language pop quiz

I pulled the following sentence, which contains an error, from an article at The Atlantic titled "Is the Ivy League Fair to Asian Americans?" Here's the sentence:

Again, the implication here seems to be that while Asian-American applicants as a group excel at tests, an important factor in admissions, their talents, skills, and other interests tend to be significantly inferior to students of other races, and having them around isn't as enriching for other students.

The nature of the error is:

(A) poor tense control
(B) faulty/illogical comparison
(C) ambiguous pronoun reference
(D) dangling or misplaced modifier

From the same article, another sentence with an error:

As I see it, we know that even well-intentioned people regularly rationalize discriminatory behavior, that society as a whole is often horrified at its own bygone race-based policies, and that race is so fluid in our multi-ethnic society that no one can adequately conceive of all the ways it is changing; knowing these things, prudence dictates acceptance of the fact that humans aren't equipped to fairly take race into consideration. [italics in original]

The nature of the error is:

(A) poor tense control
(B) faulty/illogical comparison
(C) ambiguous pronoun reference
(D) dangling or misplaced modifier




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a faulty axiological argument for the existence of God

I was alerted, on my Twitter feed, to the existence of a five-minute Prager University video by Dr. Peter Kreeft (rhymes with "strafed"), professor of philosophy at Boston College, in which Dr. Kreeft attempts to prove the existence of God by arguing that good and evil enjoy objective existence. I will lay out Dr. Kreeft's argument, phase by phase, and then demonstrate why it resoundingly fails to prove God's existence.

1. The Argument

Dr. Kreeft's argument has two principal phases:

a. Establish that all non-objective (i.e., atheistic/naturalistic) explanations for the existence of morality are unsatisfactory.

b. Conclude from the failure of all naturalistic explanations that morality has an objective basis, which must be supernatural, i.e., God.

Establishing (a) is challenge enough, but much more depends on whether Dr. Kreeft can succeed at establishing (b) satisfactorily. In the video, Dr. Kreeft breaks (a) down into five parts. This five-part argument, a systematic rejection of several naturalistic explanations for the existence of morality, begins this way:

I'm going to argue for the existence of God from the premise that moral good and evil really exist. They are not simply a matter of personal taste-- not merely substitutes for I like and I don't like.

We can therefore call this an axiological argument for the existence of God. The term axiology refers to the study of value, i.e., ethics, morals, the Good, etc. Note, too, that Dr. Kreeft is aiming to establish that good and evil are objective realities, i.e., they reside in the world, independent of any particular person's perspective.

Dr. Kreeft continues:

Before I begin, let's get one misunderstanding out of the way. My argument does not mean that atheists can't be moral. Of course: atheists can behave morally, just as theists can behave immorally.

This is an important concession, but I'm not sure how relevant it is, given what Dr. Kreeft argues later: at the end of his spiel, Dr. Kreeft seems to imply that an atheist who believes morals to have an objective basis is actually a closet theist. This comes perilously close to the claim that there are no atheists, a claim that drives most atheists crazy. (It's a bit like defining religion so inclusively that even atheists turn out to be religious. I've been guilty of making that move myself.)

Here is the transcript (all typos are my responsibility) of the rest of Dr. Kreeft's axiological argument for God's existence:

Let's start, then, with a question about good and evil. Where do good and evil come from? Atheists typically propose a few possibilities. Among these are

-evolution
-reason
-conscience
-human nature, and
-utilitarianism.

I will show you that none of these can be the ultimate source of morality.

Why not from evolution? Because any supposed morality that is evolving can change. If it can change for the good or the bad, there must be a standard above these changes to judge them as good or bad. For most of human history, more powerful societies enslaved weaker societies, and prospered. That's just the way it was, and no one questioned it. Now, we condemn slavery. But, based on a merely evolutionary model—that is, an ever-changing view of morality—who is to say that it won't be acceptable again one day? Slavery was once accepted, but it was not therefore acceptable: if you can't make that distinction between accepted and acceptable, you can't criticize slavery. And if you can make that distinction, you are admitting to objective morality.

What about reasoning? While reasoning is a powerful tool to help us discover and understand morality, it cannot be the source of morality. For example, criminals use reasoning to plan a murder, without their reason telling them that murder is wrong. And was it reasoning, or something higher than reasoning, that led those Gentiles who risked their lives to save Jews during the Holocaust? The answer is obvious: it was something higher than reasoning, because risking one's life to save a stranger was a very unreasonable thing to do.

Nor can conscience alone be the source of morality. Every person has his own conscience, and some people apparently have none. Heinrich Himmler, chief of the brutal Nazi SS, successfully appealed to his henchmen's consciences to help them do the "right" thing in murdering and torturing millions of Jews and others. How can you say your conscience is right and Himmler's is wrong, if conscience alone is the source of morality? The answer is: you can't.

Some people say human nature is the ultimate source of morality. But human nature can lead us to do all sorts of reprehensible things. In fact, human nature is the reason we need morality. Our human nature leads some of us to do real evil, and leads all of us to be selfish, unkind, petty, and egocentric. I doubt you would want to live in a world where human nature was given free rein.

Utilitarianism is the claim that what is morally right is determined by whatever creates the greatest happiness for the greatest number. But, to return to our slavery example, if 90% of the people would get great benefit from enslaving the other 10%, would that make slavery right? According to utilitarianism, it would!

We've seen where morality can't come from. Now, let's see where it does come from.

What are moral laws? Unlike the laws of physics or the laws of mathematics, which tell us what is, the laws of morality tell us what ought to be. But like physical laws, they direct and order something, and that something is right human behavior. But since morality doesn't exist physically—there are no moral or immoral atoms or cells or genes—its cause has to be something that exists apart from the physical world. That thing must therefore be above nature, or supernatural. The very existence of morality proves the existence of something beyond nature and beyond man. Just as a design suggests a designer, moral commands suggest a moral commander. Moral laws must come from a moral lawgiver. Well, that sounds pretty much like what we know as God.

So the consequence of this argument is that, whenever you appeal to morality, you are appealing to God, whether you know it or not. You're talking about something religious, even if you think you're an atheist.

I'm Peter Kreeft, professor of philosophy at Boston College, for Prager University.

2. My Critique

My first reaction to this video was that an axiological argument for the existence of God has to be one of the more bizarre attempts at proving God's existence that I've seen. St. Anselm's ontological proof for the existence of God, while flawed, strikes me as more rigorously logical than Dr. Kreeft's strange undertaking. St. Thomas Aquinas's cosmological proofs—the so-called Five Ways—also strike me as more tightly reasoned than this morality-centered approach, although they, too, are flawed.

My objections to Dr. Kreeft's arguments can be summed up thus:

1. In attempting to refute a mere subset of the total number of naturalistic arguments for the existence/ultimate source of good and evil, Dr. Kreeft has failed to address all the possible arguments and thus cannot proceed directly to the supernatural.

2. Many, if not most, of Dr. Kreeft's objections merely reject possibilities because they are distasteful, not for any rigidly logical reason. These are aesthetic objections, not logical objections.

3. Even if we consider Dr. Kreeft successful in having refuted all the naturalistic arguments for the existence/ultimate source of morality, Dr. Kreeft has failed to demonstrate that a theistic source for morality is the only remaining option. Buddhism builds its system of morality not upon theism, but upon the basic empirical fact of dukkha (suffering, unsatisfactoriness) and the relational, processual, intercausal nature of reality. No god is needed in this moral framework.

Science has also been exploring the question of morality. You might want to take a look at Robert Wright's talk with Dr. Steven Pinker over at Meaningoflife.tv (see here). Fast-forward to about minute 34, then listen as Pinker and Wright talk about the notion of objective "moral laws" (i.e., moral realism, the idea that moral laws have objective existence), which enjoy an almost Platonic status, toward which evolving organisms are converging over time—laws that govern, say, cooperative survival strategies, tendencies toward reciprocal behavior, various pancultural forms of the Golden Rule, etc. Nowhere in that discussion is God explicitly invoked.

4. At several points in his argument, Dr. Kreeft assumes what he wishes to prove. A good example of that fallacious move occurs here, early in his argument:

For most of human history, more powerful societies enslaved weaker societies, and prospered. That's just the way it was, and no one questioned it. Now, we condemn slavery. But, based on a merely evolutionary model—that is, an ever-changing view of morality—who is to say that it won't be acceptable again one day? Slavery was once accepted, but it was not therefore acceptable: if you can't make that distinction between accepted and acceptable, you can't criticize slavery. And if you can make that distinction, you are admitting to objective morality.

The notion that "slavery was once accepted, but it was not therefore acceptable" is the crucial phrase here: Dr. Kreeft is merely asserting, not arguing. He offers no support, that I can see, for his contention that slavery wasn't acceptable back in the old days: obviously it was acceptable, or it would never have been practiced! To say that slavery was never acceptable is to say it was never acceptable from a God's-eye point of view—and that's precisely where Dr. Kreeft is assuming what he wishes to prove.

5. Dr. Kreeft's argument suffers from the same problem that plagues most arguments for an objective morality: whose morality, from which culture, is the morality? There are so many moralities out there, and not all of them share certain basic tenets like "killing/murder is bad." This is Cultural Anthropology 101, folks: moralities may overlap, but as with Wittgenstein's notion of family resemblances, distant-cousin moral systems may have little to nothing in common.

6. If we assume that Dr. Kreeft has successfully made the case for theism, Dr. Kreeft still faces all the logical and moral objections to theism itself. To wit: how moral is a jealous and vindictive God? Is the petty, bloodthirsty God of the Old Testament (a God who, in Christian reckoning, sacrifices his son in the New Testament) truly worthy of worship? What about the logical problems that burden most traditional concepts of God? Divine foreknowledge is incompatible with human freedom, for example, and we associate freedom with responsible, moral action. Etc., etc.

