## Constant Change

I’m also old, cranky, and cynical. Whatever.

My frustration is not the my-flight-was-delayed-three-times-then-eventually-cancelled-and-there-are-no-more-flights-to-Cleveland-till-tomorrow-morning type. It’s not even the can’t-believe-my-boss-is-making-me-go-to-Cleveland kind of frustration. More like the why-aren’t-there-the-same-number-of-hot-dogs-and-buns-in-a-pack variety. So it’s a First World problem, to be sure, but still annoying. I’ll explain more in a moment.

But first, how ’bout a math problem to get us started?

If you make a purchase and pay with cash, what’s the probability that you’ll receive a nickel as part of your change?

Sure, if you want to get all crazy about this, then we can take all the fun out of this problem by stating the following assumptions:

- You only pay with paper currency. If you paid with coins, then the distribution of coins you’d receive as change would likely vary quite a bit.
- You never use 50¢ coins. Honestly, they’re just too obscure.
- Transaction amounts are uniformly distributed, so that you’re just as likely to receive 21¢ as 78¢ or any other amount.
- Cashiers don’t round because they dislike pennies. So, if you’re supposed to get 99¢ change, the cashier doesn’t hand you a dollar and say, “Don’t worry about it.” Instead, you actually get 99¢ change.

But stating assumptions is a form of mathematical douchebaggery, isn’t it? (As an aside, check out the definition of douchey that’s returned when you do a search. Sexist, anyone?) I prefer problems with no assumptions stated; let folks make their own assumptions to devise a model. If you and I get different answers because of different assumptions, no worries. Maybe we both learn something in the process.

Anyway, where was I? Oh, yeah…

Understanding the solution to that problem is a precursor to the issue that’s causing me frustration. I’ll give the solution in a minute, so pause here if you want to solve it on your own, but let me now allow the proverbial cat out of its bag and tell you why I’m frustrated.

At our local grocery store, there’s a **coin counting machine** that will count your change, sort it, and spit out a receipt that you can take to the customer service desk to exchange for paper currency. Walk in with a jar full of change, walk out with a fistful of fifties. Pretty nifty, right? Except the machine charges a ridiculous 8.9% fee to perform this service. No, thank you.

My bank used to have a similar coin counting machine, and if you deposited the amount counted by the machine into your account, there was no fee. The problem is that everyone was doing this to avoid the grocery store fee, so the machine broke often. The bank finally decided the machine wasn’t worth the maintenance fees and got rid of it. Strike two.

Which brings me to my current dilemma. One Saturday morning every month, we now spend 30 minutes counting coins and allocating them to appropriate wrappers. Which is fine. The problem, however, is that we run out of quarter and penny wrappers way faster than we run out of nickel or dime wrappers. Which brings me to the real question for the day:

Since pennies, nickels, dimes, and quarters are not uniformly distributed as change,

why the hell does every package of coin wrappers contain the same number for each coin type?

The Royal Sovereign Assorted Coin Preformed Wrappers is the best-selling collection of coin wrappers on Amazon, and it provides 54 wrappers for each coin type. They also offer a 360‑pack with 90 wrappers for each coin type; Minitube offers a 100‑pack with 25 wrappers for each coin type; and Coin-Tainer offers a 36‑pack with 9 wrappers for each coin type. But what **no one** offers, so far as I can tell, is **a collection of coin wrappers with a distribution that more closely resembles the distribution of coins that are received as change**.

Whew! It feels good to finally raise this issue for public consideration.

So, the question that I really wanted to ask you…

Given the distribution of quarters, dimes, nickels, and pennies that are received in change, and given the number of coins needed to fill a coin wrapper — 40 quarters, 50 dimes, 40 nickels, and 50 pennies — how many of each wrapper should be sold in a bundled collection?

To answer this question, I determined the number of coins of each type required for every amount of change from 1¢ to 99¢. The totals yield the following graph:

The number of pennies is nearly five times the number of nickels. And there are nearly twice as many quarters as dimes.

