Is Your Gödel Too Tight?

I don’t care what Stevie Nicks says, thunder does not only happen when it’s raining. And sorry, Kelly Clarkson, I’m not standing at your door because I’m sorry.

Logical fallacies are rampant in song lyrics. (Don’t even get me started.) I’m therefore hopeful that you won’t attempt to channel your inner songwriter while trying to solve the following logic puzzles, arranged roughly in order of difficulty.

Here’s Looking at You

Jack is looking at Anne, and Anne is looking at George. Jack is married, George is not. Is a married person looking at an unmarried person?

Beer is Proof that God Loves Us

Three people walk into a bar, and the bartender asks, “Would all of you like a beer?” The first says, “I don’t know.” The second says, “I don’t know.” The third emphatically replies, “Yes!”

Why was the third one able to respond in the affirmative?

Five to the Third

A five-digit number is equal to the sum of its digits raised to the third power. Alphametically,

CUBED = (C + U + B + E + D)³

What is the five-digit number?

Martin Gardner’s Children

I ran into an old friend, and I asked about her family. “How old are your three kids now?”

She said the product of their ages was 36. I replied, “Sorry, I still don’t know how old they are.”

She then said, “Well, the sum of their ages is the same as the house number across the street.”

“I’m sorry,” I said. “I still don’t know how old they are.”

Finally, she told me that the oldest one has red hair, and I finally realized their ages.

How old are my friend’s children?

If At First You Don’t Succeed…

If you take a positive integer, multiply its digits to obtain a second number, multiply all of the digits of the second number to obtain a third number, and so on, the persistence of a number is the number of steps required to reduce it to a single-digit number by repeating this process. For example, 77 has a persistence of four because it requires four steps to reduce it to a single digit: 77-49-36-18-8. The smallest number of persistence one is 10, the smallest of persistence two is 25, the smallest of persistence three is 39, and the smaller of persistence four is 77.

What is the smallest number of persistence five?

The Hardest Logic Puzzle Ever

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.

(This puzzle is attributed to Raymond Smullyan, but the twist of not knowing which word means which was apparently added by computer scientist John McCarthy.)

Σ Π :: The Sum and Product Game

This joke, or a close facsimile, has been taking a tour of email servers recently, and it’s now showing up on t-shirts, too:

…and it was delicious!

Appropriate for Pi Day, I suppose, as is the game my sons have been playing…

Eli said to Alex, “18 and 126.”

Alex thought for a second, then replied, “2, 7, and 9.”

“Yes!” Eli exclaimed.

I was confused. “What are you guys doing?” I asked.

“We invented a game,” Eli said. “We give each other the sum and product of three numbers, and the other person has to figure out what the numbers are.”

After further inquisition, I learned that it wasn’t just any three numbers but positive integers only, that none can be larger than 15, and that they must be distinct.

Hearing about this game made me immediately think about the famous Ages of Three Children problem:

A woman asks her neighbor the ages of his three children.

“Well,” he says, “the product of their ages is 72.”

“That’s not enough information,” the woman replies.

“The sum of their ages is your house number,” he explains further.

“I still don’t know,” she says.

“I’m sorry,” says the man. “I can’t stay and talk any longer. My eldest child is sick in bed.” He turns to leave.

“Now I know how old they are,” she says.

What are the ages of his children?

You should be able to solve that one on your own. But if you’re not so inclined, you can resort to Wikipedia.

But back to Alex and Eli’s game. It immediately occurred to me that there would likely be some ordered pairs of (sum, product) that wouldn’t correspond to a unique set of numbers. Upon inspection, I found eight of them:

(19, 144)
(20, 90)
(21, 168)
(21, 240)
(23, 360)
(25, 360)
(28, 630)
(30, 840)

My two favorite ordered pairs were:

(24, 240)
(26, 286)

I particularly like the latter one. If you think about it the right way (divisibility rules, anyone?), you’ll solve it in milliseconds.

And the Excel spreadsheet that I created to analyze this game led me to the following problem:

Three distinct positive integers, each less than or equal to 15, are selected at random. What is the most likely product?

Creating that problem was rather satisfying. It was only through looking at the spreadsheet that I would’ve even thought to ask the question. But once I did, I realized that solving it isn’t that tough — there are some likely culprits to be considered, many of which can be eliminated quickly. (The solution is left as an exercise for the reader.)

So, yeah. These are the things that happen in our geeky household. Sure, we bake cookies, play board games, and watch cartoons, but we also listen to the NPR Sunday Puzzle and create math games. You got a problem with that?

Math Problems for 2016

“What homework do you have to do tonight?”

I ask my sons this question daily, when I’m trying to determine if they’ll need to spend the evening doing word study or completing a math worksheet, or if we’ll instead be able to waste our time watching The Muppets or, perhaps, pulling up the animated version of Bob and Doug Mackenzie’s 12 Days of Christmas on Dailymotion.

