### Math Problems That Come Up

Math helps me interpret the world. When I first became a stay-at-home mom after nine years of teaching high school math, I saw math problems presenting themselves everywhere: at the doctor's office, in the news, cooking dinner. So I started a word file of mostly high-school-level problems, and I told myself that they might be useful some day if I ever went back to teaching. Eventually, our computer died, and I misplaced the file.

Here, I'm starting over. When I come across a math problem that would be appropriate for middle/high school students, I'll add it here.

1. Angles in a Tetrahedron  Geometry
Owen and Johnny have constructed a tetrahedron, with edges made of cardboard tubes. (It's a skeleton solid, so the faces aren't physically represented.) It was a fun project, but it's not quite regular. Owen wondered how he could make a regular tetrahedron out of, say wooden dowel rods. Let's imagine that the dowel rods fit together perfectly at each vertex, and that their diameters are large enough that you'd have to angle the ends just right in order to make the vertices work.

What do the ends of the dowel rods look like? What are the angles involved?

2. Height and Weight Percentiles Statistics, Logic
When Johnny was a small baby, we went to the doctor fairly frequently, and each time we were told his height and weight percentiles. We were also given his BMI (body mass index). I always wondered what, if anything, his height and weight percentiles could tell us about his BMI percentile. When he was very young, he was long and thin. Let's say that his height percentile was 83 and that his weight percentile was 32.  With only that information, what constraints, if any, can we put on his BMI percentile?

3. Halving the Bread Recipe Fractions
Here's a straightforward fraction computation that I use when I bake bread:

The recipe calls for 1.5 tablespoons of yeast. I don't have a half-tablespoon measuring spoon, but I do know that there are 3 teaspoons in a tablespoon, and of course I do have a teaspoon measure, as well as a half teaspoon and quarter teaspoon. When I halve the recipe, as I frequently do, how many teaspoons of yeast should I use?

4. The Surprise Quiz, or the Unexpected Hanging Paradox  Logic
This one was reportedly a favorite of my great uncle Costas:

Your teacher announces that there will be a pop quiz next week. The quiz is guaranteed to happen on one of the days of the week (i.e. Monday, Tuesday, Wednesday, Thursday, or Friday), but the day on which it will be given will be a surprise. That is, you'll arrive in class not knowing whether there will be a quiz that day.

But is this scenario possible? If Friday arrives, and there hasn't been a quiz yet, then you know that the quiz will happen that day. But then it won't be a pop quiz. So your teacher can't choose Friday to give the quiz. But then on Thursday, if the quiz hasn't happened yet, you'll know that it has to be on that day (because you've already ruled out Friday). But then it isn't a surprise. And so on.

This problem turns out to be more subtle than it appears at first glance. It has its own wikipedia article, where it's called the "unexpected hanging paradox."

5.  The Mixed Up Animals Puzzle  Combinatorics
I love the way straightforward combinatorics problems can generate such surprisingly large numbers.

We have a puzzle that looks like this:

You can scramble the pieces like this:

How many distinct arrangements of pieces are there? There are two ways of interpreting this question, so let's address both.

First, let's assume that the combinations of head-torso-legs matter, but that each three-piece animal's position on the board does not matter. For example, if the tiger and the monkey switch places, that does not count as a different arrangement.

Second, for a presumably larger answer, let's assume that the position of the pieces in relation to the board matters. So, for instance, if the tiger and the monkey switch places (or the owl and the frog-head/lion-torso/frog-legs creature), that counts as a different arrangement.

Hint: Either way you define "distinct arrangement", the answer is a very large number. More than one billion using the first definition and much more than one trillion using the second definition.

6. Surface of the Earth Observed by Satellite Integral Calculus
Owen's satellite, TRMM, observes the portion of the earth that lies within 35 degree North or South of the equator. The satellite that's set to launch soon, GPM, will cover everything within 65 degrees of the equator.

What fraction of the earth's surface is covered by TRMM, and what fraction will be covered by GPM? (We think that the answers are surprising.)

There's actually a simple formula that you can use to compute the answer, but let's assume that you don't know the formula. You'll need to use integral calculus. Once you use the calculus, the formula will become obvious.

TRMM = Tropical Rainful Measuring Mission, GPM = Global Precipitation Measurement Mission

7. Mini Meatballs vs. Regular Meatballs  Geometry, Ratios of Volumes
My Betty Crocker cookbook has a recipe for meatballs. The recipe yields 20 1.5-inch meatballs. Instructions for making mini meatballs are also given. With the same quantity of meat mixture, you shape 1-inch balls. According to the recipe, you should have 3 dozen meatballs.

Are the yields listed reasonable? How much bigger are the 1.5-inch meatballs, in volume, than the 1-inch meatballs? If the diameters are measured exactly, and the recipe does indeed yield 1.5 meatballs, how many meatballs would you expect if the diameters were 1 inch? (Answer: around 67, twice as many as predicted in the recipe.)

This might be an illustration of the principle that the ratio of the volumes of two similar solids is not generally the same as the ratio of corresponding lengths (e.g. radius, height) of those solids. In fact, the ratio of volumes is the cube of the ratio of corresponding lengths.