I think that about covers my objections to Dr. Kreeft's argument. Basically, I feel that the professor has failed to make the move from "No naturalistic explanation for morality is satisfactory" to "Only theism can explain the existence of morality." His objections to naturalistic explanations are more aesthetic than logical; he fails to answer all the naturalistic arguments for the existence of morality; he fails to provide a compelling case that theism is the only inevitable alternative in the face of naturalism's failures (cf. Buddhism and science on morality); he assumes what he wishes to prove; he fails to deal adequately with the diversity of moral systems; and finally, even if he has succeeded in making the case for God, he faces a mountain of logical and moral objections to theism itself.

That any argument for the existence of God can hold water is doubtful at best. Over the course of human history, no argument has yet proven universally acceptable, and this axiological approach strikes me as one of the stranger—not to mention weaker—attempts at supporting theism.

My thanks to my brother Sean for nudging me to write this post.


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etiological fairy tale: Why the Monkey's Rear End is Red

An adult Korean ESL student of mine wrote a cute etiological fairy tale as a writing exercise. The term "etiological" means "having to do with causes." An example of such a fairy tale might be something like Rudyard Kipling's "How the Rhinoceros Got His Skin" from his Just So Stories. Kipling's tale provides a humorous explanation for the rhino's thick, wrinkled skin. The story answers the cause-related question, "How did the rhino's skin get that way?"

Without further ado, then:

WHY THE MONKEY'S REAR END IS RED
Once upon a time, there was a monkey. When the monkey was eating a banana on a tree, a crab came to the monkey. The crab said, "Monkey, can you give a banana to me?" The monkey ignored him and continued eating his banana.

The crab really wanted that banana, so he said, "If you rest the banana on the tree and eat it, the taste would be much better." So the monkey put the banana on the tree. Suddenly, wind blew and the banana fell to the ground. The crab picked it up quickly, and he went into his small house, laughing all the way.

The monkey, furious, came down and said, "If you don't give me my banana, I will poop on your house!" Then the crab pinched the monkey's butt with his powerful claws. The monkey's rear end turned red as the crab pulled out all of his butt hair.

And that is why a monkey's rear end is red and hairless, and a crab's claws are hairy.

I will never look at crabs and monkeys the same way again.


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this week's MGRE Math Beast Challenge

From here:

Quantity A
The number of different ways all 9 letters in the word “TENNESSEE” can be arranged.

Quantity B
The number of different ways all 7 letters in the word “WYOMING” can be arranged.

(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.

Go to it! My answer will eventually appear in the comments. This appears to be a permutations and combinations problem. Ugh. The obvious answer would seem to be (A), but that very obviousness is what makes me cautious.


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answer to last week's MGRE Math Beast Challenge

MGRE has this to say about last week's Math Beast Challenge:

We are told that the 11th grade girls at Stumpville High School have an average GPA of 3.1, and the overall 11th grade average GPA is 3.05. Fortunately, the 11th grade has the same number of boys and girls, so rather than using the weighted average formula, we can simply conclude that the boys’ average GPA must be 3.0. Write on your paper something like:

11th grade boys average GPA = 3.0

(If you’re not sure about our quick inference, try this example: If a dozen people in a room each have an average of $10 and another dozen people each have an average of $20, then the average amount of money each person has is exactly $15, since the $10 group and the $20 group are the same size. Similarly, if this example had told you that a dozen people have an average of $10, another dozen people have x dollars, and the overall average is $15, then – since 15 is exactly halfway between 10 and 20 – you could confidently conclude that the other dozen people have an average of $20.)

We are told that all of the boys enrolled in Honors Chemistry are in 11th grade. From the first chart, add up the total number of boys: 46 + 52 + 52 + 50 = 200. From the bottom chart, we can see that 6% of boys take Honors Chemistry. 6% of 200 is 12, so write on your paper something like:

11th grade boys in Honors Chem = 12

We are told that these 12 boys have an average GPA of 3.8. And yet the average GPA for boys in 11th grade is only 3.0 – thus, we are expecting the rest of the boys’ GPAs to be much lower than the Honors Chemistry boys’ GPAs.

However, we CANNOT do the kind of “quick logic” we did above and assume that, since the Honors Chem 11th grade boys have an average GPA of 3.8 and the 11th grade boys in general have an average GPA of 3.0, therefore the rest of the boys have an average GPA of 2.2 (since 3.0 is exactly in the middle of 2.2 and 3.8). THIS IS A TRAP! We cannot conclude that the answer is 2.2, because the number of Honors Chem 11th grade boys and the number of other 11th grade boys are NOT THE SAME.

We must calculate a weighted average (to review Weighted Averages, see Manhattan Prep’s GRE Word Problems Strategy Guide). Remember that there are 12 boys in the 11th grade who are in Honors Chem and 40 who are not in Honors Chem:

[12(3.8) + 40(x)]/52 = 3.0

12(3.8) + 40x = 156

45.6 + 40x = 156

40x = 110.4

x = 2.76

The correct answer is B.

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house-sitting this week

I'm house-sitting for a friend this week, so blogging is going to be spotty at best. My apologies in advance.


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la sexualité: un sondage

Voici un petit sondage fait dans onze pays (y compris la France) intérrogeant les citoyens sur leur sexualité (positions sexuelles préférées, etc.). Quelques-uns des résultats vous seront un peu surprenants, j'imagine...


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hierophany

In a term coined by Romanian historian of religions Mircea Eliade (meer-CHAY-uh ell-YAH-dih), a hierophany is an eruption of the sacred into the realm of the profane (i.e., the ordinary, not the vulgar/obscene). One suddenly finds oneself standing in the presence of the holy, a fact that disrupts the normal, mundane continuity of human existence.

I'm fascinated by theistic fiction, i.e., fiction about the presence of the holy in our midst. A great example of this sort of fiction is the short story titled "The Visitation," by sci-fi author Greg Bear. You can read a truncated version of the story for yourself here.


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this week's MGRE Math Beast Challenge

From here:

The 11th-grade girls at Stumpville High School have an average GPA of 3.1, and the overall 11th-grade average GPA is 3.05. If all of the boys enrolled in Honors Chemistry are in the 11th grade and those boys have an average GPA of 3.8, what is the average GPA of all the 11th-grade boys who are not enrolled in Honors Chemistry?

(A) 2.2
(B) 2.76
(C) 2.96
(D) 3.05
(E) 3.16

Go to it! My own attempted solution will appear in the comments.


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answer to last week's MGRE Math Beast Challenge

Yes! I got the answer to last week's MGRE Math Beast Challenge problem correct! The answer was indeed 3, 5, 41, and 43. Here's MGRE's explanation:

The prime numbers less than 12 are 2, 3, 5, 7, and 11. There are five possible ages for the two children, or 5!/2!3! = (5)(4)/(2)(1) = 10 possible combinations for the children’s ages.

The prime numbers between 40 and 52 are 41, 43, and 47. There are three possible ages for the two adults, or 3!/2!1! = 3 possible combinations for the adults’ ages.

In total, there are (10)(3) = 30 possible age combinations—far too many to test each scenario looking for a prime number average age.

An alternative is to start from the resulting average age. The minimum ages of the family members are 2, 3, 41, and 43, which average to 22.25. The maximum ages of the family members are 7, 11, 43, and 47, which average to 27. The only prime number in this range is 23, which implies an age sum of (4)(23) = 92.

If the adults are 43 and 47, the sum of their ages is 90. The sum of the children’s ages would need to be 92 – 90 = 2. The minimum sum of the children’s ages is 2 + 3 = 5, so no need to continue checking these possibilities.

If the adults are 41 and 47, the sum of their ages is 88. The sum of the children’s ages would need to be 92 – 88 = 4. The minimum sum of the children’s ages is 2 + 3 = 5, so again, no need to continue checking these possibilities.

If the adults are 41 and 43, the sum of their ages is 84. The sum of the children’s ages would need to be 92 – 84 = 8. This is only possible if the children are 3 and 5.

Check: (3 + 5 + 41 + 43) = 92, so the average is 92/4 = 23, which is prime.

The correct answers are 3, 5, 41, and 43.

Interesting deductive process! I think I arrived at my answer more intuitively.


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so you think you speak Amurrican

A quick test for people who think they know American English! Select the answer that is most American and/or most grammatically correct.

1. Which is correct?

a. Thanks Fred.
b. Thanks, Fred.


2. Let's just leave this _____ .

a. between you and I
b. between you and me


3. She's a real _____ .

a. trouper
b. trooper


4. If I _____ I wouldn't have farted in the tub.

a. could have known about her phobia,
b. had known about her phobia,


5. Give this prize to _____ ate the most hot dogs.

a. whoever
b. whomever


6. Which is correct?

a. She said, "Sit down."
b. She said, "Sit down".


7. If you want to succeed in this company, _____ and don't make waves.

a. tow the line
b. toe the line


8. That was a strange proposition to Fred and _____ .

a. I
b. me


9. I try to brush my teeth _____ .

a. everyday
b. every day


10. This restaurant has a great _____ .

a. ambience
b. ambiance
c. either A or B
d. neither


11. I saw her in the woods-- _____ .

a. butt naked
b. buck naked


12. When I finally found her ring and ran up, gasping, to give it to her, she sighed and said, "_____ ."

a. Never mind
b. Nevermind


13. I'll _____ be there.

a. definately
b. definitely


14. The sky boomed with thunder and sizzled with _____ .

a. lightning
b. lightening


15. Visiting the White House is quite a _____ !

a. priviledge
b. privilege


16. I'm not _____ to being set up on a blind date.

a. adverse
b. averse


17. _____ elementary, Watson.

a. It's
b. Its


18. I felt so _____ about how disastrous her birthday party was.

a. bad
b. badly


19. Despite the chaos around him, Phineas was _____ .

a. unfazed
b. unphased


20. Which is correct?

a. I wonder where my car went.
b. I wonder where my car went?


