A Puzzle of Few Words

Winston Churchill once said, “The short words are the best, and the old words best of all.”

Today’s post contains several puzzles, the answer to each is a short word, and at least one of the answers is a very old word, indeed.

But first, a joke about words and math:

Teacher: Use the word announce in a sentence.
Student: Yes, ma’am. Announce is one-sixteenth of a pound.

The first puzzle is about the game Hangman.

In the game of Hangman, the first player thinks of a word and reveals the number of letters. The second player then guesses letters. If a guessed letter is in the word, the first player reveals the position(s) of every occurrence of that letter within the word. If the guessed letter is not in the word, then the second player receives a body part for a man who is hanging from a gallows (hence the name, Hangman). If the entire man is completed before the word is guessed, the second player has been hanged and loses.

Various versions of the game use hangmen with different numbers of body parts. The number of parts typically ranges from 7 to 13. The Hangman game at www.playhangmangames.net contains 7 body parts, and my sons play a version with 11 body parts:

Jon McLoone at the Wolfram Blog ran a simulation to determine the best words for the game of Hangman. Of course, you could just click on that link, but it might be more fun to think about the following question before you do:

What is the best word to use when playing Hangman? And does the best word change, depending on how many body parts are in the version you play?

Surprisingly, McLoone found that there is a single best Hangman word, for any game with 8 to 13 body parts. A strong hint is included at the bottom of this post.

The second puzzle is a product value puzzle, in which the product value of a word is equal to the product of the value of the letters. Specifically,

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.

One fun puzzle based on this set-up:

Find an “acre” word, which is a word with a product value of 43,560, the number of square feet in an acre.

The (unique) answer happens to be one of my favorite English words.

Turning this idea around, another variation is as follows:

Find a four-letter English word with the largest possible product value.

According to Scrabble Australia, there are 16,739 four-letter words. However, many of those would not be considered common — such as euoi (an impassioned cry), nabk (berries you’ve never tasted, from a plant you’ve never seen), and zizz (a short sleep). The last of these examples has an impressive product value of 26³ = 17,576, but it’s not the highest. Not even close, in fact; its product value is only 7.8% of the largest product value for a four-letter word.

You can explore similar puzzles with the Product Value Calculators on the MJ4MF website.

Hints for all three puzzle appear below the following joke about words and math:

A boy was told to write an essay about his favorite subject. He wrote, “I really love math,” and turned in his essay. Returning his paper, the teacher said, “Sorry, your essay needs to be at least 100 words.” So he wrote, “I really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really love math.”

Hints

The best Hangman word (according to McLoone’s analysis) has a product value of 6,760.

The defintion of the acre word is “alternating layers of silt or clay, usually of contrasting colors, that comprise an annual cycle of deposition in a body of still water.” That doesn’t help? Then just enter 43,560 into the second form on the Product Value Calculators page.

And the four-letter word with the largest possible product value? It’s an anagram of the mythological river that divides Earth and the Underworld, as well as an anagram of the 70’s rock band that sang Come Sail Away and Mr. Roboto.

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?