## Posts tagged ‘puzzle’

### What Time Is It?

Here’s a math puzzle that is rather easy. Or is it?

You look in a mirror and see the reflection of a clock. In the reflection, the clock appears to show a time of 11:51. What is the real time?

Before I share the solution, how ’bout some clock jokes?

A hungry clock goes back four seconds.

I spent 35 minutes fixing a broken clock yesterday.

At least, I think it was 35 minutes…What’s the difference between a man and a broken clock?

At least a broken clock is right twice a day.It took me four hours to eat a dozen clocks.

It was very time consuming.An alarm clock made of herbs will help you wake up on thyme.

My clock stopped at 8:23 a.m. I’m going to have a day of morning.

**Puzzle Solution**

The puzzle above is based on a math trivia question I found at Trivia Cafe.

The answer could be 12:09, if the reflection in the mirror looks like this…

Then again, the answer could be 12:11, if the reflection in the mirror looks like this…

And of course, there are all the silly possibilities — for instance, if the clock is broken, it doesn’t matter what time shows in the reflection, regardless if it’s analog or digital.

### Variations on a Theme

Three variations of one of my favorite puzzles. The first is silly; the second is doable; and, the third will take a little bit of jiggering. I don’t know where I first saw this puzzle, but I’m pretty sure the version with ten blanks is in *Gödel, Escher, Bach*.

Instructions: Place numerals in the blanks to make the sentence true.

This version is for little kids. Or is it?

There are __ zeroes and __ ones in this sentence.

I’m fairly certain there are no solutions when this is extended to three blanks, but four blanks will work:

There are __ zeroes, __ ones, __ twos, and __ threes in this sentence.

It works with seven blanks (when the greatest digit is six), but that’s not much different than the one above. The piece de la resistance is the one with ten blanks:

There are __ zeroes, __ ones, __ twos, __ threes, __ fours, __ fives, __ sixes, __ sevens, __ eights, and __ nines in this sentence.

Have fun!

### Common Problems

This is pretty cool. What is the value of the following expression?

(π^{4} + π^{5})^{1/6}

This reminds me of a series of problems that I call Maximum Mileage:

- What is the maximum possible product of two positive integers whose sum is 100?
- What is the maximum possible product of two prime numbers whose sum is 100?
- A set of positive integers has a sum of 100. What is the maximum possible product of the numbers in this set?
- A set of positive numbers has a sum of 100. What is the maximum possible product of the numbers in this set?

Bonus Question: Why does the expression at the beginning of this post remind me of that series of problems?

The following sentence might be a hint:

It is fortuitous that both conundrums incur a commonality of solution.

### Another Bad Email Math Puzzle…

It happened again. I received another email with a number trick that makes the ubiquitous claim, “This will work for everyone!” Sadly, it won’t, but it was kind of cool:

Calculate 39 × (your age) × 259.

The email said, “The result will surprise you.”

It didn’t. I suspected what the value of 39 × 259 would be, so I predicted the result. But if you don’t know the value of that product, then maybe you’ll be surprised.

The trick works well enough if you have a double-digit age. But my friend Ferdinand is 107 years old. His result was 1,080,807, and that just looks like a mess. The results for my six-year-old sons were better, albeit rather unsatisfying.

Ha-rumph. So much for the Internet providing mathematical inspiration.

Here are some similarly uninteresting puzzles that I created:

Calculate 7,373 × (your age) × 137.

Calculate 9,091 × (your age) × 11,111.

Calculate 101 × (your age) × 1,000,100,010,001.

To create more puzzles like this, enter **factor(101010…10101)** into Wolfram Alpha.

For your centenarian friends, try these:

Calculate 101,101 × (your age) × 9,901.

Calculate 3.3 × (your age) × 33.67.

Let’s not forget the little people whose age is still in the single digits:

Calculate 3 × (your age) × 37.

Calculate 41 × (your age) × 271.

And a math joke (or is it?) about age:

I’ve been good with numbers my whole life. When I turned 2, I realized that my age had doubled in one year. This concerned me… at that rate, I’d be 32 in four more years!

And another:

What goes up but never comes down?

Your age.

