Special Blog Post: Scholarly Snippets and Quantitative Quandries

While your host here at The Dark Wraith Forums slogs through finals week, a little bit of entertainment for readers is in order. Below you'll find a random collection of faint facts, interesting insights, and probing problems for thrills and excitement.

Okay, perhaps 'thrills and excitement' is a bit of an overstatement. One way or t'other, here we go.

A Math Trick on the Calculator
Grab your calculator. It doesn't have to be anything fancy, just one that does basic arithmetic operations. Don't forget to turn it on first.

Choose a three-digit number. It can be any number from 100 to 999. Key it in to your calculator.

Now, repeat the digits in the same order so you have a six-digit number displaying. For example, it you had chosen 749 to begin with, you should have 749749 now in the display window.

Remember, I don't know what three-digit number you initially chose, so I certainly don't know the six digit number, right?

But I do know that, whatever number you have on that display, it's divisible by 13. That's right, I know for a fact that it's divisible by 13. Go ahead: hit the "÷" button, then key in "13" and hit "=" to see that I'm right. You've got a whole number displaying, don't you?

But, wait; there's more. I know for a fact that the number you now have displaying on your calculator is divisible by 11. Oh, yes it is. Try it: hit the "÷" button again, then key in "11" and you'll see a whole number show up after you hit the "=" sign. Pretty neat, yes?

And here's the grand finale. The number you now have on your calculator is divisible by 7. Do it: hit the "÷" button one last time, then key in "7" and hit the "=" button.

Lo and behold, not only was I right, but you should notice a much cool something at this point. What is it you see on your calculator's display?

A Problem with Gravity
Here's a question for the Isaac Newton fan club. Suppose you're on a sailing ship cruising at a decent speed over the waters. You have a 16-pound bowling ball in your hand, and you climb up the main mast so you can stand with your bowling ball in the crow's nest. Once you get up there, you face the stern (rear) of the boat, which you might recall is flying along at a nice clip, and you drop your bowling ball. Oops. That's going to hurt if it lands on somebody's head.

If someone is standing on the deck, would it be safer to be right below the mast you'd climbed, or would it be safer to stand near the rear of the boat? What's the reason for your advice?

A Weighty Matter
Horace is on a diet, but he's dying to have some delicious pie, so he makes a decision that tomorrow, all he'll eat in a 24-hour period is half-a-pound of pie, and all he'll drink is eight ounces of tasty soda pop. His reasoning is obvious: the total weight of what he consumes tomorrow will be exactly one pound—eight ounces of pie and eight ounces of drink—so the very most he could possibly gain as the result of his one-day excess is one pound. That's all: one lousy pound. And he gets to satisfy cravings that are driving him out of his mind.

Is Horace's logic correct? If not, why not?

And Speaking of Dietary Knowledge
Myles goes to the store, and as he's walking through the frozen foods aisle, he sees a really attractive dessert. He opens the cooler door and grabs the little nugget of joy. The box reads: "97% Fat Free!"

Myles buys the item. When he gets home, his housemates start a conga line because Myles has brought home dessert.

Eugene, the analytic fellow of the group, reads the labeling and says, "And look at this! It says '97% Fat Free!'" But then he pauses. "Hey, Myles, what does '97% Fat Free' really mean?"

Myles stops what he's doing and ponders. "I think that means it's, you know, like, 97 percent of the ingredients aren't fat."

Jake, the other roommate, pipes up, "Actually, I always thought that meant that 97 percent of the ingredients by weight weren't fat."

Eugene comes back, "What if it's volume they're talking about? I'm not even sure what that would mean, though."

Myles, being the practical sort, cuts in, "I think we should eat the dessert, and then we should decide what '97% Fat Free!' means."

Everyone agrees; so they eat the dessert, and it's fabulous. Unfortunately, they get so full that they all go into the living room and fall asleep half-way through their DVD of Dark City.

By the time they wake up, just like in the movie, they've been shifted to other bodies, and all their memories of their previous lives have been erased. Among other things, that means they never return to the question of just what, exactly, that dessert box meant by the label, "97% Fat Free!" that was so much a part of their conversation the night before.

