## Posts tagged ‘puzzle’

### 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.

### Change the Vowel

The following puzzle contains a clue within each clue. The answer, of course, fits the clue, but each answer is also one of the words within the clue with its vowel sound changed. For instance, the clue “the distance from the top of your hat to the sole of your shoes” contains the word *hat*, and if you change the vowel sound from a short *a* to a long *i*, you get *height*, which fits the description.

As always, there’s a catch. Every answer to the clues below is a mathy word.

- The number of permutations of three different colored socks.

- This value is the same for 3 + 1 and 2 + 2.
- Do this with two odd numbers and you’ll get an even number.
- Three hours before noon.
- You may fail to correctly expand (
*a*+*b*)(*c*+*d*) if you don’t remember this mnemonic. - One represents this fractional portion of toes on the hoof of a deer.
- The last element of a data set arranged in descending order.
- The measure of central tendency made from the most common data points.
- The number of contestants ahead of third place if there’s a tie for first place.
- If the leader is forced to drop out of a race, the runner-up takes over this place.
- The square root is needed to calculate the length of the hypotenuse in this type of triangle.
- The graph of the declination of the Sun during a year can be approximated by this type of curve.

Answers

- six

- sum
- add
- nine
- FOIL
- half
- least
- mode
- two
- first
- right
- sine

### My 0.04 Seconds of Fame

In 2017, I attended the International KenKen Championship and filmed the final round, which I posted previously on this blog. But filmmakers Louis Cancel, Chris Flaherty, and Daniel Sullivan were there that day, too, and their cameras were significantly more sophisticated than my Samsung S8. Their footage of the competition, coupled with myriad interviews of competitors, organizers, and the inventor of KenKen himself, Tetsuya Miyamoto, has resulted in a new documentary, * Miyamoto and the Machine*, recently released by

*The New Yorker*. It tells the story of KenKen’s origins and attempts to answer the question, “Can a computer make puzzles as beautiful as those created by humans?”

Many aspects of the film will appeal to kenthusiasts, but my favorite moment occurs at 17:14. Competitor Ellie Grueskin is competing in the finals, and just over Ellie’s left shoulder is a barely visible, occasionally funny, middle-aged math guy holding — wait for it — a Samsung S8!

Yep, that’s me. I’m a star!

You have to ask yourself, what kind of monster would author such a shamelessly self-promotional post and not even provide one KenKen puzzle for the reader to enjoy? Definitely not me, so here you go.

After you solve the puzzle, definitely watch * Miyamoto and the Machine*. It’s 25 minutes well spent.

### 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.

### A No-Op KenKen for Today

This will be a short post, just to share a puzzle for today.

There’s nothing inherently special about today — though it is the 30th anniversary of The Simpsons airing on Fox, and, slightly less important, the anniversary of Wilbur and Orville Wright’s famous flight — except that (a) I introduced the students in our middle school math club to KenKen last week, and (b) today is our last meeting before the holiday break, so I thought I’d do something special and create a KenKen puzzle that used the numbers from today’s date. I had hoped to include 12, 17, 20, and 19 as the target numbers in the cages, but that effort proved fruitless. Instead, I opted for 12, 1, 7, and 19 as the target numbers, and I filled in the single-cell cage in the bottom right with its number, 3.

I rather like the result. The puzzle is not terribly difficult; and, the solution is not unique, which I figure is perfect for kids who just learned about KenKen a week ago.

If you’re not familiar with No-Op KenKen, they’re just like regular KenKen puzzles, but the operation isn’t included with the target number. Instead, you’ll need to discern the operation for each cell. (For another example of a no-op KenKen puzzle, check out Harold Reiter’s No-Op 12 Puzzle.)

Enjoy, good luck, and happy December 17!

### Silent Letter Night

Several weeks ago, Will Shortz presented an NPR Sunday Puzzle in which he stated a word and a letter, and the resulting collection would be rearranged to form a new word in which *the added letter is silent*. For instance, if Will gave RODS + W, the correct answer would be SWORD, in which the W is silent. (Note that the collection of letters is also an anagram of WORDS, but the W isn’t silent.)

At a time of year known for silent nights, it seems like a puzzle involving silent letters is completely appropriate. I’ve borrowed Shortz’s idea and extended it a bit; some of the clues in the list below have more than one silent letter added. Many items in the list are related to today’s holiday; and, because this is a math blog, the others are related to mathematics. In full disclosure, two of the answers are proper nouns.

Enjoy, and happy holidays!