### Math Problem with 6′s from Scam School

TestTube^{TM} is a new digital network from Discovery^{TM}. With shows like *Stuff of Genius*, *Blow It Up!*, and *Distort* (where “great ideas become reality”), it holds strong appeal for mathy folks.

My favorite show on TestTube^{TM} is Scam School, where magician Brian Blushwood takes you on a tour of bar tricks, street cons and scams. In the episode “Six the Hard Way,” he poses a mathematical challenge that is a variation on one you may have seen before. As Brian explains, “it’s almost poetic how simple this is.”

The puzzle is this: Form an expression with three 1′s, three 2′s, three 3′s, and so on, up to three 9′s, so that the value of each expression is equal to 6. As an example, an expression using three 7′s is shown below. Can you find expressions using the other numbers?

0 0 0 = 6

1 1 1 = 6

2 2 2 = 6

3 3 3 = 6

4 4 4 = 6

5 5 5 = 6

6 6 6 = 6

7 – 7 ÷ 7 = 6

8 8 8 = 6

9 9 9 = 6

You can watch Six the Hard Way, but be forewarned: at least one solution for each number is given, so you may want to solve the puzzle before viewing.

Also note that some folks have posted solutions in the comments below, so scroll at your own risk.

### Math Name Scramble

When you scramble the letters of “Math Name Scramble,” several excellent anagram-cum-headlines are formed. The results are just too spectacular not to have a little fun.

**Lambs Cheat Merman**

Schenectady, New York – Maybe she’s got rhythm, but Ethel Merman appears to be lacking in street smarts. When three young, corrupt sheep tempted her with a game of three-card monte, she should have politely declined. But she was insistent that identifying the proper card “shouldn’t be that difficult.” Fourteen failed attempts and $650 later, she finally accepted defeat. The ovine dealer, amused by Ms. Merman’s persistence, continually told her, “I get a kick out of you.”

**Math, Camels, Barmen**

Riyadh, Saudi Arabia – A mathematician and a camel walk into a bar. The barman says, “What is this, some kind of joke?”

Okay, enough of that silliness.

Lots of math words have interesting anagrams:

- scalene = cleanse
- thousand = handouts
- vector = covert
- algorithm = logarithm
- decimal point = I’m a pencil dot
- integral calculus = calculating rules
- innumerable = a number line

If you like anagrams, the following puzzle might be right up your alley.

The last names of ten famous mathematicians — all sufficiently scrambled, of course — are listed below.

- ACORN PIE
- A PASTRY HOG
- ASS REWRITES
- NO NUN MAVEN
- ON THREE
- RAIN MEN
- REAL GANG
- RED CHAMISE
- SCAREDEST
- THICK NERD EGO

Your task is to unscramble the names, then place them in the rows of the grid below. If you place the correct names in the correct order, another famous mathematician’s name will appear in the highlighted column.

Stumped? Don’t sweat it; lesser men have had to look at **the solution**, too.

### In Your Prime

I’ve got a prime number trick for you today.

- Choose any prime number
*p*> 3. - Square it.
- Add 5.
- Divide by 8.

Having no idea which prime number you chose, I can tell you this:

The remainder of your result is 6.

Pretty cool, huh?

I will now fill a bunch of space with quotes and jokes about prime numbers to prevent you from seeing the spoiler explanation below. But you can skip straight to the bottom if you’re not interested in the other stuff or if you just can’t control yourself.

Mark Haddon, author of *The Curious Incident of the Dog in the Night-time*, wrote the following:

Prime numbers are what is left when you have taken all the patterns away. I think prime numbers are like life. They are very logical, but you could never work out the rules, even if you spent all your time thinking about them.

(Incidentally, if you haven’t read that book, you should. Amazon reviewer Grant Cairns said it better than I could: “The integration of the mathematics into the fiction is better than any other work that I know of. The overall effect is a beautiful story that any maths fans will find hard to read without the tissue box close at hand.”)

Israeli mathematician Noga Alon said that he was interviewed on Israeli radio, and he mentioned that Euclid proved over 2,000 years ago that there are infinitely many primes. As the story goes, the host immediately interupted him and asked:

Are there still infinitely many primes?