21. She stared in frank amazement at his _____ salmon.

a. enormous, twenty inch
b. enormous twenty-inch


22. As the Titanic tilted crazily, she held _____ the railing for dear life.

a. onto
b. on to


23. Watch out for the thundering _____ !

a. hoard
b. horde


24. All that has happened has been in accordance with the _____ .

a. prophesy
b. prophecy


25. Einstein, not merely a genius, was a kind _____ he once rescued a treed cat.

a. soul;
b. soul,



How'd you do?

Answers follow; highlight the space between the brackets to see them.

[1. B; 2. B; 3. A; 4. B; 5. A; 6. A; 7. B; 8. B; 9. B; 10. C; 11. B; 12. A; 13. B; 14. A; 15. B; 16. B; 17. A; 18. A; 19. A; 20. A; 21. B; 22. B; 23. B; 24. B; 25. A]

Scale of Achievement:

25: "I am a Jedi, like my father before me."
24: "Impressive. Most impressive."
20-23: "You are not a Jedi yet."
15-19: "You will pay the price for your lack of vision."
10-14: "Scruffy-looking nerfherder!"
5-9: "Your feeble skills are no match for the power of the dark side!"
1-4: "I have a bad feeling about this."
0: "Noooooooooooo!"

What language rant topics do the above questions cover? Highlight the [bracketed area below] to see.

[1. vocative comma: always use when addressing someone!
2. pronoun case: object of preposition
3. diction (trouper = member of troupe = stalwart team player, not a soldier)
4. verb tense in conditional sentences: if (pluperfect) ➞ main (conditional past)
5. pronoun case: "whoever" is correct as subject of clause
6. US vs. UK punctuation (too many Americans forget what country they live in)
7. idioms: people put their toes up against the painted line
8. pronoun case: don't be an idiot and use a subject pronoun when an object pronoun is called for
9. adverb of frequency = every day; "everyday" = adjective meaning "ordinary"
10. spelling trivia: some words have more than one acceptable spelling
11. idioms: village idiots mishear this as "butt nekkid"
12. compounds: or, more precisely, when not to use compounds
13. spelling: there is no "a" in "definitely"!!!!!
14. spelling/diction: "lightening" comes from the verb "to lighten (a load, the sky, etc.)"
15. spelling: no "d" in "privilege"
16. diction: adverse [conditions], averse [attitude]
17. spelling/diction: it's = it is; its = possessive adjective
18. diction: with a linking verb like "feel," you need a predicate adjective, not an adverb
19. spelling/diction: only someone who had never actually read the word "to faze" would get this wrong
20. mood: "I wonder" is always declarative-- NEVER interrogative!
21. punctuation: hyphenate phrasal adjectives before a noun; no comma for non-coordinate adjectives
22. diction: the phrasal verb's infinitive form is "to hold on" not "to hold onto," which makes the "to" separate
23. spelling/diction: you'd have to be a moron not to get this one
24. spelling/diction: as above. "Prophesy" (-"sigh") is a verb; prophecy (-"see") is a noun
25. punctuation: a semicolon separates two related or contrastive clauses]

this week's MGRE Math Beast Challenge

From here:

"The Prime of Life"

In a family of four people, none of the people [has] the same age, but all are a prime number of years old. Two of the people are less than 12 years old, and the other two people are between 40 and 52 years old. If the average of their four ages is also a prime number, what are the ages of the family members?

Indicate four such ages (check 4 slots).

( ) 2
( ) 3
( ) 5
( ) 7
( ) 11
( ) 41
( ) 43
( ) 47

Go to it! My own answer will appear in the comments section.


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answer to last week's MGRE Math Beast Challenge

Well, nuts. The answer to last week's MGRE Math Beast Challenge isn't (D); it's (A). [Never pick (D), Kevin!] Here's MGRE's multi-pronged explanation:

This problem could be solved through logic, algebraically, or by plugging in numbers. For all three solutions, our first task is to simplify y – x > x – y. Notice that it has like terms that can be combined – it would be very bad to neglect to simplify this before plowing ahead with the problem!

y – x > x – y
y > 2x – y
2y > 2x
y > x

So, y is greater than x.

The logic solution is certainly the fastest. Since all of the percent changes in Quantity A and Quantity B are changes through multiplication, order doesn’t matter. Thus, the 35% increase on both sides can be ignored – it is the same on both sides, and the order in which this occurs doesn’t matter.

Additionally, the order in which the other changes occur doesn’t matter. Also, the price p is a positive number that is the same on both sides, so it can be ignored as well.

All that’s left is: Quantity A decreases a smaller percent and increases a larger percent. Quantity B increases a smaller percent and decreases a larger percent. Quantity A is definitely greater.

Or, algebraically:

Since y is greater than x, Quantity A is positive and Quantity B is negative.

Finally, plugging in numbers would also work. To make things easy, make p = 100, and make x and y easy percents, like 10 and 50, making sure y is greater than x.

Of course, we still had to simplify y – x > x – y in order to pick valid numbers, and this method is even faster if we realize we can ignore the 35% change on both sides.

See sample solution with p = 100, x = 10, and y = 50 below. (To decrease by 10%, multiply by 0.9. To increase by 50%, multiply by 1.5. To decrease by 50%, multiply by 0.5. To increase by 10%, multiply by 1.1).

Quantity A
100(0.9)(1.5) = 135

Quantity B
100(0.5)(1.1) = 55

Quantity A will be greater no matter what numbers you choose, provided that you make y > x.

The correct answer is A.

I was so close to the above conclusion, dammit. I had successfully deduced that Quantity B was the negative of Quantity A, but not that A was always positive and B was always negative. In my own explanation, I had even mentioned that it would be tempting to pick (A). I should have followed my instincts, I guess. But where did I go wrong in my math, such that (A) produced a negative result in my own calculations?


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oxen and kitties

In Zen, there's a famous series of pictures known as The Ten Ox-herding Pictures by Kakuan, a Chinese Ch'an (Zen) master. Here's a good, brief article on what the pictures mean.

And here, Dear Reader, is a paradoxically reverent parody: The Ten Cat-herding Pictures.


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le subjonctif

A good page on the French subjunctive mood is here.


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the ghost and the tinker: a study in contrasts

It's difficult to imagine a more disparate pair of movies than "Mission: Impossible-- Ghost Protocol" (MIGP) and "Tinker, Tailor, Soldier, Spy" (TTSS). While both films are members of the spy genre, their approaches to that genre differ in almost every respect. And yet, despite their diametrically opposed sensibilities, they're both thoroughly entertaining. Holding them up together for comparison will give us a chance to explore the depth of their differences, and also to ponder what it means to be entertained by a film.

Two quick, one-paragraph summaries, then, to orient the newbie.

MIGP (2011) is the fourth of the Mission: Impossible films. Tom Cruise and Simon Pegg are back as super-agent Ethan Hunt and super-techie Benji Dunn, respectively. Benji's passed his field exam, so he can now run around with Ethan while wearing disguises, speaking Russian, and even holding a gun. Benji's paired up with Jane Carter (Paula Patton), and the team soon acquires a fourth: William Brandt (Jeremy Renner, who's on a cinematic roll). The movie begins with several converging plot lines: (1) Ethan's in a Russian prison, gathering intelligence; (2) Benji and Carter, unaware of Ethan's real mission, have come to Russia to break him out; (3) Carter is fresh off a failed mission in Budapest, in which Russian nuclear launch codes have been stolen by a French assassin (who also killed Carter's boyfriend); (4) a terrorist rogue codenamed Cobalt (Kurt Hendricks, played by Michael Nyqvist) is planning to use those codes to provoke a nuclear war between Russia and the US as part of his belief that humanity is strengthened by the occasional apocalypse. That's the basic setup. What follows is essentially a chase movie: Hendricks blows up part of the Kremlin, pinning the blame on Ethan's team ("Ghost Protocol" refers to the US president's disavowal of Ethan et al.); in Dubai, Hendricks also gets the launch codes from Sabine Moreau (a disconcertingly baggy-eyed Léa Seydoux), the French assassin. The chase leads to India, where Hendricks and his sidekick Wistrom (Samuli Edelmann) break into an Indian TV station and manage to relay a missile launch command to a Russian submarine. The action-packed remainder of the film is all about stopping the missile. Does the team succeed? Well, what do you think?

TTSS (2011) is based on the John Le Carré novel of the same name (Le Carré, pulling a Stan Lee, appears at least twice in quick cameos). The story, which takes place in the 1970s, begins with a passing of the torch at the highest levels of British intelligence-- MI6, nicknamed the Circus: hoary old Control (John Hurt, looking miserable as usual) is stepping down along with his trusted lieutenant, George Smiley (Gary Oldman). Taking Control's place is puny, pugnacious Scotsman Percy Alleline (Toby Jones), whose brainchild is a network codenamed Witchcraft. Control's departure comes on the heels of a failed mission in Budapest (cf. MIGP, above), in which Control's man Jim Prideaux (Mark Strong) is shot and apparently killed. Soviet agents spread the word that Prideaux had attempted the kidnapping of a Hungarian general, but in reality the mission was predicated on Control's suspicion that the Russians have had a mole inside the Circus for years. Control dies soon after his "retirement," and government intelligence liaison Oliver Lacon (the always-smarmy Simon McBurney) asks Smiley to pursue Control's theory. Smiley enlists the aid of young Peter Guillam (Benedict Cumberbatch-- he of the infamous cheekbones) to suss out the Circus members, all of whom had been given codenames by Control: Percy Alleline (Tinker), Bill Haydon (Colin Firth as Tailor), Roy Bland (Ciáran Hinds as Soldier), and Toby Esterhase (David Dencik as Poorman). One of these men is the mole, and each one could fit the profile. The movie proceeds as a sort of whodunit, with the selling of British secrets replacing the traditional murder victim. As Smiley meticulously and inexorably deduces his way to the truth, he comes to realize that, in the larger scheme of the Cold War, the Brits are the dupes: Russia's intention, all along, has been to use Alleline's Witchcraft network to spy on the Americans, with whom Alleline has been keen to make friends. Behind these machinations is the specter of Karla, a Russian operative who has risen in the Soviet ranks and is now the puppet master making the Brits dance. Does Smiley figure out who the mole is? You get only one guess.

