## Posts tagged ‘number’

### 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.

### My Insecurity Over Security Codes

Every time I attempt to access one of my company’s applications via our single sign-on (SSO) system, I’m required to request a validation code that is then sent to my smartphone, and then I enter that code on the login page.

It’s a minor nuisance that drives me insane.

The purpose of the codes are to provide an additional level of security, but given how un-random the codes seem to be, it doesn’t feel very secure to me. This screenshot shows some of the codes that I’ve received recently:

Here’s what I’ve observed:

- Every security code contains 6 digits.
- The first 3 digits in the code form either an arithmetic or geometric sequence, or the first 3 digits contain a repeated digit.
- Similarly, the last 3 digits in the code form either an arithmetic or geometric sequence, or the last 3 digits contain a repeated digit.

As an example, one of the codes in the screenshot above is 421774. The first 3 digits form the (descending) geometric sequence 4, 2, 1, and the digit 7 appears twice in the second half of the code.

I believe the reason for these patterns is to make the codes more memorable to those of us who have to transcribe them from our phones to our laptops.

This got me thinking. The likelihood of someone correctly guessing a six-digit code is 1 in 1,000,000. But what is the likelihood that someone could correctly guess a six-digit code if it adheres to the rules above?

If you’d like to answer this question on your own, stop reading here. To put some space between you and my solution, here’s a security-related joke:

“I don’t understand how someone stole my identity,” Lily said. “My PIN is so secure!”

“What’s your PIN?” Millie asked.

“The year of Knut Långe’s death,” Lily replied.

“Who is Knut Långe?”

“A King of Sweden who usurped the throne from Erik Eriksson.”

“And what year did he die?”

“1234.”

(Incidentally, Data Genetics reviewed 3.4 million stolen website passwords, and they found that 1234 was the most popular four-digit code. The researchers claimed that they could use this information to make predictions about ATM PINs, too, but I don’t think so. All this shows is that 1234 is the most commonly *stolen* password, and therefore this inference suffers from survivorship bias. Without having data on all the codes that were *not* stolen, it’s impossible to make a reasonable claim. But, I digress.)

To determine the number of validation codes that adhere to the patterns I observed, I started by counting the number of arithmetic sequences. With only 3 digits, there are 20 possible sequences:

- 012
- 024
- 036
- 048
- 123
- 135
- 147
- 159
- 234
- 246
- 258
- 345
- 357
- 369
- 456
- 468
- 567
- 579
- 678
- 789

But each of those could also appear in reverse (210, 975, etc.), giving a total of 40.

There are far fewer geometric sequences; in fact, only 3 of them:

- 124
- 139
- 248

And again, each of those could appear in reverse, giving a total of 6.

Finally, there are 10 × 9 × 8 = 720 three-digit numbers with no repeated digits, which means there are 1,000 ‑ 720 = 280 numbers with a repeated digit. (Here, “number” refers to any string of 3 digits, including those that start with a 0, like 007 or 092.)

Consequently, there are 40 + 6 + 280 = 326 possible combinations for the first 3 digits and also 326 combinations for the last 3 digits, which gives a total of **326 × 326 = 106,276 possible validation codes**.

That means that it would be about 10× more likely for a phisher to correctly guess a validation code that follows these rules than to guess a completely random six-digit code. But said another way, the odds are still significantly against a phisher who’s trying to steal my code. And quite frankly, if someone wants to exert that kind of effort to pirate my access to Microsoft Word online, well, I say, go for it.

### Do You Have Mathopia?

When I was young, we spent a lot of time on highways, driving to and from our summer cottage. I’d see a Pennsylvania license plate like the one below, which at the time had five digits and one letter. Most people, I suspect, would be unimpressed. But not me. I’d say to my parents, “How cool is that license plate? If *p* = 26 and the cracked bell were an equal sign, it would be 23 × 26 = 598.”

My mom would respond with, “If you say so,” or a shrug. She had failed algebra in high school and would regularly and disgustedly declare, “How the hell can *x* = 6, when *x* is a letter and 6 is a number?”

My father — who dropped out of school to join the Navy at age 15 and had never taken an algebra course — would simply grunt.

Neither of them saw the beauty in numbers. I, on the other hand, couldn’t *not* see it. I wasn’t **mad** about this. I was just **sad** that they couldn’t share my joy.