But I realize that’s a **theoretical** result that may not match what happens in practice, since this assumes that the amounts of change from 1¢ to 99¢ are uniformly distributed (they aren’t) and that cashiers don’t round down to avoid dealing with pennies (they do). In fact, when I made a purchase of $2.59 yesterday, instead of getting one penny, one nickel, one dime, and one quarter as change, the cashier gave me one penny, **three** nickels, and one quarter, in what was clearly a blatant attempt to skew my data.

So for an **experimental** result, I counted the pennies, nickels, dimes, and quarters in our home change jar. The results were similar:

The ratio of pennies to nickels is closer to three, but the ratio of quarters to dimes is still roughly two.

Using a hybrid of the theoretical and experimental results, and accounting for the fact that only 40 quarters and nickels are needed to fill a wrapper whereas 50 pennies and dimes are needed, it seems that an appropriate ratio of coin wrappers would be:

**quarters : dimes : nickels : pennies :: 17 : 8 : 6 : 19**

Okay, admittedly, that’s a weird ratio. Maybe something like **3:2:1:4**, to keep it simple. Or even 2:1:1:2. All I know is that 1:1:1:1 is completely insane, and this nonsense has got to stop.

Hello, Royal Sovereign, Minitube, and Coin-Tainer? Are you listening? I’ve completed this analysis for you, free of charge. Now **do the right thing**, and adjust the ratio of coin wrappers in a package accordingly. Thank you.

Wow, that was a long rant. Sorry. If you’ve made it this far, you deserve some comic relief.

How many mathematicians does it take to change a light bulb?

Just one. She gives it to a physicist, thus reducing it to a previously solved problem.If you do not change direction, you may end up where you are heading. – Lao Tzu

The only thing that is constant is change. – Heraclitus

Turn and face the strange ch-ch-ch-changes. – David Bowie

A Buddhist monk walks into a Zen pizza parlor and says, “Make me one with everything.” The owner obliged, and when the pizza was delivered, the monk paid with a $20 bill. The owner put the money in his pocket and began to walk away. “Hey, where’s my change?” asked the monk. “Sorry,” said the owner, “change must come from within.”

As for the “probability of a nickel” problem that started this post, here’s my solution.

For change amounts from 1¢ to 25¢, there are ten values (5‑9 and 15‑19) for which you’ll receive a nickel as part of your change.

This pattern then repeats, such that for change amounts from **25 n + 1** to

**25**, where

*n*+ 25*n*is the number of quarters to be returned, you’ll receive a nickel when the amount of change is

**25**, where

*n*+*k**k*∈ {5, 6, 7, 8, 9, 15, 16, 17, 18, 19}. For 0 ≤

*n*< 4, there are 40 different amounts of change that will contain a nickel, so the probability of getting a nickel as part of your change is 40/100, or 40%.

## Math of the Rundetaarn

As we were exiting the Rundetaarn (“Round Tower”) in Copenhagen, Denmark, I noticed a man wearing a shirt with the following quotation:

Find what you love, and let it kill you.

The only problem is that the shirt attributed the quotation to poet Charles Bukowski, when apparently it should have been attributed to humorist Kinky Friedman. For what it’s worth, my favorite Friedman quote is, “I just want Texas to be number one in something other than executions, toll roads, and property taxes.” But this ain’t a post about Kinky Friedman, or even Charles Bukowski. So, allow me to pull off the sidewalk and get back on the boulevard.

Whoever said it, the quotation hit me as drastically appropriate. I suspect that **math will someday kill me**… likely as I cross the street while playing KenKen on my phone, oblivious to an oncoming truck. As I exited the Rundetaarn, I was thinking about all the math that I had seen inside — much of which, I suspect, would not have been seen by many of the other tourists.

The Rundetaarn, completed in 1642, is known for the 7.5-turn helical ramp that visitors can walk to the top of the tower and, coincidentally, to one helluva view of the city. That leads to Question #1.

Along the outer wall of the tower, the winding corridor has a length of 210 meters, climbing 3.74 meters per turn.

What is the (inside) diameter of the tower?

Above Trinitatis Church is a gift shop that is accessible from the Rundetaarn’s spiral corridor. The following clock was hanging on the wall in that little shop:

I have no idea who the bust is, but the clock leads to Question #2.

