## Archive for October, 2011

### Trig or Treat

On Mischief Night, I spent a long time explaining our Halloween decorations to my neighbor. But I understood why he was confused. After all, they didn’t *look* like Halloween decorations.

There was the WITCH of Agnesi taped to our front door…

…the SKELETON of a cube hanging from our tree…

…and Napier’s BONES drawn on the sidewalk in chalk.

After I set our neighbor straight, I went for a walk in the woods.

I came across a mathematician stirring a cauldron in the middle of an open clearing. One by one, she added the eye of a newt, two whiskers from a black cat, a pinch of wolf’s bane, a dash of bat’s blood, three hairs from a mermaid, the ear of a troll, and finally a dollup of dragon’s breath. I asked her, “Why don’t you just add everything at the same time?”

“Oh, no,” she said. “To make an effective potion, the ingredients must be integrated by parts.”

Finally, a math equation for today:

### Math for Figger Filbert*

A well-known problem:

A man walks 1 mile south, 1 mile east, and 1 mile north. He arrives at the same place where he started, and then he sees a bear. What color is the bear?

The answer, of course, is white. It’s a polar bear. These three moves will let a person return to the same place if he starts at the North Pole. (The person could also return to the same place if he starts at an infinite number of points near the South Pole, too. He could start at a point so that when he walks 1 mile south, he is at a point such that the east-west circle on which he is standing has a circumference of 1 mile. Then, he can walk 1 mile east to return to the same spot. Finally, he can walk 1 mile north, and he’s back where he started. Then again, he could also start at a point so that he can walk 1 mile south to a point where the circumference of the east-west circle is 1/2 mile, do that loop twice, then walk 1 mile north. Or find points where the circumference is 1/3 mile, 1/4 mile, 1/5 mile, etc. You get the idea. However, since there are no bears in Antarctica, the answer to my original question is still correct.)

Two points about this:

- In answer to the question, “Are there polar bears in Antarctica?” there is only one correct answer:
*Only if they are bipolar.* - I really don’t care to receive silly comments about how a bear trapper could capture a grizzly and take him to Antarctica, or how a brown bear might mistakenly meander north to the Arctic Circle.

Here is a similar question:

A man runs 90 feet, turns left, runs another 90 feet, turns left, runs another 90 feet, and turns left. He is now headed home, and two men with masks are waiting for him. Who are they?

If you don’t know the answer to this riddle, remember that today is the first day of the World Series. My prediction? The Rangers will win easily. It’s not really a fair fight. I mean, members of the Lone Star State’s law enforcement agency with opposable thumbs and automatic weaponry versus defenseless birds? Seriously, if the Rangers don’t win, then we need to seriously reconsider the theory of natural selection.

If you watch the first game of the World Series tonight, remember to enjoy the game. Please don’t get caught up trying to figure out if it converges or diverges.

Here are a few baseball-related math puzzles:

- A baseball player has four at-bats in a game. At three different times during the game, his batting averages for the entire season (rounded to three decimal places) have no digits in common. What was his average at the end of the game?
- During a little league game, the visiting team scored 1 run per inning, and the home team scored 2 runs per inning. What is the final score of this seven-inning game?
- During the first half of the season, Derek batted .100, but his average was .300 during the second half of the season. Similarly, Alex batted .200 the first half of the season and .400 the second half of the season. Both players ended up with the same number of total at-bats, yet Derek had a higher batting average for the entire season. How is this possible?

* *Figger Filbert* is a term for baseball fans who are obsessed with statistics. Such fans are easily identified; they will make statements like, “Did you know that Albert Pujols is batting .275 when facing married pitchers in suburban ballparks that only sell popcorn on the mezzanine level?” It’s a synonym for *number nut*.