And of course there’s this moldy oldie:

Several professionals were asked how many odd integers greater than 2 are prime. The responses were as follows:

Mathematician:3 is prime, 5 is prime, 7 is prime, and by induction, every odd integer greater than 2 is prime.

Physicist:3 is prime, 5 is prime, 7 is prime, 9 is experimental error, 11 is prime, …

Engineer:3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, …

Programmer:3 is prime, 5 is prime, 7 is prime, 7 is prime, 7 is prime, …

Marketer:3 is prime, 5 is prime, 7 is prime, 9 is a feature, …

Software Salesperson:3 is prime, 5 is prime, 7 is prime, 9 will be prime in the next release, …

Biologist:3 is prime, 5 is prime, 7 is prime, the results for 9 have not yet arrived…

Advertiser:3 is prime, 5 is prime, 7 is prime, 11 is prime, …

Lawyer:3 is prime, 5 is prime, 7 is prime, there is not enough evidence to prove that 9 isnotprime, …

Accountant:3 is prime, 5 is prime, 7 is prime, 9 is prime if you deduct 2/3 in taxes, …

Statistician:Try several randomly chosen odd numbers: 17 is prime, 23 is prime, 11 is prime, …

Professor:3 is prime, 5 is prime, 7 is prime, and the rest are left as exercises for the student.

Psychologist:3 is prime, 5 is prime, 7 is prime, 9 is prime but tries to suppress it, …

Card Counter:3, 5, and 7 are all prime, but I prefer 21.

*Explanation of the Prime Number Trick*

We are trying to show that (*p*^{2} + 5) mod 8 = 6. This is equivalent to showing that (*p*^{2} ‑ 1) mod 8 = 0, or that (*p* + 1)(*p* ‑ 1) is divisible by 8.

Because *p* > 3 and is prime, then either *p* = 1 mod 4 or *p* = 3 mod 4. Consequently, it must be the case that (a) *p* + 1 = 2 mod 4 and p ‑ 1 = 0 mod 4 or (b) *p* ‑ 1 = 2 mod 4 and *p* + 1 = 0 mod 4. That is, both numbers will be even, and at least one of them will be a multiple of 4. For either (a) or (b), the product (*p* + 1)(*p* ‑ 1) will be a multiple of 8. Q.E.D.

### Rectangles for Mathemagicians

Depending who you ask, *mathemagician* has at least two different definitions:

- A person who enjoys both math and magic. (Wikipedia)
- A person who is so good at math that the answers to math problems seem to come to them magically. (Urban Dictionary)

When professor Art Benjamin told Stephen Colbert that he was a mathemagician, Colbert asked, “What does that mean? Were those two words by itself not nerdy enough?”

Below is a math puzzle involving magic. To be precise, magic rectangles. But first, a little warm-up…

What do you call a quadrilateral with four right angles that’s been in a car accident?

A wrecked angle.

For the last several years, I’ve had the pleasure of creating puzzles for the Daily Puzzle Challenge at the NCTM Annual Meeting. A new set of four or five puzzles appears in each day’s challenge. The following puzzle, which appeared on Friday’s Daily Puzzle Challenge, involves rectangles and is my favorite puzzle from this year’s meeting.

A magic rectangle is an *m* × *n* array of the positive integers from 1 to *m* × *n* such that the numbers in each row have a constant sum and the numbers in each column have a constant sum (although the row sum need not equal the column sum). Shown below is a 3 × 5 magic rectangle with the integers 1-15.

Below are three arrays that can be filled with the integers 1-24, but only two of them can be filled in such a way as to form a magic rectangle. Construct two magic rectangles below; for the array that cannot be used to construct a magic rectangle, can you explain why not? More generally, can you determine what types of rectangles can be used to construct magic rectangles and which cannot?

### Math Rebus Puzzles

What word or phrase is represented by the following picture?

Puzzles like this are sometimes called Pictogram Puzzles, Word Picture Puzzles, or Hieroglyphs.

Kevin Stone at BrainBashers calls them Brain Bats.

Lots of folks refer to them as Rebus Puzzles. And while I don’t think that’s exactly right (what I think of as a rebus can be found here), that’s the word I’m going to use, too, because it’s the shortest.