When I asked this question last night, though, the answer was surprising:

We have to do our reading, but we already completed your math problem.

My problem? I had no idea what this meant. So they explained:

It’s not a problem you gave us. It’s one we got from [our teacher], and it says, “This problem was written by Patrick Vennebush.”

I was puzzled, but then it dawned on me. I asked, “Does it have a monkey at the top with the word BrainTEASERS?”

“Yes!”

“Which problem?”

“It’s about the word CAT.”

I knew the problem immediately. It’s the Product Value 60 brainteaser from Illuminations:

Assign each letter a value equal to its position in the alphabet (A = 1, B = 2, C = 3, …). Then find the product value of a word by multiplying the values together. For example, CAT has a product value of 60, because C = 3, A = 1, T = 20, and 3 × 1 × 20 = 60.

How many other words can you find with a product value of 60?

As it turns out, there are 14 other words with a product value of 60. Don’t feel bad if you can’t find them all; while they’re all allowed in Scrabble™, the average person won’t recognize half of them.

You can see the full list and some definitions in this problem and solution PDF.

This problem resurfaced at the perfect time.

With 2016 just around the corner, no doubt many math teachers will present the following problem to students after winter break:

Find a mathematical expression for every whole number from 0 to 100, using only common mathematical symbols and the digits 2, 0, 1, and 6. (No other digits are allowed.)

And that’s not a bad problem. It gets even better if you require the digits to be used in order. For instance, you could make:

• 2 = 2⁰ + 1⁶
• 9 = 2 + 0 + 1 + 6
• 36 = (2 + 0 + 1)! × 6

But that problem is a bit played out. I’ve seen it used in classrooms every year since… well, since I used it in my classroom in 1995.

So here are two versions of a problem — the first one being for younger folks — using the year and based on the Product Value 60 problem above:

How many words can you find with a product value of 16?

How many words can you find with a product value of 2016?

There are 5 words that have a product value of 16 and 12 words that have a product value of 2016 (spoiler: those links will take you to images of the answers). As above, you may not recognize all of the words on those lists, but some will definitely be familiar.

Number Words and Learned Helplessness

How about some number word puzzles? Here’s a well-known puzzle that you’ve likely seen before:

What is the first positive integer that, when spelled out, contains the letter a?

And here’s a modification of that puzzle that you may find a little more difficult:

What is the first positive integer that, when spelled out, contains the letter c?

And taking it one step further:

What letters are never used in the spelling of any positive integer?

Who says that math isn’t useful in English class?

One more problem in a similar vein:

Pick any positive integer you like, and count the letters when that number is spelled out. Now count the letters when the resulting number is spelled out. Continue ad infinitum. What do you get?

Maybe those weren’t your cup of tea. Perhaps anagrams are more to your liking, so here are two (related) puzzles for you.

Try to make an anagram for each of the following three words.

• whirl
• slapstick
• cinerama

Too tough? Then try these three words instead.

• bat
• lemon
• cinerama

If you had trouble with the first set, you’re in good company. There are no anagrams for the words whirl or slapstick.

These two sets of words were used by Charisse Nixon, a pyschologist at Penn State–Erie, who gave the first set of words to half her class and the second set of words to the other half. She instructed them to find an anagram of the first word on their list; those students who had received the second set were successful. Nixon then instructed them to find an anagram of the second word on their list; again, those students who had received the second set were successful. When she then instructed them to find an anagram of the third word on their list — of which there is exactly one, American — those who hadn’t found anagrams for the first two words were less successful than their peers, even though the final challenge was identical.

Afterwards, students who received the first set of words admitted to feeling confused, rushed, frustrated, and stupid.

Nixon was studying learned helplessness, a condition in which a person suffers from a sense of powerlessness, often arising from persistent failure.

This has implications the math classroom. Students who perform at a fourth-grade level but are asked to participate in an eighth-grade class are surely as confused and frustrated as the subjects in Nixon’s experiment. Students need to occasionally feel success, or else they’ll shut down. If you’re a teacher, you don’t need me or a psychological research study to tell you that. So the question is, how can you get students to feel success? That is, what can you do to prevent learned helplessness?

My suggestion is to look for acceptable and accessible entry points.

Consider the following problem, which might be seen in a middle school classroom:

What is the maximum possible product of a set of positive integers whose sum is 20?

As written, that problem contains three words — maximum, product, and integers — that may confound some students. For middle school students who do understand the terminology, finding an appropriate strategy might be daunting.

In my opinion, the following is a better way to present this problem so that all students have an entry point:

Find some numbers with a sum of 20. Now, multiply those numbers together. Compare your result with a partner. Whose result was greater? Can the two of you work together to find a product that’s greater still?