What

sequence of geometric transformationswere required to convert a regular clock into this clock?

And to Question #3.

Do the hands on this clock spin

clockwiseorcounterclockwise?

And to Question #3a.

What is the “error” on the clock?

A privy accessible from the spiral corridor in the Rundetaarn has been preserved like a museum exhibit. Sadly, I have no picture of it to share, but a sign next to the privy implied that the feces deposited by a friar would fall 12 meters into the pit below.

That leads to Question #4.

What is the

terminal velocity of a depositwhen it reaches the bottom of the pit? (Or should that be “turd-minal velocity”?)

The first respondent to correctly answer all of these questions will earn inalienable bragging rights for perpetuity.

## How Wide and How Deep?

In 2002, William Schmidt described the U.S. math curriculum as “a mile wide, an inch deep,” and it’s been bugging the sh*t out of me ever since.

I mean, I get what he and his co-authors were saying: The curriculum contains too many topics, so they can’t be covered with sufficient depth.

But if a mile is **too wide** and an inch is **too shallow**, then what dimensions *would be* appropriate?

One-inch wide and a mile deep would be problematic, too. That’d be like spending an entire year teaching kids to count to 10.

I suppose we could opt for a square curriculum instead. A curriculum that is a mile wide and an inch deep has an area of 5,280 × 1/12 = 440 square feet, so the conversion would look something like this, with the thin line representing a mile by an inch and the square representing 21 feet by 21 feet:

Sorry, it’s not to scale because of space limitations.

The square curriculum doesn’t feel quite right, either. The only way I know to make this problem tractable is to look at data.

In the late 1990’s, I was a standards weenie. I was fascinated by the variety from state to state. Because I didn’t have a girlfriend (and ostensibly didn’t want one, either), I would read state standards documents **for fun**. At the time Schmidt coined his phrase, Florida had more than 80 standards in each grade, and Utah subjected students to over 130 standards each year. As I recall, the average state had more than 100 standards at each grade level.

Today, Common Core represents a significant reduction in the number of standards. There are approximately 30 standards per grade for K‑8, and closer to 40 standards per course in high school.

Which means that if the curriculum used to be a mile wide, then the current curriculum is closer to ⅓ × 5,280 = 1,760 feet wide.

But if it’s ⅓ as wide, then it needs to be 3 times as deep. Which means the current curriculum is 1,760 feet wide by 3 inches deep, so it looks something like this:

Doesn’t feel like much of an improvement, does it? And the phrase “3 inches deep” doesn’t inspire confidence that the curriculum now has the depth it needs.

So, I give up. I don’t know what the proper dimensions ought to be. I just know that Schmidt’s phrase was hyperbole for dramatic effect, and it worked.

**What do you think are the proper dimensions of the math curriculum?**

Here’s a puzzle about width and depth:

How much dirt is in a hole that measures 4¾ feet × 5¼ feet?

And I know a joke about width, but you need to be able to read CSS:

.yomama { width: 99999999px; }

** **

## 3 Questions to Determine if You’re a Math Geek

Yesterday morning on Cooley and Kevin, a local sports radio show, the hosts and producer each posited three questions that could be used to determine if someone is **a real man**. (The implication being, if you can’t answer all three, then you ain’t.) I didn’t like that many of the questions focused on sports, but I’m not surprised. I was, however, surprised by some of the non-sports questions. What do you think?

**Thom Loverro (guest host):**

- Who wrote
*The Old Man and The Sea*? - What was the name of the bar owned by Humphrey Bogart in
*Casablanca*? - Name three heavyweight boxing champions.

**Kevin Sheehan (regular host):**

- Who was Clark Kent’s alter ego?
- Name one of the two fighters in the “Thrilla in Manila.”
- Who won the first Super Bowl?

**Greg Hough (producer):**

- Name one James Bond movie and the actor who played James Bond in it.
- Who did Rocky beat to win the title?
- With what team did Brett Favre win a Super Bowl?

During the rounds of trivia, Loverro remarked, “If you can name three heavyweight champs but haven’t seen *Casablanca*, then you’re still in puberty.”

This made me wonder:

What three questions would you ask to determine if someone is

a real woman?