On almost every level, with almost every aspect of filmmaking you can think of, these two movies are diametrically opposed. It matters little which aspect I begin with, so I'll plunge in with a discussion of how each film handles its main villain, then go from there.

Visibility of the main villain. The big bad guy in MIGP is Kurt Hendricks, an insane genius who, thanks to his espionage training and his time with the Swedish Special Forces, does his own infiltration work despite his advanced age.* The movie is at pains to build Hendricks up as physically imposing, and he enjoys quite a bit of screen time. Hendricks is, you might say, a very hands-on baddie, as physical as he is intellectual, always one step ahead of the IMF** team. By contrast, in TTSS, the main villain-- Karla-- is never seen directly: we receive only glimpses. His presence is nonetheless felt thanks to a marvelous script that makes him into a pervasive, Sauron-like phantom. When the normally taciturn Smiley has a drink and opens up to young Peter Guillam about his long-ago encounter with Karla, we learn that Karla never said a word during the encounter. The irony, here, is that this is normally Smiley's tactic: our protagonist quietly gathers data and makes his careful deductions before acting. I was very impressed with how the script made Karla a real presence, a real threat, throughout the movie. The search for the mole inside the Circus, which occupies most of the film, is actually a sideshow: the main event is the battle of wills and wits between Smiley and Karla.

Pace and visuals. Here as well, MIGP and TTSS stand in contrast with each other. MIGP is a young person's movie: its scenes are, for the most part, brightly and unsubtly lit, and the script propels us forward at breakneck speed. The Russian prison has its stark fluorescent lights; the dramatic Kremlin explosion (I wonder what Russian audiences thought of that) occurs in the daytime; the Burj Khalifa scenes-- even the sandstorm!-- were all the opposite of murky; even the interior and exterior scenes in India made use of strong color contrasts. TTSS, meanwhile, is drab and subdued: London is stereotypically gray (so gray that it was grey); the scenes in Budapest are either interior shots or cloudy exteriors; most of the London interiors are wan and shadowy. TTSS's pace is different, too; this isn't an action movie so much as a thinker's movie, and there were moments when I felt that the film had been directed by Clint Eastwood. The camera work is stately and unpretentious; there are no violent smash cuts to get our blood pumping, no complicated chase scenes. TTSS's antiquated costume design does a marvelous job of evoking the Cold War era,*** and most of its intrigue comes from ambient hints, subtle facial expressions, and layered dialogue. Which leads me to...

Expository dialogue. MIGP's script is written like a condensed version of the TV series "24." Most of its dialogue is expository-- not so much about revealing character as about keeping the viewer abreast of the rapidly changing circumstances. TTSS, on the other hand, uses dialogue both to develop character and to provide the watchful viewer with hints as to what is to come. While some of TTSS's dialogue is occasionally expository, the story requires the viewer to do his own thinking. When a character like field agent Ricky Tarr (Tom Hardy) delivers a long spiel, it's not so much the content of Tarr's discourse that matters as what Smiley makes of it.

Music. MIGP's soundtrack comes courtesy of the talented Michael Giacchino (juh-KEE-noh), who also scored "The Incredibles" and 2009's "Star Trek." I thoroughly enjoyed Giacchino's versatility in his scoring of "The Incredibles," a movie that mixed the superhero and spy genres. The music for that film had a cool retro feel at times, but easily transitioned into the more grandiose strains that we expect when titans are dueling-- all without losing that lighthearted tone that is a Giacchino trademark. I think this style worked less well with "Star Trek": I have a hard time forgiving Giacchino for creating such a catchy theme, then beating that theme to death in almost every scene. And the lightheartedness that worked so well for a Pixar animation didn't work nearly as well for a science-fiction blockbuster. I didn't hate the soundtrack for "Star Trek," but I did feel that it revealed Giacchino's limits. His score for MIGP confirmed those limits: I've begun to realize that Giacchino is a director's go-to guy if the movie in question isn't particularly deep. That said, the best musical moment, for me, was the soundtrack's soaring tribute to the majestic Burj Khalifa. The worst moments were the intros to Russia and India: both were painfully stereotypical. I cringed.

By contrast, the soundtrack for TTSS was marked by its thoughtful, slow-jazz leitmotifs. Original music was provided by Alberto Iglesias, who is not, as far as I can tell, related to singer Julio Iglesias, whose version of Charles Trenet's "La Mer" is what we hear during the movie's conclusion. TTSS's music is subtle, the opposite of bombast; it never dominates a scene. I don't know much about Iglesias's career in the movie business, but I can see him being in demand among noir directors.

Our protagonists, and how the good guys win. Because MIGP and TTSS occupy such different cinematic universes, it's nearly impossible to imagine a crossover film in which the respective protagonists have a chance to match wits. Ethan Hunt's kinetic modus operandi involves a lot of running, jumping, climbing, and hand-to-hand combat (I'm trying to remember whether he fired a single shot in the entire film); George Smiley, meanwhile, is like the spider that sits at the center of its web, immovable, patiently testing the vibrations and allowing all enemies and information to flow toward him. Smiley's style isn't merely the result of his age; it's a function of his personality. While Hunt and Smiley both recognize the need for teamwork, their management styles differ. Hunt's unspoken motto seems to be the Marines' "Improvise, adapt, overcome"; his team spends much of its time coping with faulty technology, and with an enemy who seems able to anticipate their every move. At the end of the film, Hunt even notes that "the only thing that functioned properly on that mission was this team." The socially awkward Smiley, meanwhile, isn't nearly so chummy with his underlings. At one point he tells his right-hand man, Peter Guillam, that he's sending the younger man "into the lion's den" and that Guillam will, if caught, have to disavow any knowledge of Smiley's activities, just as Smiley will do of Guillam's. It's a far cry from the ethic of "no man left behind," but Smiley's stance makes sense given the circumstances.

The IMF team's struggles involve playing catch-up against a clever enemy; Smiley and his men, meanwhile, gather their data and pounce only when they're absolutely sure. The closest Smiley gets to seeing any real action is when he removes his shoes while in the London-based Russian safe house and pads softly across the floor, gun in hand. In the end, when Smiley gets his man, there's no need even to fire it.

As I mentioned at the beginning, I found both of these movies, MIGP and TTSS, quite entertaining. MIGP isn't particularly cerebral; it's all about the chase-- the action, the adrenaline, the humor, the suspense. Kurt Hendricks, MIGP's villain, has a simple agenda: he wants to instigate a nuclear war as a way to pare down and purify the human race, for strength is born through struggle. TTSS, though, is all about the cerebral. It's not obvious what Karla wants, and Karla's presence is more inferred than revealed. Far from leading us viewers by the nose and keeping us abreast of the plot twists through clear-cut camera work and detailed expository dialogue, TTSS obliges us to deduce, interpret, surmise, and conjecture-- right along with the characters themselves. We're given hints, phrases, and shadowy implications. Much of the important information is non-verbal. The movie doesn't treat the audience as stupid, and it definitely rewards multiple viewings. The plot of TTSS is beautifully put together, and it's certainly the more profound of the two movies.

But both films are ably directed (Brad Bird for MIGP; Tomas Alfredson for TTSS), and both understand economy of expression: not a single moment is wasted in either film. How is it possible to be almost equally entertained by two such different stories? I imagine it's because the eyes and the brain need different sorts of food. Sometimes the eyes-- and the adrenal glands-- demand good, heart-pounding action; sometimes the brain harrumphs and demands a good puzzle. Entertainment comes in all shapes and sizes; surely there's room in this world for two very different approaches to the spy genre!

---

*This required a rather significant suspension of disbelief, but the story asks us to take on faith that the old, plump Hendricks is the physical match of Tom Cruise.

**IMF stands for Impossible Mission Force. Hard to say with a straight face.

***One of TTSS's main costume designers, Jacqueline Durran, said that she had deliberately chosen 1960s-style clothing as an exaggerated way to evoke the 1970s. I'd say her trick worked.


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this week's MGRE Math Beast Challenge

From here:

An item originally cost p dollars, where p > 0.

y – x > x – y

Quantity A
The price of the item if the original price were decreased by x%, increased by 35%, and then increased by y%

Quantity B
The price of the item if the original price were increased by x%, decreased by y%, and then increased by 35%


(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.

Go to it! My own answer will eventually appear in the comments.


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answer to last week's MGRE Math Beast Challenge

The answer to last week's MGRE Math Beast Challenge was indeed (B). But MGRE shows how to arrive at this answer without using Heron's Formula. To wit:

To find the area of triangle PQR we need a base and a height. If we consider side PR the base of the triangle, then QS, which is at a right angle to PR, is the height. We know that both triangle PQS and triangle SQR are right triangles, but to find the height QS we’ll need to know the length of either PS or SR. Unfortunately we don’t know either one, so we’ll have to name variables for several legs of the triangle.