It's up to you, then. What does a label mean by "97% Fat Free"? You'd be surprised at how many people don't know, and you'd be even more surprised by what the right answer implies about the "healthfulness" of the food in the box.

The Myth of the Monkeys and Their Typewriters
Ever hear the one about how, if you put a bunch of monkeys in a room with a bunch of typewriters and have them just randomly pound out letters, you'll eventually get something Shakespeare wrote, provided you give them long enough—as in, say, an infinite amount of time?

Well, it's not true: the monkeys will very likely never produce even a modest page of Shakespearean literature, much less an entire play or sonnet. Random events don't work that way.

Unfortunately, probability theory—which underpins statistics—is so woefully under-taught that all kinds of myths are running through high-stakes statistical analysis, these day; and all kinds of utterly useless results are being pumped out that swiftly come to be unassailable gospel. Moreover, many real results from probability theory are actually kind of counter-intuitive, as we'll see in the next three little snippets.

Phony Data
Doris is the manager of the accounting department of a medium-size firm. Her most reliable confidant in the company has told her that one of two managers, Phil or Ted, has been playing games with expenses in his department for years. The informant refuses, however, to say which one of the guys it is.

In front of Doris are the expense reports for the two departments. Having a really good eye for detail, she notices a difference between the numbers in the two reports. All of the expense amounts in both reports are in the range from $100 to $999, but that's where the similarity ends. The expense numbers in the report from Phil's department are pretty uniformly distributed: there doesn't seem to be any particular bias toward any starting number. A figure in the $800 range is about as likely to appear as a figure in the $200 range. The numbers appear to be quite randomly distributed.

However, the expense numbers from Ted's department—and Ted has quite a few more expenses listed, so the total expenses of the departments come out to be about the same—are heavily weighted toward numbers in the $100, $200, and $300 range. As a percentage of all the numbers, figures in the $700, $800, and $900 are much less common.

Doris does some checking, and she learns that the two departments are virtually identical in their operational details, so this isn't some cost-control thing where Ted is managing department expenses differently from Phil.

This makes Doris very suspicious. Ted's numbers are "skewed" to the low end, while Phil's numbers are "uniform" in their distribution, and there's no good reason why there should be such a difference. She calls Ted into her office, and she tells him point-blank that she thinks he phonied up his numbers to make his per-item expenses look better. Ted goes bananas and says, "Are you crazy? Everyone knows Phil's been phonying the numbers he sends to you for years!"

Ted seems so earnest in his righteous indignation that Doris tells him she'll have to think about it over night.

Based just on the numbers Doris was reviewing, was she justified in hauling Ted into her office, or should she have dragged Phil in there?

Remember: your decision upon whom to suspect is based only on the numbers in those reports.

Birth Dates in a Crowd
Suppose you're in a room with other people. How many people would have to be in that room for there to be a 50-50 chance that someone shared your birth date (month and day)?

A)   Fewer than 25 people.
B)   Between 26 and 50 people.
C)   Between 51 and 200 people.
D)   Between 201 and 365 people.
E)   More than 365 people.

A Coin-Flipping Problem
You have a quarter in your hand, and you know it's "fair" in the sense that the odds of flipping it and getting heads is the same as the odds of getting tails. The coin has been flipped ten times, and every time, it's come up heads. What should you call on the eleventh flip?

A)   Heads.
B)   Tails.
C)   It doesn't matter.

And Finally, a Finance Problem
Ten years ago, you took out a 30-year, fixed-rate mortgage loan for $200,000 that carries an annual interest rate of 6.9%. Your monthly payments (ignoring escrows, mortgage insurance, and other nonsense) are $1,317.20. After ten years, what percentage of the original principal (the $200,000) will you have paid off?

A)   Less than 15%.
B)   Between 15% and 33%.
C)   Exactly 33%.
D)   More than 33%.

In Conclusion
There you have it: a nice collection of little facts and problems to take your mind off the monotony of the holidays. Offer answers to the questions, your insights, and/or your thoughts in the comments. Correct answers will be provided in a follow-up post later this week.



The Dark Wraith hopes readers enjoy this kind of intellectual stimulation.