- TO + W =
- TON + K =
- TOGS + H =
- GEE + D =
- TIN + G + H =
- SIN + G =
- SIN + E =
- CORD + H =
- HOLE + W =
- HEART + W =
- TINNY + E =
- COINS + E =
- NOELS + M =
- PILES + E + L =
- FRAME + T =
- REDACT + E + S + S =
- RACISMS + H + T =

### 12 Math Games, Puzzles, and Problems for Your Holiday Car Trip

It doesn’t matter if you’re one of the 102.1 million people traveling by car, one of the 6.7 million people traveling by plane, or one of the 3.7 million people traveling by train, bus, or cruise ship this holiday season — the following collection of games, puzzles, and problems will help to pass the time, and you’ll be there long before anyone asks, “Are we there yet?”

**Games**

*1. Street Sign Bingo*

Use the numbers on street signs to create expressions with specific values. For instance, let’s say you see the following sign:

The two numbers on the sign are 12 and 3, from which you could make the following:

12 + 3 =

1512 – 3 =

912 × 3 =

3612 ÷ 3 =

4

Of course, you could just use the numbers directly as 3 or 12. Or if your passengers know some advanced math, they could use square roots, exponents, and more to create other values.

Can you split the digits within a number and use them separately? Can you concatenate two single-digit numbers to make a double-digit number? That’s for you and your traveling companions to decide.

To play as a competitive game, have each person in the car try to get every value from 1‑20. It’s easiest to keep track if you require that players go in order. To make it a cooperative game, have everyone in the car work collectively to make every value from 1 to 100.

As an alternative, you could use the numbers on license plates instead.

*2. Bizz Buzz Bang*

Yes, I’ve played this as a drinking game. No, I don’t condone drinking while driving. No, I don’t condone under-age drinking, either. Yes, I condone playing this game with minors while driving. (See what I did there?)

The idea is simple. You pick two single-digit numbers, *A* and *B*. Then you and your friends start counting, one number per person. But each time someone gets to a number that contains the digit *A* or is a multiple of *A*, she says, “Bizz!” Every time someone gets to a number that contains the digit *B* or is a multiple of *B*, she says, “Buzz!” And every time someone gets to a number that meets both criteria, he says, “Bang!”

For example, let’s say *A* = 2 and *B* = 3. Then the counting would go like this:

one, bizz, buzz, bizz, five, bang, seven, bizz, buzz, bizz, eleven, bang, buzz, …

When someone makes a mistake, that round ends. Start again, and see if you can beat your record.

You can use whatever numbers you like, or modify the rules in other ways. For instance, what if bizz is for prime numbers and buzz is for numbers of the form 4*n* + 3? Could be fun!

*3. Dollar Nim* (or any other variation)

Nim is a math strategy game in which players take turns removing coins from a pile. Different versions of the game are created by adjusting the number of coins that can be removed on each turn, the number of coins in the pile originally, how many piles there are, and whether you win (normal) or lose (misère) by taking the last coin.

A good aspect of Nim is that you don’t actually need coins. To play in the car, players just need to keep track of a running total in their heads.

A version of Nim dubbed **21 Flags** was played on *Survivor Thailand* several years ago. There were 21 flags, and each team could remove 1, 2, or 3 flags on each turn. The team to remove the last flag won. This is a good first version to play in the car, especially for young kids who have never played before. Then mix it up by changing the initial amount and the number that can be removed on each turn.

Our family’s favorite version, **Dollar Nim**, is played by starting with $1.00 and removing the value of a common coin (quarter, dime, nickel, penny) on each turn. (While discussing this post with my sons, they informed me that they much prefer **Euro Nim**, which begins with 1€, but then the coin values to be removed are 50c, 20c, 10c, 5c, 2c, and 1c. It’s essentially the same game, but they like the different coin amounts.)

**Three-Pile Nim** consists of not just one but three piles with 3, 5, and 7 coins, respectively. On each turn, a player must remove at least one coin, and may remove any number of coins, but all removed coins must be *from the same pile*.

Finally, **Doubling Nim** is played as the name implies. On the first turn, a player may remove any number of coins but not the entire pile. On every turn thereafter, a player may remove any number of coins up to double the number taken on the previous turn. For instance, if your opponent removes 7 coins, then you can remove up to 14 coins.

For every version of Nim, there is an optimal strategy. We’ve wasted hours on car trips discussing the strategy for just one variation. Discussing the strategies for all the versions above could occupy the entire drive from Paducah to Flint.

*4. Guess My Number
*

One person picks a number, others ask questions to try to guess the number.

