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.

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