But . . .

Going a bit deeper: What are some possible ways that the cookbook might have made such a large miscalculation? Is 36 vs. 67 really so unreasonable? If we assume that the recipe does yield 20 1.5-inch meatballs, how large would we make the diameters of the meatballs in order to get exactly 36 meatballs?

Answer to the last question: About 1.2 inches. So, at first thought, the cookbook seemed to be way off, because they under-predicted the yield of the mini meatballs. But when you're making meatballs, it's difficult to tell the difference between 1 inch and 1.2 inches. Such a small increase in diameter results in nearly twice as much volume.

8. Typical Snowfall in Washington D.C.  Mean, Median
Source: The Washington Post, Capital Weather Gang, "Officially snowy: D.C. edges above average, for only 4th time in 25 years." Feb. 18, 2014

So far this year, D.C. has had 15.5 inches of snow, which is slightly more than the 30-year mean of 15.4".

Here's a cool fact: Even though this year's snow total (so far) is basically average, this winter is the 4th snowiest winter in 25 years. Only four winters in 25 years have had above average snowfall.

This is a great example of how the mean gets skewed by the few winters with huge amounts of snow. It also is a good example of the need to be careful about what we mean when we say, "average." We tend to think that the "average" gives us an idea of what is "typical." But in this case, 80% of the data points fall below average.

9. Ratio of Arc Length to Chord Length  Geometry, Trigonometry

Suppose that the only information you have about an arc of a circle is the ratio of the length of an arc to the length of the chord whose endpoints coincide with the endpoints of the arc. Does that ratio uniquely determine the measure of the arc (i.e. in degrees or radians, not the arc length)? With no information except the aforementioned ratio (i.e. arc length: length of chord), how might one determine the measure of the arc?

Here's the same problem using different words:

You have an arc, that is a piece of a circle, and it's not straight. It will always be a longer path from one endpoint to the other than the straight segment joining those same endpoints. But how much longer? Let's say you're able to compare the length of the arc with the length of the segment, i.e. the chord. Specifically, you have the ratio of the arc length to the segment length. With that information alone, can you determine how much of the circle your arc represents? For example, if your ratio is close to one, then you'd expect your arc to be only a small portion of the circle. How could you calculate the precise measure of the arc in degrees or radians? (The circle measures 360 degrees, the semi-circle measures 180 degrees, and so on.)

It turns out that the problem is not as trivial as you might think, and it boils down to an equation that cannot be solved analytically.

Personal note: The arc length/chord length problem was something that I thought about when I was a teenager. I thought that it might eventually come up in one of my math courses (pre-calculus or something), but it never did.

10. Leap Year  Thinking through things logically

How does Leap Year affect the timing of the beginning of each season?

For example, spring begins at the precise moment of the year when the sun crosses the plane of the earth's equator (and thus the earth's axis neither tilts away from nor toward the sun). We call that moment the vernal equinox. How would we expect the precise timing of the equinox of a given leap year, 2016 for example, to compare with the timing of the equinox of subsequent years? Consider at least the four years following a leap year. (Remember the rule that every end-of-century year that is not divisible by 400 is not a leap year. For example, 2100 will not be a leap year, but 2400 will be.)

Once you've figured it out, you may look up the dates and times of equinoxes for several years in this Wikipedia article.

11. Costco Membership Arithmetic, Percents

When I joined Costco a few years ago, I had to choose between the Gold Star membership costing \$50/year and the Executive membership costing \$100/year.
Executive members get a 2% "reward" (i.e. a refund equal to 2% of total purchases) on Costco purchases. How much money would I have to spend at Costco each year to make the Executive membership worth it?

12. Spot-it   I'm not sure how to categorize this problem, but it's a fun one.

My son and I were introduced to the game Spot-it. It is a deck of 55 circular cards, each with 8 different pictures. Between every pair of cards there is exactly one pair of matching pictures.

 Cards in the game of Spot-it.

Can you find a method for creating a deck of your own where every pair has exactly one match?

Start with a smaller set of symbols on each card, say, four per card. Using only the 26 letters of the alphabet, for example, how big can you make your Spot-it deck?

There are several websites that tackle this problem, but don't visit them until you've thought about the problem yourself fist!

Here's a good starting place.

13. Making Tortillas  Exponents, Powers of Two

I recently bought masa, corn flour, and used it for the first time to make tortillas. The directions say to form a dough by mixing the corn flour, water, and a little salt, then to divide the dough into 16 equal pieces, shaping each piece into a ball.

Why is the number 16 particularly convenient here? Why is it easier to divide the dough into 16 equally sized balls than into, say, 15 balls or 20 balls?

14. Bernard Baruch's Quotation  Logic, Contrapositive

Here is a quotable quote from Bernard Baruch: "Be who you are and say what you feel, because those who mind don't matter, and those who matter don't mind."

It's catchy and has a nice ring to it, but is it logically redundant? Consider the following statements:

Those who mind don't matter.
Those who matter don't mind.