On my commute this morning, I saw a truck with the number 12448 on the tailgate. I mentally added two symbols and formed the equation 12 × 4 = 48.

When my boss told me that he was retiring on January 4, I remarked, “What a great choice! The numbers 1, 4, and 16 are all square numbers, and 1, 4, 16 forms a geometric sequence.”

The truth is, it’s not really possible for me to look at a number — whether it’s a license plate, calendar, billboard, identification card, lottery number, bar code, serial number, road sign, odometer, checking account, confirmation number, credit card, phone number, phone bill, receipt total, frequent flyer number, VIN, TIN, PIN, ISBN, or any of a million other numbers — and not try to figure out some way to give it meaning beyond just its digits.

I’m not the only one with this affliction. All mathy folks have **mathopia** — a visual disorder that causes us to see the all things through a mathematical lens.

G. H. Hardy had mathopia. He looked for a special omen in 1729, the number of the taxicab he took to visit his sick friend Srinivasa Ramanujan. Upon arriving, he mentioned that he hoped it wasn’t a bad omen to have taken a cab with such a dull number. Ramanujan had mathopia, too. He replied that 1729 was actually “an interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

Jason Padgett, whose latent mathematical powers suddenly appeared after he sustained a brain injury, has mathopia. He explained how he sees the world:

I watch the cream stirred into the brew. The perfect spiral is an important shape to me. It’s a fractal. Suddenly, it’s not just my morning cup of joe; it’s geometry speaking to me.

This is the way that math people work. We see numbers and patterns everywhere, sometimes even when they’re not really there. Or, maybe, when they’re not **meant** to be there. And while I am not trying to imply that I’m anything close to Hardy or Ramanujan or Padgett, I do think that they and I shared one characteristic — the burden, and the blessing, of seeing the world through math-colored glasses.

World Sight Day, celebrated on the second Thursday of October each year — in other words, *today* — seems like a good day to bring awareness of mathopia to the masses. It doesn’t hurt that today is 10/13/16, a date forming an arithmetic sequence, in which all three numbers are Belgian-1 numbers. (See, I can’t turn it off.)

**Do you have mathopia? What do you see when you encounter a number?**

### Demitri Martin and Me

As I was watching *If I* by Demetri Martin, I realized something.

I

loveDemitri Martin, because IamDemetri Martin.

Not literally, of course. I didn’t inhabit his body and take over his soul. (Would if I could!) Nor is this blog a ruse that appears to be written by Patrick Vennebush when it is, in fact, written by Demitri Martin. I just mean that he and I are about as similar as two people can be without entering the world from the same womb. Check out this list:

Demitri Martin |
Patrick Vennebush |

He’s weird. (In a good way.) | I’m weird. (No disclaimer.) |

He did Mensa puzzles as a kid. | I did Mensa puzzles as a kid. |

He uses convoluted mnemonics to remember numbers. | I use convoluted mnemonics to remember numbers. |

He uses drawings and visual aids during stand-up performances. (See below.) | I use drawings and visual aids during math presentations. (See below.) |

He was influenced by Steven Wright, Emo Philips, Eddie Izzard, and Mitch Hedberg. | I watched every Steven Wright performance on cable television when I was a teenager; my favorite joke is from Emo Philips; I own every Eddie Izzard CD; and one of my great regrets is that I never saw Mitch Hedberg perform live. |

He was slated to play Paul de Podesta in Moneyball but was replaced by Jonah Hill. |
I wasn’t in Moneyball, either. |

He was born in a prime number year (1973). | I was born in a prime number year (1971). |

He won a Perrier Comedy Award. | I sometimes drink Perrier while watching Comedy Central. |

He once attended class wearing a gorilla suit. | I had no fashion sense in college. |

He is extremely allergic to nuts. | I’m not allergic to them, but I really don’t like crazy people. |

One of Demetri’s drawings:

Oh, sure, I could list hundreds of other similarities between Demitri and me, but I think the list above is enough to see that the coincidence is uncanny. I mean, we practically live parallel lives.

Demetri used to sneak Mensa puzzle books — not muscle mags or girlie mags — into school to read during class. One of the puzzles purportedly from his *Mensa Presents Mighty Mindbusters* book:

If a crab-and-a-half weigh a pound-and-a-half, but the half-crab weighs as much again as the whole crab, what do half the whole crab and the whole of the half-crab weigh?