In this case we’ll let x represent the length of PS and y represent the height QS. Note that if PS has length x, then SR has length 21 – x, so we do not need to (and should not) name a variable for length SR. Updating our diagram yields the following:



Apply the Pythagorean Theorem to each right triangle.
From triangle PQS: x² + y² = 100
From triangle SQR: (21 – x)² + y² = 289

We solve each in terms of y²:
y² = 100 – x²
y² = 289 – (21 – x)²

Then set the two expressions equal to y² equal to each other:
289 – (21 – x)² = 100 – x²

Now simplify:
189 – (21 – x)² = -x²
189 = (21 – x)² – x²
189 = (441 – 42x + x²) – x²
189 = 441 – 42x
42x = 252
x = 6

Now that the value of x is known, solve for y:
100 = 36 + y²
64 = y²
8 = y
(Or just recognize that PQS is a 6 – 8 – 10 right triangle.)

Finally, we can solve for the area of triangle PQR:

(1/2)bh = (1/2)(21)(8) = 84

The correct answer is B.

{Incidentally, this problem is named for Heron’s Formula, which is an alternative to the (1/2)bh triangle area formula. For a triangle with side lengths a, b, and c, the semiperimeter is defined as s = (a + b + c)/2. The area of the triangle equals √(s(s-a)(s-b)(s-c)). Note that this doesn’t require a right triangle, nor does it require knowledge of any heights measured perpendicular to any base. You will NOT need to know this for the GRE, as the original solution above attests.}

I was glad to see the above explanation, because I was honestly stumped as to how to approach the problem. I had thought that Heron's Formula would be the only way to solve it.


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"spiritual, not religious"?well, that's a punch in the face for you, then!

Quite possibly a new book for my collection: Dispirited: How Contemporary Spirituality Makes Us Selfish, Stupid, and Unhappy by Dr. Dave Webster, professor of religion, philosophy, and ethics at the University of Gloucestershire, England. Excerpt:

When someone tells me that they are “Not religious, but very spiritual,” I want to punch them in the face.

Hard…

[Interviewer (Webster himself, really)] What’s the most important take-home message for readers?

[Dave Webster] That the idea of being “spiritual, but not religious” is, at the very least, problematic. As I suggest in the book, mind-body-spirit spirituality is in danger of making us stupid, selfish, and unhappy.

Stupid—because its open-ended, inclusive and non-judgemental attitude to truth-claims actually becomes an obstacle to the combative, argumentative process whereby we discern sense from nonsense. To treat all claims as equivalent, as valid perspectives on an unsayable ultimate reality, is not to really take any of them seriously. It promotes a shallow, surface approach, whereby the work of discrimination, of testing claims against each other, and our experience in the light of method, is cast aside in favour of a lazy, bargain-basement-postmodernist relativism.

Selfish—because the ‘inner-turn’ drives us away from concerns with the material; so much so that being preoccupied with worldly matters is somehow portrayed as tawdry or shallow. It’s no accident that we see the wealthy and celebrities drawn to this very capitalist form of religion: most of the world realizes that material concerns do matter. I don’t believe that we find ourselves and meaning via an inner journey. I’m not even sure I know what it means. While of course there is course for introspection and self-examination, this, I argue, has to be in a context of concrete social realities.

Finally, I argue that the dissembling regarding death in most contemporary spirituality—the refusal to face it as the total absolute annihilation of the person and all about them—leaves it ill-equipped to help us truly engage with the existential reality of our own mortality and finitude. In much contemporary spirituality there is an insistence of survival (and a matching vagueness about its form) whenever death is discussed. I argue that any denial of death (and I look at the longevity movements briefly too) is an obstacle to a full, rich life, with emotional integrity. Death is the thing to be faced if we are to really live. Spirituality seems to me to be a consolation that refuses this challenge, rather seeking to hide in the only-half-believed reassurances of ‘spirit’, ‘energy’, previous lives, and ‘soul’.

I can tell already that I'm going to disagree with some or most of the author's contentions, but the book still sounds fascinating.


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when and how the French subjunctive is used

Before I get into how the French subjunctive (le subjonctif) is formed, I'd like to go over when and how it is employed. Off the top of my head, I remember the following contexts:

1. émotion: Je suis contente que tu sois là.

2. antériorité: Fais-le avant qu'il n'arrive.

3. nécessité: Il faut qu'il sache combien elle l'aime.

4. désir/souhait: Que la Force soit avec toi. Je veux que tu viennes.

5. conséquence: J'ai trois jobs pour que tu puisses aller à l'université.

6. unicité: ...la seule personne que je connaisse...

7. doute: Je ne crois pas que ce soit vrai.

8. faits contrefactuels: Bien que je sois américain, je sais parler français. Qu'il soit mort ou vivant, peu importe.

The subjunctive is a mood, not a tense-- a fact you can see reflected in the above list. The French tend to use the subjunctive in more contexts than we use it in English, but it's not foreign to us:

It's important that you be on time. (not "you are on time")

May there be peace on earth. (not "may there is")

(etc.)

In a subsequent post, I'll go over how the French subjunctive is formed.


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don't overcorrect!

I find myself increasingly annoyed by the smug idiots who say, "Decimate does not mean destroy: it means remove or kill a tenth of."

Folks, decimate may have had that latter meaning as its only meaning when the word first came into existence, but nowadays, it's perfectly fine to say The bomb decimated the city. This might not imply total destruction, but the word in modern English can mean anything up to near-total destruction. It's a perfectly fine descriptor of what a bomb can do.

People who make such foolish "corrections" are mistaking original meanings for proper meanings. Notions of proper change with the times. The next time someone tries to tell you you're misusing decimate, ask him whether he thinks his best friend is a nice person. When he says "yes," look shocked and ask him whether he really believes his friend is foolish and stupid. This is, after all, the older-- and therefore proper!-- use of the word nice.

Right?

I once heard a man of Scottish extraction claim that no self-respecting Scotsman would ever use the term "Scotch-Irish." "Scotch is a drink!" he said, to much polite laughter from a crowd that knew no better. "The proper term is "Scots-Irish." But he was wrong: "Scotch" is a perfectly serviceable term in the perfectly legitimate expression "Scotch-Irish." Wikipedia has an interesting write-up on this expression, and notes that "Scotch-Irish" is current only in North America, while "Scots-Irish" is a term of more recent invention, and is also confined to North American usage.

This makes our man wrong twice over: if "Scotch-Irish" isn't heard outside of North America (the demographic in question is apparently referred to as "Ulster Scots" in the UK), then how does calling oneself "Scots-Irish" prove that one is a self-respecting Scotsman? This terminological quibble seems to have little, if anything, to do with the monikers used in the Old Country, and that reinforces the point I'm making in this post: in trying to sound smart, don't sound stupid.

(Click this link to see the etymology of nice.)


ADDENDUM: Here's an interesting article on the "singular they."


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this week's MGRE Math Beast Challenge

From here:



Go to it! My own response will appear in the comments.


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answer to last week's MGRE Math Beast Challenge

The answer to last week's challenge is indeed (A)! That means Charles and I are both right. But here's the thing: I had anticipated that MGRE would take Charles's tack and use the plug-and-chug method, but instead they went full-on algebra, as I did, and offered their own version of plug-and-chug only at the very end as an afterthought, and only as a way to check their algebra. To review, then-- here's what Charles had written in his comment:

Yeah, I got A, too, although my process was not nearly as detailed. I just took 15 as a possible number of women at the party, subtracted 8, and multiplied by 4 to get 28 men originally at the party. Since the question then says that 35 men left the party, I knew that the original number of women had to be greater than 15, so the answer was A.

It took me about five times as long to write the above paragraph as it did to work out the answer. I never did figure out how many women were originally at the party.

Disgustingly simple. My own approach, you may recall, went for the algebra:

Let m = original # of men.

Let w = original # of women.

1st phase: we have m and w.
[All men & women are present.]

2nd phase: we have m and (w - 8).
[Eight women have left.]

3rd phase: we have (m - 35) and (w - 8).
[Thirty-five men have left.]

Given (per what we know of the second phase, and what the word problem tells us):

m = 4(w - 8)

And for the third phase:

(w - 8) = 2(m - 35)

At this point, it's a matter of systems of equations.

m = 4w - 32 (2nd phase)

2m = w + 62 (3rd phase)

Multiply the first equation by 2:

2m = 8w - 64

Match it up with the other equation and solve:

2m = 8w - 64
-(2m = w + 62)

=

0 = 7w - 126

7w = 126

w = 18

The original number of women was 18, so Quantity A is greater.

I'm going with (A).

Not simple, but definitely thorough. And here, finally, is how MGRE tackled the problem:

This problem can be solved with a system of two equations.

First, “after 8 women leave, there are four times as many men as women.” Thus, once 8 is subtracted from the number of women, there is a 4 to 1 male/female ratio:

m/(w - 8) = 4/1

Cross-multiply and simplify:

m = 4(w – 8)
m = 4w – 32

Then, 35 men leave, and the 8 women don’t come back, resulting in a 1 to 2 male/female ratio:

(m - 35)/(w - 8) = 1/2

Cross-multiply and simplify:

2(m – 35) = (w – 8)
2m – 70 = w – 8

We now have two equations in two variables.
1st equation: m = 4w – 32
2nd equation: 2m – 70 = w – 8

Since the 1st equation is already solved for m, simply plug into the 2nd equation for m:

2(4w – 32) – 70 = w – 8
8w – 64 – 70 = w – 8
8w – 134 = w – 8
8w = w + 126
7w = 126
w = 18

Since 18 is more than 15, the correct answer is A.

