## Posts tagged ‘Will Shortz’

### 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.

### A Week of KenKen, Day 1: Introduction

Welcome to **A Week of KenKen** (AWOKK). Every day this week, the MJ4MF blog will feature a new post about the number puzzle that Sudoku wishes it could be. That’s right — seven days, nothing but KenKen.

Here’s a list of the posts that you’ll see in coming days:

- Day 1: Intro (that’s
*today*!) - Day 2: The KENtathlon
- Day 3: KenKen Times
- Day 4: My KenKen Puzzles
- Day 5: Harold Reiter’s Puzzles
- Day 6: KenKen Glossary
- Day 7: KenKen Puzzle for 2016
- Day 8: KenKen in the Classroom

If the Beatles got nothin’ but love, babe, eight days a week, then I can certainly have a week with eight days of KenKen. Deal with it.

Today is an introduction, for those of you unfamiliar with KenKen. Here are the rules of the puzzle:

- For an
*n*×*n*grid, fill each row and column with the numbers 1 through*n*. A number may not be repeated in any row or column. - Each heavily outlined set of cells, called a
*cage*, contains a mathematical clue that consists of a number and an arithmetic operation: +, –, ×, or ÷. The numbers in that cage must combine (in any order) to produce the target number using the mathematical operation indicated. - Cages with just one cell should be filled with the target number.
- A number may be repeated within a cage, provided it’s not in the same row or column.

*The New York Times* crossword puzzle editor and *Weekend Edition* puzzlemaster Will Shortz explains KenKen in this short video:

Ready to try for yourself? Here’s a simple puzzle, which is dubbed an “easy” puzzle from the KenKen website:

Too easy? Here’s a slightly more interesting one that I created:

Did that whet your appetite for more? If you were a kid who could’ve held out for several minutes to get two marshmallows, then check back tomorrow for the next installment. But if you were a kid who just couldn’t wait and would’ve gobbled that single marshmallow immediately, then here’s your instant KenKen gratification:

- kenken.com
- Illuminations KeKen app (provided by NCTM, iPhone only)
- MathDoku Pro (my personal favorite, but Android only)

Till tomorrow, happy solving!

### 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!

### 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.