He said that solving problems from those books was validating.

When I got one right, I’d be like, “Yes! I

amsmart! These other idiots don’t know how much the crabs weigh.” But I do. Because I just spent Saturday working it out.

I solved puzzles like this, too. I don’t know if they made me feel smart, but I enjoyed the way I felt when I figured out a particularly tough one.

From the way he describes it, such puzzles may have had the same effect on both of us.

Whatever the reason, I spent a lot of time as a kid doing these puzzle books. And it came to shape the way I see the world. So now, as an adult, I see the world in those terms. For example, to me a phone number is always a sentence or an equation. Like my friend Becky…

He goes on to say that he remembers Becky’s phone number using a convoluted, mathematical mnemonic:

That is, he converts the first three digits into an expression that is equal to an expression formed by the last four digits. He concludes that it’s “much simpler,” but it’s unclear how.

Now that’s some crazy, messed-up sh*t.

And I’d probably think it even weirder… if I didn’t do it, too.

One night many years ago, my roommate Adam asked for the number of the local pizza shop. I replied, “3^{3}, 1^{3}, 20^{3},” because that’s how I saw it. Adam looked at me like I was nuts, and he was probably onto something.

My friend AJ’s street address is 6236, which I remember as 6^{2} = 36.

My street address growing up was 1331, which I associated with the third row of Pascal’s triangle. (It also happens to be 11^{3}, but I didn’t know that at the time.)

I chose the four digits of my PIN because… no, wait, that wouldn’t be prudent.

My co-worker Julia’s extension is 2691. I used to remember this as 2 + 6 = 9 – 1, until I recognized a more elegant geometric mnemonic: the sequence 2, 6, 9, 1 forms an isosceles trapezoid on my office phone’s keypad — or it would, were the buttons equally spaced.

I can’t explain why I do this. Perhaps, as Demetri says, it’s the influence of all those puzzle books. Or maybe it’s just that the mental conversion to an equation gives the number meaning, making it more memorable. Or perhaps it’s that I’m wired to see the world through a mathematical lens, despite not wearing glasses.

Larry McCleary, author of *The Brain Trust Program,* claims that numbers are difficult to remember because “most of us don’t have any emotional attachment to particular numbers.” Mr. McCleary, I’d like you to meet my friend Demetri…

Demitri and I are both into anagrams.

Even when I walk down the street, things look a little different. The signs… the letters dance around. It becomes a little puzzle for me. So, say MOBIL, the gas station — that becomes LIMBO. STARBUCKS becomes RACKS BUST. CAR PHONE WAREHOUSE… AH, ONE SOUR CRAP — WHEE!”

Yeah, I do that, too…

My first car was a CHEVROLET IMPALA, which transforms to COMPARATIVE HELL. Our neighbor’s son is CARSON, whom I jokingly call ACORNS. And I can’t see a STOP sign without also thinking of OPTS, POST, POTS, and TOPS.

If you’re reading this, you likely have some things in common with Demetri, too. **What number mnemonics do you use, or what anagrams to do you see?**

### Heavy Cookies, Undervalued Coins, and Misconceptions

Simple question to get us started…

Which is worth more?

And **of course** the answer is, “The quarters, because 50¢ is more than 20¢,” right? But not to a kindergarten student or a pre-schooler who hasn’t yet learned how much coins are worth. A young student might argue, “Four is more than two.”

Why didn’t the quarter follow the nickel when he rolled himself down the hill?

Because the quarter had more cents.

Recently, I was asked to review an educational video for kindergarten math that had a similar question.

The video stated, “Can you tell the green, yellow, and orange cookies are heavier? That makes sense, doesn’t it? Because there are **more** of them!”

Uh, no.

This is the same logic that would lead one to claim that the value of four nickels is greater than the value two quarters because there are more nickels. It’s a huge misconception for students to focus on **number** rather than **value**. So it’s very frustrating to see this video reinforce that misconception.

For example, if each green, yellow, or orange cookie weighs 3 ounces, but each blue or purple cookie weighs 5 ounces, then the left pile would weigh 6 × 3 = 18 ounces, and the right pile would weigh 4 × 5 = 20 ounces, so the right side would be heavier. (Then again, are there really 6 cookies on the left and 4 on the right, or are some cookies hidden? Hard to tell.)