Although we are not asked for the number of men, note that we could easily generate it by plugging w = 18 into either equation:

m = 4w – 32
m = 4(18) – 32
m = 40

This would allow us to check our answer. If we begin with 18 women and 40 men, and then 8 women leave, we would have 10 women and 40 men, which indeed would be a 1 to 4 ratio of women to men. If 35 men then leave, we would have 10 women and 5 men, which indeed would be a 2 to 1 ratio of women to men.

The correct answer is A.

on profiling

I normally blog about religion on Fridays, since religious studies is one of my fields of interest. In this case, though, I'm going to provide a link to a long conversation between atheist thinker Sam Harris and security expert Bruce Schneier. When it comes to national security, Harris is pro-profiling; his belief is that it's ridiculous for US airline security to waste its time "randomly" plucking, say, 80-year-old white grandmothers from the line for a pat-down (or four-year-old East Asian kids, for that matter) when the demographic from which suicide bombers come is known to all. Schneier rejects Harris's view and defends the current approach to airline security. Harris responds that the type of profiling he advocates is not based on a correlation between a certain demographic and terrorism, but is, rather, based on a causal relationship, in which religion is the basic cause. (See why I want to link to this exchange?) Specifically, Harris writes:

And I am not proposing a mere correlation between extremist Islam and suicidal terrorism. I am claiming that the relationship is causal. There are many ways to see this, and not too many ways to credibly deny it (though Robert Pape keeps at it by skewing his data with the Tamil Tigers).

The first sign of a religious cause comes from what the terrorists say of themselves: al Qaeda and its sympathizers have not been shy about discussing their motives in public. The second indication is what they say when they think no one is listening. As you know, we now have a trove of private communications among jihadists. The fine points of theology are never far from their thoughts and regularly constrain their actions. The 19 hijackers were under surveillance by German police for months before September 11, 2001 (read Perfect Soldiers). Islam was all that these men appeared to care about.

And we should recall how other people behave when subjected to military occupation or political abuse. Where are the Tibetan Buddhist suicide bombers? They have the suicide part down, because they are now practicing a campaign of self-immolation—which, being the incendiary equivalent of a hunger strike, is about as far from suicide bombing as can be conceived. And where is that long list of Palestinian Christian suicide bombers you’ve been keeping in your desk? Now would be a good time to produce it. As you know, Palestinian Christians suffer the same Israeli occupation. How many have blown themselves up on a bus in Tel Aviv? One? Two? Where, for that matter, are the Pakistani, Iraqi, or Egyptian suicide bombers killing for the glory of Christ? These Christian communities are regularly attacked by suicidal jihadists—why don’t they respond with the same sort of violence? This is practically a science experiment: We’ve got the same people, speaking the same language, living in the same places, eating the same food—and one group forms a death cult of aspiring martyrs and the other does not.

I'm still going through the exchange. It's been a fascinating read thus far, and I encourage you to read the whole thing.


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quirks with the imperfect tense in French

You may recall my post on how to form the imperfect tense in French. I mentioned some freakiness with the double-i:

Nous étudiions. (We were studying.)

Nous skiions. (We were skiing.)

Another quirk to watch out for has to do with the letters c and g. I believe I've explained this in another post, but to reiterate:

When placed in front of the vowels a and o, the consonants c and g undergo a slight change if they're to be pronounced softly (i.e., the "s" sound for c, the "zh" sound for g): the c gains a cedilla, and the g takes on an extra e after it.

Examples: le français, nous mangeons

This is relevant when forming the imperfect tense. Take commencer, for example:

je commençais (note the cedilla)
tu commençais
elle commençait
nous commencions (note the lack of cedilla, because the c is followed by an i, not a or o)
vous commenciez
ils commençaient (cedilla again!)

Now watch what happens with manger:

je mangeais (note the additional e)
tu mangeais
il mangeait
nous mangions (no e!)
vous mangiez
elles mangeaient (e again!)

Keep these changes in mind as you master l'imparfait!


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sentence equivalence!

Here's a GRE Sentence Completion problem from Manhattan Prep's blog.

The exhibit is not so much a retrospective as a __________ ; the artist’s weaker early work is glossed over and any evidence of his ultimate dissolution is absent entirely.

Select two correct answers.

(A) paean
(B) philippic
(C) tirade
(D) panacea
(E) eulogy
(F) crescendo

In the revised GRE's Sentence Completion section, the object of the game is to select TWO words that are each capable of (1) completing the sentence correctly and (2) giving the sentence a similar meaning. In other words, the words you select need to be either synonyms or almost-synonyms. Two antonyms might conceivably complete the sentence, but this would violate criterion (2). To get around this problem, the GRE Sentence Completion questions are designed so that a pair of antonyms can't be selected without one or the other word in the selected pair sounding ridiculous in context.

Have at it, then click the above link for the answer and explanation. I was able to answer correctly despite not knowing the meaning of "philippic," which is not a word I'd normally expect to see on the GRE.


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this week's MGRE Math Beast Challenge

From here:

Everyone at a party is either a man or a woman. After 8 women leave, there are four times as many men as women. After 35 men leave (and the 8 women do not return), there are twice as many women as men.

Quantity A
The number of women originally at the party

Quantity B
15

(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.

Go to it! My answer will appear in the comments.


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answer to last week's MGRE Math Beast Challenge

The correct answer was indeed (B)!

Here's MGRE's explanation:

This problem gives us an equilateral triangle and a circle, and tells us that the perimeter of the equilateral is 1.25 times the circumference of the circle.

This is license to plug in. Since both an equilateral and a circle are regular figures—that is, all equilaterals are in the same proportion as all other equilaterals, and all circles are in the same proportion as all other circles—we can be certain that we only need to plug in one set of values in order to be sure of the answer. Because we have regular figures and a way to relate them (triangle perimeter = 1.25 × circumference), we will not need to repeatedly try different values as we often do on Quantitative Comparisons.

We could say the radius of the circle is 2, so the circumference is 4π. Then, the perimeter of the equilateral triangle is (1.25)(4π) = 5π . This isn’t ideal, because then we are stuck with π in the calculations for the triangle, where it is unnecessarily awkward.

We could say circumference = 4 and therefore the perimeter of the triangle = (1.25)(4) = 5. But then the side of the equilateral triangle is 5/3, a non-integer, which is inconvenient. We can avoid this by using larger numbers (multiplied by a factor of 3).

So our smartest numbers are circumference = 12 and therefore the perimeter of the triangle = (1.25)(12) = 15, so each side of the triangle is 5. The height of an equilateral triangle is always (√3)/2 the side length. You can always derive this yourself by splitting the equilateral triangle into two 30–60–90 triangles, with known side ratio 1x : (√3)x : 2x. So, the area of the triangle is

(1/2)bh =

(1/2) • (5) • ((5)((√3)/2)) =

(25√3)/4 =

approx. 10.625. (Use the calculator.)

In the circle, circumference = 12 = 2πr, so

r = 12/(2π) = 6/π. The area of the circle is

π(r^2) = π((6/π)^2) = 36/π = approx. 11.46.

(Use the calculator and the approximation 3.14 for π.)

The correct answer is B.

I have to say... I like my method a lot better.


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monks who gamble with other people's money

This link, about gambling Buddhist monks in Korea, comes from my brother David. I'd heard the stories for years: monks who sneak out to eat meat, monks involved in sexual indiscretions... it comes as little surprise to learn that, yes, there are monks who gamble:

SEOUL -- Six leaders from South Korea's biggest Buddhist order have quit after secret video footage showed some supposedly serene monks raising hell, playing high-stakes poker, drinking and smoking.

The scandal erupted just days before Koreans observe a national holiday to celebrate the birth of Buddha, the holiest day of the religion's calendar.

The head of the Jogye order (external link to Jogye Order of Korean Buddhism's site), which has some 10 million followers, or about a fifth of the country's population, made a public apology on Friday, vowing "self-repentance."

South Korean TV networks aired shots of eight monks playing poker, some smoking and drinking, after gathering at a luxury lakeside hotel in late April for a fellow monk's memorial service.

"The stakes for 13 hours of gambling were more than 1 billion won ($875,300)," Seongho, a senior monk who uses one name, told Reuters on Friday.

Seung Sahn, founder of the Korean monastic order Kwaneum, was lauded for his wisdom, wit, and humor. With his energetic marketing of Korean Seon (i.e., Zen) Buddhism, he was able to spread his school all over the world. Unfortunately, as is true with many men in power, he abused his authority and was caught in a sex scandal involving several women.

It amazes me that there are people within Seung Sahn's Kwaneum Order who have tried to justify his unethical, precepts-breaking behavior. The women were willing! they say. That's a defense? I wonder, now, what defense would be given on behalf of the Jogye Order monks just caught gambling. "If Kwaneum-bosal (bosal = bodhisattva) can do it,* then so can we"?

My view: if you take precepts, whether in Buddhist monasticism or Catholic monasticism or any other sort of clerical endeavor, you're supposed to adhere to them. Such people have chosen to take on the yoke of higher standards. If they can't abide by those standards because of their own human failings, there's no need for us to defend them. In the above case, the Jogye Order did the right thing by apologizing. We can only hope that this will translate into more stringency within the order.

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*The Lotus Sutra says the Bodhisattva of Compassion (Avalokiteshvara, Kuan-shih-yin [Chn.], Kanseon/Kannon [Jpn.], Kwaneum/Kwansaeeum [Kor.]) can assume any form to save beings from suffering. If I remember correctly, my old Buddhism prof said there are stories of the bodhisattva assuming the form of a gambler to rescue gamblers from their destructive habits. This would be in consonance with the Lotus Sutra's emphasis on the concept of upaya, i.e., skillful means. One does what one can to bring people to enlightenment. If it helps to appear to mortals as a gambler, then so be it.


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contrastive tenses: l'imparfait et le passé composé

First a quick review of the French imperfect tense (l'imparfait), then a little demonstration of how the tense is used contrastively.