The second statement is the contrapositive of the first statement. To see this more easily, let's rewrite both sentences as if-then statements. We'll assume that "those" is short for "those people."

If a person minds, then he/she does not matter.
If a person matters, then he/she does not mind.

If A, then not B.
If B, then not A.

Or, if we let C stand for he/she does not matter, then the negation of C will be he/she does matter. So then we have

If A, then C.
If not C, then not A.

Since any conditional statement is logically equivalent to its contrapositive, the statement, "Those who mind don't matter," means exactly the same thing as, "Those who matter don't mind." Logically speaking, Baruch is repeating himself. (Of course, I'm not complaining! Poetic license allows him to repeat himself in whatever way he pleases. But it's a good exercise to think about what each part of his sentence actually means.)

I came up with my own set of redundant statements recently, quite by accident. I had attended a catered event which caused some of the guests to be sick afterwards. Initially suspecting food poisoning, I was imagining how we might find the source. I reasoned that we could ask all the guests which foods they ate. Then, hypothetically, suppose we found the following:

Everyone who got sick ate the chicken.

And then, suppose we also found the following to be true:

Everyone who didn't eat the chicken didn't get sick.

It took me a few minutes to realize that these would not be two separate pieces of evidence. The two statements, in fact, have exactly the same meaning.

15.  Genetic Genealogy

My sister recently had her DNA analyzed. She learned that her genetic origins are 99.2% European and 0.8%  North African or Middle Eastern. Of course, since she and I have the same set of parents, we share ethnic make-up. (Frankly, I'm a little disappointed that we aren't more multi-ethnic. Oh well.)

Here's the question: If you look at our family tree, how far would you need to go back before you might be able to find one ancestor who is roughly 100% North African/Middle Eastern? For example, it's pretty easy to see that we don't have a grandparent or even a great-grandparent who is fully North African/Middle Eastern, but what about a great-great-grandparent? Or a third-great-grandparent? How many "greats" would it take to find a North African/Middle Eastern ancestor?

Answer: If  0.8% of our DNA is of  North African/Middle Eastern origin, that is equivalent to having a fifth-great-grandparent who is 100%  North African/Middle Eastern. Of course, we don't really know that there is just one line of  North African/Middle Eastern blood in our ancestry. It's possible that we have two unrelated sixth-great-grandparents who did not produce children together, but whose descendants eventually paired up in the family tree.  There are other, more complicated possibilities as well. Two seventh-great-grandparents and one sixth-great-grandparent, each of whom married Europeans, would do it.

1. This is a math problem I had to solve yesterday.
I was making a pop-up book that would open up as six evenly spaced pages around a semicircle. The pages were 4.75 inches wide. I wanted a band to go around the outside of the pages, so that when open it would form a section of the surface of a regular prism. Approximately how long did the band need to be? Is the approximation close enough that I can neglect the error? Is the approximation going to be too big, or too small?
I wanted to make a floor for the surface that would open up to make a semicircle. The floor would need to fold up so that it would fit between the evenly spaced page-walls. what should the folded surface look like? Without doing any measuring with a ruler, how could I get it to fold in that way?
The math I do for my work is largely linear algebra and 2D and 3D applied geometry, and always sitting in front of a computer with mathematical software.
Although I often need to understand concepts from calculus class to do my work (the concept of a function, adding, composing, and multiplying high-dimensional functions, the concept of the derivative, function minimization, basis functions, and power series are all things that often come up) , I have never needed to solve an integral or take a derivative of a function for work, and if I did I would use software to solve it.

1. Doug,

First of all, I can't visualize your pop-up book very well. It sounds really cool, so I wish that I could.

On the use of software to do calculus: Do you believe that the training you had in solving integrals and taking derivatives was important in helping you develop the conceptual understanding that you need to do your current work? Would you have benefited from relatively less time spent on computation?

2. If math instruction time is a zero-sum game, then yes, I would have benefited from more time spent on getting intuitive understanding of ideas, visualization and lots of examples at the cost of getting proficient at calculating integrals. I can only really speak for myself in this, though.
I will post photos of my pop-up Tardis soon. I'll try to describe it so you can picture it, though.
When the book is open, it depicts a cylindrical room (a version of this room http://www.kasterborous.com/wordpress/wp-content/uploads/2010/2013/08/Royal-TARDIS-visit.png) , something like a gazebo with a tall pillar in the center. Only half of the cylinder is actually represented.
Imagine a regular book with six pages. Each page is identical and has had some large parts cut out of it, but is essentially still a rectangular page. When the book is open to 180 degrees, these pages are equally spaced at the outer edge of the pages, so that from the top they form half of a twelve pointed asterisk.
On the bottom of the book is a paper fan with six ridges. It has been glued to the pages in such a way that the pages fall into each of the valleys of the fan. When the book is closed, the fan folds up.
There is a band glued around the outside that helps the pages to maintain their proper spacing, like a fence around the gazebo.

3. Oh, now I get it. Thanks for that cool problem, Doug. I really want to see your pop-up book!