The simplest version is using “greater than” and “less than” questions. Is it greater than 50? Is it less than 175? And so forth. Using this method, the guessers can reduce the number of possibilities by half with each question, so at most, it should take no more than *n* guesses if 2^{n} > *m*, where *m* is the maximum possible number that the picker may choose. For instance, if the picker is required to choose a number less than 100, then *n* = 7, because 2^{7} = 128 > 100. Truthfully, this version of the game gets boring quickly, but it’s worth playing once or twice, especially if there’s one picker and multiple guessers. And a conversation about the maximum number of guesses need can be a fun, mathy way to spend 15 minutes of your trip.

A more advanced version excludes “greater than” and “less than” questions. Instead, the guessers can ask other mathematical questions like, “Is it a prime number?” or “Do the digits of the number differ by 4?” Those questions imply, of course, that the answer must be yes or no, and that’s typical for these types of guessing games. If you remove that restriction, though, then guessers could ask questions that reveal a little more information, like, “What is the difference between the digits?” or “What is the remainder when the number is divided by 6?” With this variant, it’s often possible to identify the number with two strategic questions.

**Puzzles**

Here are three puzzles that can lead to hours of conversation — and frustration! — on a car trip. Before you offer one to your crew, though, put forth the disclaimer that anyone who’s heard the puzzle before must remain mum. No reason they should spoil the fun for the rest of you. (The puzzles are presented here without solution, because you’ll know when you get the right answer. You can find the answer to any of them online with a quick search… but don’t do that. You’ll feel much better if you solve it yourself.)

*5. Dangerous Crossing*

Four people come to a river in the middle of the night. There’s a narrow bridge, but it’s old and rickety and can only hold two people at a time. They have just one flashlight and, because it’s night and the bridge is in disrepair, the flashlight must be used when crossing the bridge. Aakash can cross the bridge in 1 minute, Britney in 2 minutes, Cedric in 5 minutes, and Deng in 8 minutes. When two people cross the bridge together, they must travel at the slower person’s pace. And they need to hurry, because zombies are approaching. (Oh, sorry, had I failed to mention the zombie apocalypse?) What is the least amount of time that all four people can cross the bridge?

And how can you be sure that your method is the fastest?

*6. Weight of Weights*

Marilyn has a simple balance scale and four small weights, each weighing a whole number of grams. With the balance scale and these weights, she is able to determine the weight of any object that weighs between 1 kg and 40 kg. How much does each of the four weights weigh?

*7.* *Product Values*

Assign each letter a value equal to its position in the alphabet, i.e., A = 1, B = 2, C = 3, …, Z = 26. Then for any common word, find its product value by multiplying the value of the letters in the word. For instance, the product value of CAT is 60, because 3 × 1 × 20 = 60.

- Find as many common English words as you can with a product value of 60.
- Find a common English word that has the same product value as your name. (A little tougher.)
- Find a common English word with a product value of 3,000,000. (Zoiks!)

**Problems**

*8. The Three of Life*

This one looks so innocent!

What’s the probability that a randomly chosen number will contain the digit 3?

But spend a little time with it. And prepare for. Mind. Blown.

*9. Hip to Be (Almost) Square*

No calculators for this one.

There are four positive numbers — 1, 3, 8, and

x— such that the product of any two of them is one less than a square number. What is the least possible value ofx?

No spreadsheets, either.

*10. Hip to Be Square (Roots)
*

Just some good, old-fashioned algebra and logic to tackle this one.

Which of the following expressions has a greater value?

or

Surprised?

*11. Coming and Going*

Potentially counterintuitive.

At the holidays, Leo drove to his grandma’s, and the traffic was awful! His average speed was 42 miles per hour. After the holidays, however, he drove home along the same route, and his average speed was 56 miles per hour. What was his average speed for the entire trip?

And, no, your first guess was most likely not correct.

*12. The Year in Numbers*

A moldy oldie, to be sure, but this puzzle is always a crowd-pleaser for those who haven’t seen it before.

Use the digits of the new year — 2, 0, 1, and 9 — and any mathematical operations to form the integers from 1 to 100. For instance, you can form 1 as follows: 2 × 0 × 9 + 1 = 1.

For an added challenge, add the restriction that you must use the four digits *in order*.

**Bonus: A Book**

My sons get sick when they read in the car. That’s why I love the two books *Without Words* and *More Without Words* by James Tanton. Literally, there are no words! Each puzzle is presented using a few examples, and then students must follow the same rules to solve a few similar, but more challenging, puzzles.

Wherever you’re headed during the holiday break — driving to your relatives’ house, flying to Fort Lauderdale, or just relaxing at home — I hope your holidays are filled with joy, happiness, and lots of math!