### C’mon, Have a (Magic) Heart

The Zen of Magic Squares, Circles and Stars by Clifford Pickover is chock full of magic arrangements. On page 55, Pickover discusses Dürer’s method for creating a 4 × 4 magic square:

- Starting with the upper left corner and proceeding horizontally to the right, number the squares of a 4 × 4 grid with the consecutive integers 1‑16.
- Starting with the lower right corner and proceeding horizontally to the left, number the squares of a different 4 × 4 grid with the consecutive integers 1‑16.
- From the first grid, keep the integers that occur on the main diagonals. From the second grid, keep the integers that do not occur on the main diagonals.

A visual representation of the process might help to clarify:

The result is a 4 × 4 magic square. In fact, it is a slightly modified version of the magic square that appears in Albrecht Dürer’s *Melencolia I*. (Note that if the other 8 numbers from each grid were combined in a similar fashion, they would form a magic square, too.)

Serendipitously, my sons and I recently completed an art project that can be combined with Dürer’s method to form a “magic square heart.” The project my sons completed is as follows:

- Draw a square with a semicircle on top. Repeat to create two of these figures, preferably on paper of two diffferent colors, and cut them out.
- Cut from the bottom of each figure to the diameter of the semicircle, to divide the squares into equal‑width strips.
- Finally, “weave” the strips to form a checkerboard pattern.

This idea can be combined with Dürer’s method to create a magic square heart. But instead of dividing the squares into equal‑width strips, divide them into three strips whose widths are in the ratio 1:2:1. Then, draw the outlines for 16 squares, and number the squares as described in Dürer’s method above. The two pieces will look like this:

Then, weave the three strips into a pseudo‑checkerboard pattern. When woven together, the result will be the following magic square heart:

To complete this project with students, you can use the template below.

**Magic Heart Template:**

http://mathjokes4mathyfolks.com/mj4mf-magicheart.pdf

That said, it’s my belief that students will have maximum mathematical fun if they are allowed to create the heart from scratch. It’s an exercise in geometric construction to draw a square with a semicircle on top; weaving the strips into the appropriate configuration can lead to a discussion of geoemtric symmetry; investigating the patterns formed by the numbers can lead to a discussion of numerical symmetry; and, investigating the square to find that the rows, columns, and diagonals have a constant sum may inspire young minds in the same way that it inspired Albrecht Dürer.

### More Number Picking

In a previous post, I mentioned the Pick-a-Number game that the folks at NPR’s Planet Money were running:

Pick a number between 0 and 100. The goal is to pick the number that’s closest to

halfthe average of all guesses. For example, if the average of all guesses were 80, the winning number would be 40.

If everyone picked randomly, you would expect the mean to be approximately 50, in which case the winning number would be 25. So, you’d choose 25, right? But if everyone uses that same logic, then the mean would be 25, and the winning number would be 12.5. So, you’d choose 12.5, right? But if everyone used that same logic…

Well, you get the point.

When making your choice, it starts to feel like a game against Vizzini, the Sicilian from *Princess Bride*.

Only a great fool would reach for what he was given. I am not a great fool, so I can clearly not choose half the expected mean. But you must have known I was not a great fool, so I can clearly not choose half of half the expected mean…

Well, the results are in, and you can view them (and an explanation) here.

I take a minimal level of pride in receiving one of 772 honorable mentions for my guess of 12. (Don’t look for my name in the list, though. I used my son’s name as a pseudonym.)

Here’s a very simple pick-a-number game:

Pick a number between 12 and 5.

Make your pick before reading the next paragraph.

Did you pick 7? Most people do. My theory is that the magnitude and order of the numbers matters. Because the larger number is given first, and because the difference between the numbers falls within the appropriate range (12 – 5 = 7), it’s the “obvious” choice.

The trick would probably work equally well if the set-up were, “Pick a number between 19 and 6.” I suspect the most common choice would be 13.

Of course, this is just pop math psychology.

Speaking of “picking” and “numbers,” here’s a line a friend of mine used on an attractive waitress:

How can it be it that I’ve memorized the first 100 digits of π, yet I don’t know the 7 digits in your phone number?

For the record, I condone neither hitting on a waitress nor using that line.

