Scientist* in 1999. You know the one, which purported “that randomizing letters in the middle of words had little or no effect on the ability of skilled readers to understand the text.”

For iancstne, you utadnnersd this snenctee celalry eevn tughoh the ltetres of msot wodrs are out of oedrr.

It is inaccurate to say that there is “no effect,” however. A follow-up study showed that college students experienced a 12% decrease in overall reading speed when confronted with sentences containing transposed letters. Quite a few of the other statements in the email that we received — for instance, that there was a study done at Cambridge (there wasn’t) — were also inaccurate. Even the email itself is misleading, ostensibly written to enhance the desired effect and further prove its point; in truth, almost half of the words contain only two or three letters and are spelled correctly.

But don’t blame Graham Rawlinson for all of that. That’s just the way things work in cyberspace.

Okay, enough already. How ’bout some math?

How many permutations exist for the word PERMUTATIONS?

And last but not least, some permutation jokes…

Combinatorists do it in every possible permutation, but they do it discretely.

What do you get if you add Daytona Beach, Pismo Beach, Palm Beach, and South Beach in various permutations?

Sums of beaches.