### 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

### WODB, Quora Style

The following puzzle was recently posted on Quora:

Which of the following numbers don’t belong: 64, 16, 36, 32, 8, 4?

What I liked about this puzzle was the answer posted by Danny Mittal, a sophomore at the Thomas Jefferson High School for Science and Technology. Danny wrote:

64 doesn’t belong, as it’s the only one that can’t be represented by fewer than 7 binary bits.

36 doesn’t belong, as it’s the only one that isn’t a power of 2.

32 doesn’t belong, as it’s the only one whose number of factors has more than one prime factor.

16 doesn’t belong, as it’s the only one that can be written in the form

x, where^{y}xis an integer andyis a number in the list.8 doesn’t belong, as it’s the only one that doesn’t share a digit with any other number in the list.

4 doesn’t belong, as it’s the only one that’s a factor of all other numbers in the list.

I suspect that Danny has visited Which One Doesn’t Belong or has read Christopher Danielson’s *Which One Doesn’t Belong*. Or maybe he’s just a math teacher groupie and trolls MTBoS.

But then Jim Simpson pointed out the use of “don’t” in the problem statement, which I had assumed was a grammatical error. Jim interpreted this to mean that there must be two or more numbers that don’t belong for the same reason, and with that interpretation, Jim suggested the answer was 32 and 8, since all of the others are square numbers.

Don’t get me wrong — I don’t think this is a great question. But I love that it was interpreted in many different ways. It could lead to a good classroom conversation, and it makes me consider all sorts of things, not the least of which is standardized assessments. How many times have students gotten the wrong answer for the right reason, because they interpreted an item on a state exam or the SAT differently than the author intended? And how many times have we bored students with antiseptic questions, only because we knew they’d be free from such alternate interpretations? Both scenarios make me sad.

### 2017 KenKen International Championship

If you like puzzles and ping pong, then Pleasantville, NY, was the place to be on December 17.

More than 200 Kenthusiasts — people who love KenKen puzzles — descended on Will Shortz’s Westchester Table Tennis Center for the 2017 KenKen International Championship (or the KKIC, for short). Participants followed 1.5 hours of solving KenKen puzzles with a pizza party and several hours of table tennis.

The competition consisted of three rounds, with the three puzzles in each round slightly larger and more difficult than those from the previous round. Consequently, competitors were given 15, 18, and 20 minutes to complete the puzzles in the first, second, and third rounds, respectively.

Competitors earned 1,000 points for each completely correct puzzle, and 0 points for an incomplete or incorrect puzzle. In addition, a bonus of 5 points was earned for every 10 seconds in which a puzzle was turned in before time was called. So, let’s say you got two of the three puzzles correct and handed in your answers with 30 seconds remaining in the round; then, your score for that round would be

The leader after the written portion was John Gilling, a data scientist from Brooklyn, whose total score was 10,195. And if you’ve been paying attention, then you know what that means — Gilling earned 9,000 points for completing all of the puzzles correctly, so his time bonus was 1,195 points… which is the amount you’d earn for turning in the puzzles 2,390 seconds (combined) before time was called. The implication? Gilling solved all 9 puzzles from the written rounds — which contained a mix of puzzles from size 5 × 5 to 8 × 8 — in just over 13 minutes.

Wow.

As a result, Gilling, the defending champion, earned a spot in the Championship Round against Tess Mandell, a math teacher from Boston; Ellie Grueskin, a high school senior at The Hackley School; and Michael Holman, a technology consultant. In the final round, each of them attempted a challenging 9 × 9 puzzle, which was displayed on an easel for the crowd to see. Solving a challenging 9 × 9 is tough enough; having to do it as 200 kenthusiasts follow your every move is even tougher.

So, how’d they do? See for yourself…

When the dust settled, Gilling had successfully defended his title. For his efforts, he received a check for $500. But more importantly, he retained bragging rights for one more year.

If you think you’ve got what it takes to compete with the best KenKen solvers, try your hand at the 9 × 9 puzzle that was used in the final round. In the video above, you saw how fast Gilling solved it to win the gold. But even the slowest of the four final-round participants finished in under 15 minutes.

Again, wow.

Finally, I’d be failing as a father if I didn’t mention that my sons Alex and Eli competed in the Delta (age 10 and under) division. Though bested by Aritro Chatterjee, a brilliant young man who earned a trip to the 2017 KKIC by winning the UAE KenKen Championship, Eli took the silver, and Alex brought home the bronze. They’re shown in the photos below with Bob Fuhrer, the president of Nextoy, LLC, the KenKen company and host of the KKIC.

#proudpapa

For more KenKen puzzles, check out www.kenken.com, or see my series of posts, A Week of KenKen.