### Preparing for Mid-Terms

The day before mid-term exams, the calculus professor allowed 10 minutes at the end of class for questions.

When one student asked the professor how many problems would be on the exam, the professor replied, “I think you will have a lot of problems on the exam.”

“Well, sir,” the student continued, “do you have any suggestions for what I can do to prepare?”

“Yes,” he said. “Just study the old exams. The mid-term exam will have the same types of problems, just the numbers will be different. But not all of the numbers will be different. Both π and *e* will be the same, of course, and there’s a reason it’s called Planck’s *constant*…”

Before dismissing the class, the professor warned that there would be no acceptable excuses for missing the exam.

Upon hearing this, the class clown said, “What about sexual exhaustion?”

“I’m sorry, Jason,” said the professor. “You’ll just have to write with your other hand.”

### 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.

### Pick a Number

I’ve told you about my favorite game before.

Planet Money from National Public Radio is currently conducting an experiment using a similar game.

Pick a number between 0 and 100. The goal is to pick the number that’s closest to

halfthe average of all guesses. For example, if the average of all guesses were 80, the winning number would be 40.

You can be part of the experiment until 11:59pm ET on Monday, October 10. The winner and an explanation will be posted on the Planet Money blog on Tuesday, October 11.

### What (Math) is in a Name?

One of my favorite online tools is the Mean and Median app from Illuminations. This tool allows you to create a data set with up to 15 elements, plot them on a number line, investigate the mean and median, and consider a box-and-whisker plot based on the data. Perhaps the coolest feature is that you can copy an entire set of data, make some changes, and compare the modified set to the original set. For example, the box-and-whisker plots below look very different, even though the mean and median of the two sets are the same.

It’s a neat tool for learning about mean and median, and I plan to use this tool in an upcoming presentation.

**Exceptional, Free Online Resources for the Middle Grades Classroom**

*G. Patrick Vennebush*

Thursday, October 20, 12:30-2:00pm

Room 401 (Atlantic City Convention Center)

For classroom use, I like to use this app with real sets of data. However, the app requires all elements of a data set to be integers from 1-100. Can you think of a data set with a reasonable spread that has no (or at least few) elements greater than 100? If so, leave a comment.

Recently, and rather accidentally, I found a data set that works well. Do the following:

Assign each letter of the alphabet a value as follows: A = 1, B = 2, C = 3, and so on. Find the sum of the letters in your name; e.g., BOB → 2 + 15 + 2 = 19.

Now imagine that every student in a class finds the sum of the letters in their first name. For a typical class, what is the range of the data? What is the mean and median?

The name with the smallest sum that I could find?

ABE → 1 + 2 + 5 = 8

The name with the largest sum?

CHRISTOPHER → 3 + 8 + 18 + 9 + 19 + 20 + 15 + 16 + 8 + 5 + 18 = 139

The Social Security Administration provides a nice resource for investigation, Popular Baby Names. Using a randomly selected set of 2,000 names and an Excel spreadsheet, I found the mean name sum to be 62.49, and 96% of the names had sums less than 100. Of the 80 names with sums greater than 100, many (such as Christopher, Timothy, Gwendolyn, Jacquelyn) have shortened forms (Chris, Tim, Gwen, Jackie) for which the sum is less than 100.

As it turns out, the frequency with which letters occur in first names differs from their frequency in common English words. The most common letter in English words is *e*, but the most common letter in names is *a*. The chart below shows the frequency with which letters occur in first names.

Because of this distribution, the average value of a letter within a first name is 10.54, which is slightly less than the 13.50 you might expect. This is because letters at the beginning of the alphabet, which contribute smaller values to the name sum, occur more often in names than letters at the end of the alphabet.

The chart below shows the distribution for the number of letters within first names. The mean number of letters within first names is 5.92 letters, and the median is 6. (In the data set of 2,000 names from which this chart is derived, no name contained more than 11 letters.)

Do you know a name that has more than 11 letters or has a name sum greater than 139 or less than 8? Let me know in the comments.