One possible question might be, “Name two of the three actresses who tortured their boss in the movie *Nine to Five*.” Then I remembered that women don’t play the same stupid games that men do. And I realized that strolling too far down that path will lead to hate mail or a slap or both. So, let’s move on.

It also made me wonder if there are three questions you could ask to determine if someone is **a real math geek**. Sure, you could use the Math Purity Test, but that’s 63 questions. A 95% reduction in the number of items would be most welcome.

So, here are my three questions:

- What’s the eighth digit (after the decimal point) of π?
- What’s the punch line to, “Why do programmers confuse Halloween and Christmas?”
- Name seven mathematical puzzles that have entered popular culture.

And my honorable mention:

- What’s the airspeed velocity of an unladen swallow?

One of my initial questions was, “Have you ever told a math joke for your own amusement, knowing full well that your audience either wouldn’t understand it or wouldn’t find it funny?” But I tossed that one, because it’s a yes/no question that was personal, not factual. Eventually, which questions were kept and which were discarded came down to one simple rule: If nothing was lost by replacing a question with, “Are you a math dork?” then it should be rejected.

How’d I do? Opinions welcome. **Submit new or revised questions for determining one’s math geekiness in the comments. **

## WODB, Philly Style

Given the subject line, you might think I’d ask which of the following doesn’t belong:

So many jokes to be made, so little time.

But actually, I was referring to something completely different.

On Thursday morning, I gave a talk to 850 enthusiastic teachers at the School District of Philadelphia‘s Summer Math Institute. That may be the largest group to which I’ve ever spoken; it certainly exceeds the 600+ to whom I delivered my Punz and Puzzles talk at the North Carolina Council of Teachers of Mathematics conference, and it likely exceeds the number of people who heard me sing a karaoke version of Liz Phair’s *Girls! Girls! Girls!* after a half bottle of tequila — although that’s a story for another time. (Yes, I know all the words. But I’ve said too much.)

I was going to begin my talk in Philadelphia with the following warm-up question,

Quadrilateral

MATHis similar toATHM. What can you say aboutMATH?

because I wanted to ask the follow-up question,

What can you say about math?

taking advantage of the double entendre caused by *MATH* (a geometric figure) and *math* (an academic subject). Clever, no?

But I was worried that a high-school level geometry question might overshoot my audience of K‑8 and Algebra I teachers. So I was looking for an alternative.

That’s when Jen Silverman — to whom I owe a huge thanks and several pints — suggested that I do a *Which One Doesn’t Belong* using the letters M, A, T, and H. Based on her suggestion, I created this:

It led to a great discussion, both mathematical and otherwise. So, here’s my challenge to you:

**Which letter doesn’t belong?
**

*Post your choice and explanation in the comments.*

If you’re not familiar with *Which One Doesn’t Belong*, then check out http://wodb.ca or follow @WODBMath.

I had always thought that the #WODB movement began with Christopher Danielson’s “better shapes” book, *Which One Doesn’t Belong* (forthcoming).

But then I found this activity sheet in *Navigating through Problem-Solving and Reasoning in Prekindergarten and Kindergarten*, which was published by NCTM in 2003:

So WODB is at least 13 years old, probably more. Anyone know exactly when or where it started? I’d guess Lola May, though that’s purely speculative.

Huge props to Karl Fisch, who posted the funniest WODB to date:

## As If Your Life Depends on It

It was early Wednesday morning (or late Tuesday night, depending on how you look at it) of Finals Week. Yes, I should have been studying — or sleeping; it was 3 a.m., after all — but I was young and in love, and wandering through the quads and into unlocked academic buildings on Penn State’s campus with my girlfriend held far more appeal than the problems and theorems in my linear algebra textbook. I remember a light snowfall and how beautiful she looked in the lamplight. I remember my surprise when I pushed on the main door to Sparks Building and it opened. But what I remember most from that night is a quote that a psychology professor had borrowed from a student’s paper and taped to her office door:

Many things depend on many things.

Indeed.