As far as I’m concerned, the only acceptable answer is that the pile of green, yellow, and orange cookies must be heavier — assuming, of course, that the balance scale isn’t malfunctioning — because the pans are tipped in that direction.

All of this reminds me of the poem “Smart” by Shel Silverstein.

SMARTMy dad gave me one dollar bill

‘Cause I’m his smartest son,

And I swapped it for two shiny quarters

‘Cause two is more than one!And then I took the quarters

And traded them to Lou

For three dimes — I guess he don’t know

That three is more than two!Just then, along came old blind Bates

And just ’cause he can’t see

He gave me four nickels for my three dimes,

And four is more than three!And I took the nickels to Hiram Coombs

Down at the seed-feed store,

And the fool gave me five pennies for them,

And five is more than four!And then I went and showed my dad,

And he got red in the cheeks

And closed his eyes and shook his head–

Too proud of me to speak!

### 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

multiplythe prices together and declared that the total was $7.11. Although I knew the prices should have beenadded, 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!

### Think of a Number

I love to create math games almost as much as I love to play them.

My favorite professional project was leading the development of Calculation Nation. And my favorite game on the site is neXtu, though other games on the site may promote more sophisticated mathematical thinking.

I have many reasons to love my wife, not least of which is her creation of the game Dollar Nim. While I can’t take credit for the rules, I will take credit for its analysis and its popularization. (What do you call a wife who makes up a game that gets you a publication credit? A **keeper**!)

Recently, I’ve been frustrated by the lack of games for teaching algebra. I’ll give props to the good folks at Dragonbox, which uses a game environment to teach algebra. But I’m not yet convinced that it leads to deep algebraic understanding; even they admit “to transfer to pencil and paper, children must be explained how to rewrite equations line by line.” They also claim that “in-house preliminary tests indicate a very high level of transfer to pencil and paper,” but that’s the fox watching the henhouse.

So I’ve been thinking about games I can play with my sons that will allow them to engage in algebraic thinking. But I don’t want them to know they’re engaging in algebraic thinking. I have two criteria for all math games:

- The game mechanics depend on mathematics. The math is not tangential to the game; it
**is**the game. - Kids don’t realize (or at least they don’t care) that it’s a math game, because it’s fun.

It pains me to write that second criterion, because math **is** fun. But I know not everyone shares that opinion. So I do my best to disguise any math learning in the game and then, when they least expect it — BOOM! — I drop the bomb and show them what they’ve learned.

So here’s a game I recently devised.

- Player A chooses a number.
- Player B chooses two operations for Player A to perform on the number.
- Player A performs those operations and then tells the result to Player B.
- Player B then tries to identify Player A’s number.

These rules leave something to be desired, since Player B could simply ask A to “multiply by 1” and then “add 0,” in which case finding A’s number would involve no work whatsoever. To be a stickler, an additional rule could impose that either addition or subtraction can be used exactly once and that no operation can involve either 0 or 1. In a middle school classroom, I suppose I would state such a rule explicitly; for playing this game with my seven-year-old sons, I opted not to.

We played this game three times on the car ride to school yesterday. One game went like this:

- I thought of a number (14).
- Eli asked me to add 3 to my number.
- Alex asked me to multiply by 3.
- I told them the result: 51.

Eli then guessed that my number was 16. He had subtracted 3, then divided by 3.

“No!” said Alex. “You added 3 first, so you need to subtract 9.”

“Why 9?” Eli asked. “Daddy only added 3.”

“But he multiplied by 3, so if you subtract first, you have to subtract 3 × 3.”

Eli then realized that my number was 14.

He thought for a second. “Oh,” he said. “I should have divided by 3 **first**, then subtracted.”

Wow, I thought. This is going even better than I hoped.

Though they didn’t use the proper terminology, the boys had a great discussion about “undoing” operations by performing inverse operations in reverse order. In 10 minutes, they taught themselves how to solve a two-step equation:

3*x* + 3 = 51

Grace Kelemanik once said that she knew she was being effective when she didn’t have to say a word. She’d watch from the back of the room as students carried the conversation and guided one another to correct mathematical thinking.

I will never claim to be half the educator that Grace Kelemanik is. But yesterday morning, I was pretty darn effective.

**I’d love to hear about math games you’ve played with kids, whether you invented them or not.**