Whatever. Enjoy the **MJ4MF Rebus Quiz**. Note that each picture is somehow mathematical, even if the answer isn’t.

**Download the MJ4MF Rebus Quiz (PDF)**

If you’re a classroom teacher, the dread day-before-break is fast approaching. The MJ4MF Rebus Quiz is a great activity for students who have too much energy to sit still and too little focus to learn anything. (Permission is granted for the MJ4MF Rebus Quiz to be used with students for non-commercial educational purposes.)

The answer key and a copy of the exam can be found at http://mathjokes4mathyfolks.com/rebus.html.

### 5 Math Strategy Games to Practice Basic Skills

The summer is a great time for kids to hike, bike, swim… and forget everything that they learned during the school year.

The son returned to school after summer break. At the end of the first day, his mother received a call from the teacher about his poor behavior. “Now, just one minute,” said the mother. “He had poor behavior all summer, yet I never called you once!”

In *Outliers*, Malcolm Gladwell purports that poor kids lose ground to affluent kids during summer break. Their experiences and academic progress during the school year are similar, he contends, but their out-of-school experiences during the summer are very different. Though minor at first, the cumulative effect of those summer losses becomes noticeable as children get older.

The following are five games/puzzles that can be used with young kids to prevent summer losses and, possibly, even elicit some summer gains. Each has the characteristics that I love about a good game for young kids: It requires students to use and practice basic skills, but there is a higher purpose for doing so.

**1. KenKen**

This is a game that’s kind of like SuDoku, but a million times better. If you don’t know the game, check it out at www.kenken.com. My sons noticed me playing it one afternoon and asked what it was. I explained, and they asked if they could do it with me. We now solve three or four games every afternoon. I used to help them a lot, but now they pretty much know all of their math facts up through 7 × 7. How do you not love a game that helps four-years-olds learn the times table?

**2. Wormhole**

This is a puzzle, not a game, and you can learn all about it at Math Pickle. The general idea is that you start with a sequence of numbers in a flower-like pattern. You then multiply two adjacent numbers, subtract 1, and divide by the number below. The cool and surprising part is that every intermediate result is an integer, so there are no ugly decimals for kids to deal with. And by the twelfth ring of petals, every result is 0. Happens every time.

**3. Squares of Differences**

The good folks at Math For Love reminded me of this great problem, and Josh Zucker discussed it at length on the NYTimes Numberplay blog. Draw a square, and put a positive integer at each vertex. Then at the midpoint of each side, write the difference of the numbers at the two adjacent vertices. Now connect the midpoints to form a rotated square inside the original square, and repeat. It seems that if you continue this process long enough, you’ll eventually get all 0′s. But does that always happen?

By the time kids test this conjecture with three or four attempts, they’ve done a hundred subtraction problems without even realizing it.

**4. Decimal Maze**

The Decimal Maze (PDF) comes from the lesson Too Big or Too Small on Illuminations. Trying to obtain the maximum value while traversing a maze with decimal operations, students learn about the effects of multiplying and dividing by decimals that are greater or less than 1. The activity is good for upper elementary and middle school students, but I’ve used modified versions with very young kids. For instance, a modified maze for kids in first grade uses single-digit positive integers while limiting the operations to just addition and subtraction; for older kids, a maze could include fractions or powers instead of decimals.

**5. Dollar Nim**

As I mentioned in a previous post, my wife created a great game that I call Dollar Nim. The idea is simple. Imagine you have 100¢, and on your turn you can remove 1¢, 5¢, 10¢, or 25¢. Players alternate turns; the player to reduce the amount to 0¢ is the winner. The optimal strategy is not obvious, and kids practice a whole lot of subtraction, especially as it relates to making change.

More generally, any one-pile nim game is great for the purpose of having kids practice subtraction without realizing it.

I hope you find some free time this summer to enjoy these games. I’ll leave you with a joke/truth about summer school.

I never understood the concept of summer school. The teacher’s going to go up there and go, “OK, class. You know that subject you couldn’t grasp in nine months? Well, we’re going to whip it out in six weeks.” – Todd Barry