I don’t remember that girlfriend’s name. And I remember very little from my linear algebra course. But I’ll never forget that quote, and I’ve repeated it many times in business meetings.

de·pen·dentn.what hangs from de necklace

Dependence is a topic that rears its head frequently in mathematics, from algebra to probability, and it’s useful in a variety of contexts.

Football, for instance. Redskins safety David Bruton showed his understanding of dependence during a recent radio interview:

I’m between 225 and 230 [pounds], depending on what I had for lunch.

And measurement. Comedian Ron White understands dependence, too:

Now, I’m between 6’1″ and 6’6″, depending on which convenience store I’m leaving.

Some things aren’t really dependent at all…

The economy depends on economists in the same way that the weather depends on forecasters.

And some things are subjective…

Your true value depends entirely on what you are compared with.

Some things depend on whom you ask…

A teacher said to her student, “Billy, if both of your parents were born in 1967, how old are they now?”

After a few moments, Billy answered, “It depends.”

“On what?” the teacher asked.

“On whether you ask my mother or my father.”

And other things on your perspective…

How long a minute feels depends on what side of the bathroom door you’re on.

The location of an animal?

Where can you find polar bears?

Depends on where you lost them!

But the better answer to that joke is, “Just check their polar coordinates!” (You’re welcome.)

This post wouldn’t be complete without an obligatory old-person joke…

An old man is flirting with a woman at the senior center. He asks her, “If I took you out for a night of wining, dining and dancing, what would you wear?”

The old woman replies shyly, “Depends.”

And finally, one last math joke…

How many math professors does it take to plaster a wall?

Depends how hard you throw them.

## Flag Day Math

Tuesday, June 14. Flag Day. It’s nearly impossible for mathy folks to **not** tell this joke today.

Several engineers were attempting to measure the height of a flag pole. They only had a measuring tape, and they were getting quite frustrated trying to slide the tape up the pole. They could get the tape no more than a third of the way up the pole before it would bend and fall down.

A mathematician asks what they’re doing, and they explain. The mathematician offers to help. She removes the pole from the ground, sets it down, and measures it easily. She then returns the measuring tape to the engineers, and walks off.

When she leaves, one engineer says to the others, “That’s just like a mathematician! We need to know the height, and she gives us the length!”

Those who know it will also tell this one, or a variant.

How do statisticians determine which banner to hoist?

They take a flag poll.

And then there are jokes about specific flags.

I’m about as motivated as the guy who designed the Japanese flag.

Honestly, I want to stop. But I can’t. Just one more…

What’s the best thing about Switzerland?

I don’t know, but the flag’s a big plus.

Okay, seriously… I didn’t invite you here today to listen to bad jokes. (Well, that’s not the **only** reason, anyway.)

I invited you here today to have a little Flag Day fun with math. The projectionist Shahee Ilya has **converted the flag of every country into a pie graph** based on its colors. For example, the Austrian flag has two red stripes and one white stripe, so it is converted to a pie graph as follows:

Pretty cool, huh?

What follows are pie graphs for ten flags. Even if you are geographically challenged, I assure you that you’ve heard of all ten countries represented below. **Can you name the country whose flag was used to create each pie graph?**

Stumped by the challenge? Here’s a hint: The countries whose flags are represented above are **the ten most populous countries on Earth**. (Admittedly, had someone asked me to name the ten most populous countries prior to writing this post, I would have been lucky to identify half of them.)

And just to put some space between the pie graphs above and the countries whose flag they represent below (i.e., the answers), I include for your enjoyment one of the most hideous puns you’ll ever see, modified from an even worse version at Six Puns:

During a recent heat

wave, apollrevealed that beads of sweat had amassed(mast) on the secretary’s forehead and a virus wasripplingthrough the officestaff. Although the boss knew that the secretary was very sick, he saw no reason toban herfrom the office. Instead, he wrote a note withpennant(pen and) paper, and heflagged the issue to be addressed with thestandardprotocol.

If you tolerated that, you certainly deserve the answers…

Nigeria | Pakistan |

Russia | United States |

Indonesia | Japan |

Bangladesh | Brazil |

China | India |

Click on over to **shaheeilyas.com/flags** to see the pie graph for every country in the world. Clicking on the pie graph will reveal the flag and country name.