Even a struggling middle school student could start this activity. Surely he could find some numbers with a sum of 20. Certainly, he could multiply them without a problem.

Why is this a better presentation? The wording is simplified. There is encouragement to work with a partner. It feels more like a collaborative game than a traditional math problem. It sounds — dare I say it? — like fun.

When a struggling student is able to get into a problem, and they’re able to make some strides in the right direction, and they’re rewarded by your positive encouragement, they attain some level of success. Maybe they won’t solve the problem entirely, but who cares? For many students, trying is progress.

And for students who are having trouble finding any success, perhaps the following words of encouragement will help.

If at first you don’t succeed, call it version 1.0.

If at first you don’t succeed, destroy all evidence that you ever tried.

If at first you don’t succeed, blame someone else and seek counseling.

If at first you don’t succeed, then skydiving is not for you.

If at first you don’t succeed, get new batteries.

If at first you don’t succeed, try two more times so your failure is statistically significant.

 

Ring Me Up!

When my college roommate contracted crabs, he went to CVS to buy some lice cream. As you can imagine, he didn’t want to announce to the world what he was buying or why, so he put the box on the counter with a notepad, a bottle of aspirin, a pack of cigarettes, a bag of M&M’s, and a tube of toothpaste — hoping the cream would blend in. The attractive co-ed clerk at the register rang him up without a second look.

As he walked out of the drug store thinking he had gotten away with it, he opened the cigarettes, put one to his lips, and realized he had nothing with which to light it. He returned to the checkout and asked the clerk for a pack of matches.

“Why?” she asked. “If the cream doesn’t work, you gonna burn ’em off?”

Ouch.

My luck with clerks wasn’t much better. At a grocery store, I placed a bar of soap, a container of milk, two boxes of cereal, and a frozen dinner on the check-out counter. The girl at the cash register asked, “Are you single?”

I looked at my items-to-be-purchased. “Pretty obvious, huh?”

“Sure is,” she replied. “You’re a very unattractive man.”

I did, however, have an exceptional experience at a convenience store. This is what happened.

I walked into a 7-11 and took four items to the cash register. The clerk informed me that the register was broken, but she said she could figure the total using her calculator. The clerk then proceeded to multiply the prices together and declared that the total was $7.11. Although I knew the prices should have been added, not multiplied, I said nothing — as it turns out, the result would have been $7.11 whether the four prices were added or multiplied.

There was no sales tax. What was the cost of each item?

As you might have guessed, that story is completely false. (The one about me being called ‘unattractive’ is a slight exaggeration. The one about my roommate, sadly, is 100% true.) The truth is that I learned this problem from other instructors when teaching at a gifted summer camp.

It may not be true. It is, however, one helluva great problem.

But it has always bothered me that the problem is so difficult. I’ve always wanted a simpler version, so that every student could have an entry point. Today, I spent some time creating a few.

Use the same set-up for each problem below… walk into a store… take some items to check-out counter… multiply instead of add… same total either way. The only difference is the number of items purchased and the total cost.

I’ve tried to rank the problems by level of difficulty. Below, I’ve given some additional explanation — but not the answers… you’ll have to figure them out on your own.

• (trivial) Two items, $4.00.
• (easy) Two items, $4.50.
• (fun) Two items, $102.01.
• (systematic) Two items, $8.41.
• (perfect) Three items, $6.00.
• (tough) Three items, $6.42.
• (rough) Three items, $5.61.
• (insane) Four items, $6.44.
• (the one that started it all) Four items, $7.11.

---

Editor’s Notes

trivial — C’mon, now… even my seven-year-old sons figured this one out!

easy, fun, systematic — All of these are systems of two equations in two variables. Should be simple enough for anyone who’s studied basic algebra. All others can use guess-and-check.

perfect — Almost as easy as trivial, and the name is a hint.

tough — But not too tough. Finding one of the prices should be fairly easy. Once you have that, what’s left reduces to a system of equations in two variables.

rough — Much tougher than tough. None of the prices are easy to find in this one.

insane — Gridiculously hard, so how ’bout a hint? Okay. Each item has a unique price under $2.00. If you use brute force and try every possibility, that’s only about 1.5 billion combinations. Shouldn’t take too long to get through all of them…

the one that started it all — As tough as insane, and not for the faint of heart. But no hint this time. Good luck!

 

Two Simple Math Games

In his article “What Is the Name of This Game?” author John Mahoney discussed the mathematics of the following game:

Cards numbered 1-9 are placed face up on a table. Two players alternate picking up one card at a time. The winner is the first player who has exactly three cards with a sum of 15.

You can play this game with nine cards removed from a deck of cards, or you can play online by going to http://illuminations.nctm.org/deepseaduel. The online version is a one-player game, but it has modifications that use different numbers of cards, different values on the cards, and different required sums.