To form the imperfect tense, remove the "-ons" ending from the first-person plural conjugation of a verb to create a stem:

parler: parlons: parl-
choisir: choisissons: choisiss-
vendre: vendons: vend-
prendre: prenons: pren-
appeler: appelons: appel-
lire: lisons: lis-
vouloir: voulons: voul-
savoir: savons: sav-
devoir: devons: dev-

(irregular form) être: ét-

(etc.)

Depending on person and number, add these endings:

je: -ais
tu: -ais
il/elle/on: -ait
nous: -ions
vous: -iez
ils/elles


In French, the imperfect tense is equivalent to the past progressive tense in English: was ...ing. So:

Je parlais = I was speaking
Tu choisissais = You were choosing
Elle vendait = She was selling
Nous prenions = We were taking
Vous appeliez = You were calling
Ils lisaient = They (masc.) were reading
Je voulais = I was wanting (to)...
Tu savais = You knew (a bit awkward to translate this is as "You were knowing")
Il devait = He had to...


As in English, French verb tenses can be used contrastively. Here's an English example of a contrast between the past progressive and the simple past tense:

I was sleeping when my cell phone rang.

In French, the same contrast is expressed with l'imparfait and le passé composé. To wit:

Je dormais quand mon portable a sonné.

The imperfect tense is used for the "background action," i.e., for actions or events that occur over a period of time. The passé composé, like the simple past tense in English, is used for the "interrupting action," i.e., for actions or events that tend to be sudden and of very short duration. In the above examples, sleeping is the background action; the phone's ringing is the interrupting action.

What if I gave you a problem like this:

Je (regarder) la télé quand le martien (frapper) à la porte.

You'd ask yourself, first, what the background action was: watching TV or the Martian knocking? Obviously, watching TV occurs over a longer period of time than a sudden knock, so regarder should be in the imperfect. Thus:

Je regardais la télé quand le martien a frappé à la porte.

Try this one, which may be a bit more difficult:

Mes copains (arriver) quand je/j' (être) dans la salle à manger.

What's the background action? My being in the dining room or my friends' arriving? It helps to remember that, technically speaking, an arrival happens in a single moment-- the moment the arriving person or thing stops moving. It's only at the very instant that my friends are at the door that I can say they have arrived. Knowing this, we can say that:

Mes copains sont arrivés quand j'étais dans la salle à manger.
My friends arrived when I was in the dining room.

Try your hand at the following sentences.

1. Maman (parler) au téléphone quand notre chat (miauler). (miauler = to meow/mew)

2. Nous (conduire) quand nous (percuter) le cerf. (le cerf = the deer; percuter = to hit, crash into)

3. Quand il (casser) son crayon, je/j' (étudier).

4. Robert et Maxine (skier) quand le bâtiment (exploser). (bâtiment = building)

5. Tu (être) où quand le vol (avoir lieu)? (vol = theft; avoir lieu = to take place)


Final note: The imperfect tense can lead to strange spellings, especially the double-i in the nous form:

Nous étudiions (the imperfect stem of étudier is étudi-)
Nous skiions (the imperfect stem of skier is ski-)

Beware!


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the poetry of Stephen R. Donaldson

Two of my favorite poems are by fantasy/SF writer Stephen R. Donaldson, whom I will always associate with his Chronicles of Thomas Covenant the Unbeliever series: two trilogies and a tetralogy-in-progress. The first poem I'll place here is about death and bereavement:

Death reaps the beauty of the world--
bundles old crops to hasten new.
Be still, heart:
hold peace.
Growing is better than decay:
I hear the blade which severs life from life.
Be still, peace:
hold heart.
Death is passing on--
the making way of life and time for life.
Hate dying and killing, not death.
Be still, heart:
make no expostulation.
Hold peace and grief
and be still.

--Stephen R. Donaldson, 1977
Lord Foul's Bane, Chapter 17, "End in Fire"

The second poem, about the last defense of nature, requires a bit of explanation. The being reciting this poem is a Forestal, a powerful spirit of the woods whose function is the guardianship of forest life: trees, plants, and forest creatures. This particular Forestal, Caer-Caveral, is also trying his best to hold the beautiful region of Andelain together. Andelain is the heart of the Land, but like the rest of the Land, it is under attack by an invention of the Despiser (a satanic/Sauron-like figure): the Sunbane, a curse that drives the Land's natural cycles into unnatural frenzy, forcing earth and sky into a cruel series of rapid changes: desert, rain, pestilence, fertility, etc.-- each phase lasting only a few days, then quickly changing, in random sequence, to a new phase in under a day. The Sunbane, a violation of the natural Law, is ripping the earth apart, and Caer-Caveral knows that even he cannot win against its onslaught. This song, then, is his lament.

Andelain I hold and mold within my fragile spell,
While world's ruin ruins wood and wold.
Sap and bough are grief and grim to me, engrievement fell,
And petals fall without relief.
Astricken by my power's dearth,
I hold the glaive of Law against the Earth.

Andelain I cherish dear within my mortal breast;
And faithful I withhold Despiser's wish.
But faithless is my ache for dreams and slumbering and rest,
And burdens make my courage break.
The Sunbane mocks my best reply,
And all about and in me beauties die.

Andelain! I strive with need and loss, and ascertain
That the Despiser's might can rend and rive.
Each falter of my ancient heart is all the evil's gain;
And it appalls without relent.
I cannot spread my power more,
Though teary visions come of wail and gore.

Oh, Andelain! forgive! For I am doomed to fail this war.
I cannot bear to see you die-- and live,
Foredoomed to bitterness and all the gray Despiser's lore.
But while I can I heed the call
Of green and tree; and for their worth,
I hold the glaive of Law against the Earth.

--Stephen R. Donaldson, 1977
The Wounded Land, Chapter 12, "The Andelainian Hills"

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this week's MGRE Math Beast Challenge

From here:

The perimeter of an equilateral triangle is 1.25 times the circumference of a circle.

Quantity A
The area of the equilateral triangle

Quantity B
The area of the circle

(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.

Go to it! My answer will appear in the comments.

By the way, sorry about the lack of posting last week. I think my sickness lingered on a bit longer than I had thought.


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answer to last week's MGRE Math Beast Challenge

Correct! The answer is indeed 14.286%. Strangely enough, MGRE's initial reasoning is rather tortured. Instead of following their own advice about solving a problem quickly through the plug-and-chug method, they went for the brute-force approach and solved the problem with variables. It's only at the end that the gurus discuss plug-and-chug as an afterthought. Here's what MGRE wrote:

When calculating a percent change “from (original) to (new),” be careful to use the ratio (change/original), not (change/new) or (new/original).

Create some variables:
x = attendance in 2010
y = attendance in 2011
z = attendance in 2012

In 2012, attendance was greater than in 2011, and even greater than it had been in 2010. So, x < y < z.

The question asks for the percent change from 2010 to 2011, or [(y - x)/x]•100%. This can be rewritten as [(y/x)-1]•100%, so what we really need to find is y/x.

We are given two percent changes from one year to another, but watch out! The “from (original)” year is different for each percent given.

“In 2012, attendance at an annual sporting event was 5% greater than it was in 2011”:
Note that “5% greater than” a number means “105% of” that number.
z = 105% of y
z = 1.05y

“In 2012, attendance at an annual sporting event was ... 20% greater than it was in 2010”:
Note that “20% greater than” a number means “120% of” that number.
z = 120% of x
z = 1.2x

To find y/x, we’ll first set both z equivalents equal:
z = 1.05y = 1.2x
y/x = 1.20/1.05

The answer is [(y/x) - 1]•100% = (1.14285714 - 1)•100% = 14.285714%. Rounded to the nearest 0.001%, the final answer is 14.286%.

Alternatively, we could pick numbers. A smart number for z would be a multiple of 120 and 105 (reflecting 5% and 20% increases from an easy base of 100).

z = (105)(120)
y = (100)(120)
x = (105)(100)

The answer is [(y/x) - 1]•100% = {[(100•120)/(105•100)] - 1}•100% = 14.285714%. Rounded to the nearest 0.001%, the final answer is 14.286%

For what it's worth, my own method took me 45 seconds with an on-screen calculator.


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this week's MGRE Math Beast Challenge

From here:

In 2012, attendance at an annual sporting event was 5% greater than it was in 2011 and 20% greater than it was in 2010. What was the percent increase in attendance from 2010 to 2011?

Give your answer to the nearest 0.001%.

Go to it! I'm no longer sick, so my answer will definitely appear in the comments sometime during the next 36 hours. As you can tell, this is a "grid-in" problem, i.e., you have to write the correct answer, not select an answer from multiple choices.


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answer to last week's MGRE Math Beast Challenge

I was still sick last week, so I failed to answer this rather complicated question. Here is MGRE's answer:

Rebecca began with 288 friends, evenly divided among 12 months. Thus, she had 24 friends with birthdays in each month.
Let’s make a simple chart:

[chart 1]

Now we’ll simply calculate and record all the changes. First, “the number of Rebecca’s friends with birthdays in the last quarter of the year increased by 25%.”

Thus, October, November, and December’s totals collectively increased by 25%, so from 3(24) = 72 to 72(1.25) = 90. We don’t actually know if each month increased by 25% (from 24 to 30) or if their total simply increased by 25% (for instance, maybe the entire increase of 18 occurred in one month, bringing that month’s total to 42, and the number of people with birthdays in the other two months remained at 24). The new total is 90 either way, and this question is ultimately about the total. However, one of the later constraints in this problem mentions “the month with the largest number of birthdays,” so let’s put the increase all in one month, as it might ultimately be the relevant month.