Can you find the winning strategy for this game? (Hint: The strategy is described in the linked article above.)

Here’s a modification of the game that seems interesting, too.

Use cards with the following numbers: 1, 2, 3, 4, 6, 9, 12, 18, 36. The winner is the first player who has exactly three cards with a product of 216.

The optimal strategy for this game is different than the strategy from the original game. Can you find it?

Note: For the original game, there are eight sets of three cards with a sum of 15:

{1, 5, 9}, {1, 6, 8}, {2, 4, 9}, {2, 5, 8},
{2, 6, 7}, {3, 4, 8}, {3, 5, 7}, {4, 5, 6}

For the modification, there are 36 sets of three cards with a product of 216.

Perhaps that fact will help you identify the optimal strategy.

1 2 Find a Gr8 Name?

While listening to a recent episode of NPR’s You Bet Your Garden, host Mike McGrath said that 10-10-10 fertilizer is a marketing ploy. “No plants want nitrogen, phosphate, and potash in equal proportions,” McGrath said.

I’m not much of a gardener, despite my love of rose (curves), stems and leaves, (square) roots, and (factor) trees. But it struck me as numerically interesting that fertilizer manufacturers sell a product that has the wrong mixture of nutrients. Why would they do that?

Well, money, for one. Products with nice, round numbers tend to be purchased more than others, according to marketing researchers Dan King and Chris Janiszewski. A product with a name like 10-10-10 is more appealing to an average consumer than, say, 9-12-15 or 5-12-13, even though the latter might be more appealing to Pythagoreans.

Consumers will more often choose brands whose names contain likable numbers, of which there are several types:

• Small numbers, such as 1, 2, 3, …, 9.
• Round numbers, like 1, 10, or 1,000.
• Numbers that are frequent sums or products, such as 10 or 24.

It’s easy enough to recognize numbers of the first two types. The third category is a bit loosey-goosey, though, so I would improve the definition as follows: likable numbers of the third type can be represented as a product in more than two ways. For instance, 44 is a likable number because it can be represented in three different ways: 1 × 44, 2 × 22, and 4 × 11; but, 57 is not because it can only be represented in two ways, 1 × 57 and 3 × 19.

King and Janiszewski go on to say that consumers are further influenced if the operands of the number are included in advertisements. In their paper The Sources and Consequences of the Fluent Processing of Numbers, they state,

“…not only is a Volvo S12 more liked than a Volvo S29, but liking is further enhanced when an advertisement for a Volvo S12 includes a license plate with the numbers 2 and 6. The operands 2 and 6 make 12 more familiar because they encourage the subconscious generation of the number 12.”

Though some of it sounds like hooey to me, this theory of number relevance is appealing, mainly because it implies that humans are hard-wired for mathematics. (It also makes me think that I chose a good name for my book.)

Upon hearing about likable numbers in products, I tried to think of a well-known product for each likable number up to 100. As you can see from the list below, I had limited success. (Note that I relied entirely on memory. Sure, I could have used Google to find companies like Take 2 Interactive or products like 32 Poems Magazine, but if likable numbers make a brand more attractive, then shouldn’t I be able to remember the name?)

1: One-a-Day, Mobil 1, A-1
2: Intel Core 2 Duo, Dos Equis
3: 3M, Three Musketeers
4: Number 4 Hair Care, 4-H
5: 5-Hour Energy, Five Alive, Chanel No. 5
6: Motel 6, Six Flags
7: 7-11, Monistat 7, 7-Up
8: Super 8, V-8, Sulfur 8
9: 9 West, 9 Lives
10: Tanqueray 10, Oxy 10, Pac 10
12: K12, Big 12
16: 16 Handles
18:
20: Mad Dog 20/20, Commodore Vic 20
24: 24-Hour Fitness, Claritin 24
25:
28:
30: 30 Rock
32:
36:
40: WD-40
42:
44: Vicks Formula 44
45: Colt 45
48:
50:
52:
54:
56:
60:
63:
64: Commodore 64
66:
68:
70:
72:
75:
76:
78:
80:
81:
84:
88: 88 Rice Bowl
90: P90X
92:
96:
98:
99: 99 Designs
100: 100 Grand Bar

I was also able to think of a few product names that include likable numbers greater than 100:

• RU-486
• Saab 900
• 2000 Flushes
• Atari 2600

And of course, there are many successful products whose names contain numbers that are not likable, too:

• Thirteen (WNET, New York City)
• X-14
• Product 19
• Select 55 Beer
• Heinz 57
• Vat 69
• Bacardi 151
• Formula 409
• Levi 501

If you can fill in any of the gaps from the likable numbers product list, please leave a comment. Or if you can think of any other products with numbers in the name, likable or not, feel free to leave a comment for those, too.