[chart 2]

Next, “the number of friends with birthdays in each month beginning with “J” increased by one-third.” To increase a number by one-third, multiply by one and one-third (this is faster than multiplying by one-third and then adding it back to the original): 24 (4/3) = 32

[chart 3]

Next, “the number of people with birthdays in February was increased by 12.5%.” Since 12.5% is 0.125, multiply by 1.125 to ADD 12.5% percent to the original number in one step: 24(1.125) = 27

[chart 4]

Next, “the number of people with birthdays in March became 166.6666...% of the new number of people with birthdays in February.”

166.6666...% of 27 is simply one hundred percent of the number, plus another two-thirds. Since 2/3 of 27 is 18, the new total for March is 45. Or, in the calculator: 27(1.666666666...) = 45. (Actually, putting this in the calculator will yield 44.9999999...., since you didn’t actually type in infinity 6’s. This is fine! The answer is 45.)

[chart 5]

Next, “the number of people with birthdays in April became five less than 75% of the new number of people with birthdays in February and March combined.”

February + March = 27 + 45 = 72
75% of 72 = 72(0.75) = 54.
We need the number 5 less than that: 54 – 5 = 49.

[chart 6]

Now, “the number of people with birthdays in May increased by 1, and the number of people with birthdays in August became one less than 20% greater than the new number of people with birthdays in May.”

May is now simply 24 + 1 = 25.
August is one less than 20% greater than 25. In the calculator: 25(1.2) = 30, then one less, or 29.

[chart 7]

Finally, “September’s total increased to 6% less than one more than the new total for the month with the largest number of birthdays.”

The month with the largest number of birthdays is April, with 49. Remember that even if the 25% increase in the total for the last quarter of the year occurred in a single month, that month (October in our chart) would only have 42 people.

One more than 49 is 50.

September’s total is 6% less than 50. To decrease a number by 6%, take 94% of it (this is faster than finding 6% and subtracting it from the original): 50(0.94) = 47

[chart 8]

To calculate the final answer, simply add the “AFTER” row of the chart:

32 + 27 + 45 + 49 + 25 + 32 + 32 + 29 + 47 + 42 + 24 + 24 = 408

The correct answer is B.

I don't have time to do it right now, but I'll be adding the charts later tonight.


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don't know much about history

My goddaughter suddenly canceled her geometry tutoring session with me because of a "music thing" (a performance, apparently) she hadn't known about until the last minute. This left me with some free time, so I spent some of it taking part of an AP World History exam. The College Board has a PDF that contains, among other things, thirty multiple-choice questions. Knowing full well that I'm terrible at history, I decided to see how I'd fare.

Final score: 23/30. Not horrible, but also not enough to rate more than a mediocre 3 on the exam. For the most part, I used a combination of guesswork and common sense-- strategies available to anyone who's facing a multiple-choice test.* Some of the questions were easy to figure out because they weren't exclusively history-oriented: they could just as easily have appeared in the SAT's Reading Comprehension section.

Here are the seven questions I got wrong:

8. Inca and Aztec societies were similar in that both

(A) developed from Mayan civilization
(B) acquired empires by means of military conquest
(C) independently developed iron technology
(D) depended entirely on oral record keeping


(The map below applies to question #10.)

10. The map above shows what significant economic developments?

(A) Trade connections that linked the Hellenistic and Maurya
empires to African cities from 300 through 150 B.C.E.
(B) Trading networks that promoted the growth of new cities
from 600 C.E. through 1450 C.E.
(C) Chinese dominance of Indian Ocean trading networks
because of the voyages of Zheng He in the 1400s C.E.
(D) Changes in Indian Ocean trading networks that resulted
from technological innovations from 1450 C.E. through 1750
C.E.


12. The Columbian Exchange involved which of the following new
connections in the era 1450–1750?

(A) European food to the Western Hemisphere; Western
Hemisphere diseases to Europe; African population to Europe
(B) Western Hemisphere technology to Africa; African food to
Europe; European population to the Western Hemisphere
(C) European technology to Africa; Western Hemisphere
population to Africa; African food to the Western Hemisphere
(D) African population to the Western Hemisphere; Western
Hemisphere food to Europe and Africa; African and European
diseases to the Western Hemisphere


14. Which of the following is most likely to have influenced
eighteenth-century population trends in both Europe and
China?

(A) A sharp decline in average global temperatures
(B) Introduction of Western Hemisphere crops
(C) Innovation in birth control measures
(D) Improvement in surgical procedures


16. In recent decades, many world historians have challenged the
commonly held view that Europeans controlled the largest
share of world trade in the seventeenth through the eighteenth
centuries. Which of the following evidence from the period
would best support this historical reinterpretation?

(A) Prices for Chinese goods were much higher in Europe than
in China.
(B) European trading companies often backed their long-distance
trading ventures with the threat of military force.
(C) Asian trading companies dominated trade in the Indian
Ocean region.
(D) European merchants transported only a fraction of the
goods shipped globally.


19. Which of the following statements is true about both the
Mughal and Ottoman empires in the sixteenth century?

(A) In both empires the majority of the people were Muslims.
(B) Both empires had powerful navies that engaged European
navies.
(C) Both empires expanded through the use of gunpowder
weapons and extensive bureaucracies.
(D) Both empires gave little monetary support to artistic and
cultural endeavors.


22. In contrast to initial industrialization, the second Industrial
Revolution in the last half of the nineteenth century was
particularly associated with the mass production of which of
the following?

(A) Textiles, iron, and coal
(B) Textiles, automobiles, and plastics
(C) Airplanes, ships, and radios
(D) Electricity, steel, and chemicals

Feel free to try your hand at these questions by leaving a comment. If you want, I can supply answers, but only to the curious, and only after they've tried to respond. History buffs will probably find the above questions easy.

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*Long-time readers know I consider multiple choice to be the worst possible testing format.


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this week's Math Beast Challenge problem

From here:

Rebecca had 288 Facebook friends, and noticed that an equal number of these friends had birthdays in each of the twelve months of the year. Then, Rebecca approved many friend requests at once. After doing so, the number of Rebecca’s friends with birthdays in the last quarter of the year increased by 25%, the number of friends with birthdays in each month beginning with “J” increased by one-third, the number of people with birthdays in February was increased by 12.5%, the number of people with birthdays in March became 166.6666...% of the new number of people with birthdays in February, the number of people with birthdays in April became five less than 75% of the new number of people with birthdays in February and March combined, the number of people with birthdays in May increased by 1, and the number of people with birthdays in August became one less than 20% greater than the new number of people with birthdays in May. Finally, September’s total increased to 6% less than one more than the new total for the month with the largest number of birthdays. Assuming no one de-friended her, after approving all her friend requests, how many Facebook friends did Rebecca then have?

(A) 396

(B) 408

(C) 453

(D) 512

(E) 696

Good Lord. Given the time it takes to read this problem, you'd run out of time to do the rest of the Quant section if this were an actual GRE! Anyway, go to it! I'm still sick, so my answer may or may not appear in the comments section below.


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answer to last week's MGRE Math Beast Challenge

Having gotten disgustingly sick on Monday, I never answered last week's MGRE Math Beast Challenge. Here's MGRE's explanation, which-- strangely enough-- doesn't seem to require a very deep knowledge of how standard deviations work.

The main challenge in this problem is working through the math language to figure out what the question is really asking.

First, we are given a function: f(x) = 0.27(-3.12x – 4)

Let’s distribute: f(x) = -0.8424x – 1.08

The ugliness of these numbers is a good clue that this is more of a logic problem than a straight math problem.

We are then told that Set P consists of n distinct values that are inputted into f(x). Keep in mind that n here is just the number of numbers in the set. So, if Set P were 10, 11, 12, n would simply be 3. We are also told that the values are distinct (so the set could not be 1, 1, 1, 1, 1, for instance).

Set Q consists of all the results you get from plugging the values in Set P into the function.

Let’s try a simple example. What if Set P = {1, 2}?

f(1) = -0.8424(1) – 1.08
f(1) = -1.9224

f(2) = -0.8424(2) – 1.08
(2) = - 2.7648

Thus, if Set P = {1, 2}, then Set Q = {-1.9224, -2.7648}. Note that the values in Set P are further apart (exactly 1 apart), while the values in Set Q are closer together (less than 1 apart). Thus, the standard deviation of Set P is greater. But will this always be true?

At this point the answer is either A or D. We could try other possibilities – Set P could be nearly anything, after all. But a bit of logic might prove helpful.

The standard deviation of a distinct set increases when every item in the set is multiplied by a value > 1 or < -1.

The standard deviation of a distinct set decreases when every item in the set is multiplied by a value between -1 and 1, not inclusive.

The standard deviation of a distinct set does not change when every item in the set has the same value added to it (or subtracted from it).

Thus, in the function f(x) = -0.8424x – 1.08, x undergoes two changes:

It is multiplied by a number between -1 and 1.
It has a value subtracted from it.

When you perform both these changes to every item in Set P, the first change will cause the standard deviation to decrease – that is, the numbers get closer together. The second change makes no difference to the standard deviation.

Thus, no matter what numbers you pick, Set P will always have a greater standard deviation than Set Q (said another way, running at least two distinct numbers through this particular function yields output numbers that are closer to one another than the input numbers were).

Notice that the problem specified “distinct” values? If that one word were removed from the problem, the answer would become D. Why? Without the word “distinct,” Set P could be something like {10, 10, 10}, which has a standard deviation of zero. Putting {10, 10, 10} through the function would yield {-9.504, -9.504, -9.504}, which also has a standard deviation of zero. Since it would then be possible for Quantity A to be larger but also possible for the quantities to be equal, the answer would become D. Watch out for the “distinct trap” in standard deviation problems!

The correct answer is A.

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