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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taking this coming week off

I'm sick with something that might be bronchitis, and I've got a big proofreading project to deal with, so I do believe I'm going to take the week off. I've written quite a bit since I began this blog last year, so there's plenty of material here for the curious reader. Feel free to rummage, and for what it's worth, keep in mind that I might put up a post or two this week.


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on labyrinths

Zen teacher Lorianne DiSabato writes about what it means to walk a labyrinth.

Walking a labyrinth underscores the idea that taking care of today—the next step—is enough to get you there and back safely, without undue worry or exertion. Don’t worry about the destination, which will come in due time: just keep going. It’s a lesson that we need every day, everywhere, regardless of whether we live with a labyrinth near.

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nasal vowels (les voyelles nasales)

If you're one of those French learners who mispronounces "vingt" as "vent," then you need help with your nasal vowels. A very good resource to help de-confuse you can be found here. In the meantime, practice that classic phrase: un bon vin blanc. That covers all four nasal vowels.


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seven non-errors

This Cracked.com article says nothing I haven't heard already, but you might enjoy it:

7 Commonly Corrected Grammar Errors (That Aren't Mistakes)

(Credit for this find goes to people on my Twitter feed.)


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this week's MGRE Math Beast Challenge problem

From here:

f(x) = 0.27(-3.12x – 4)

Set P consists of n distinct values, where n > 1. Set Q consists of the n values that result from inputting the n distinct values from Set P into f(x).

Quantity A
The standard deviation of Set P

Quantity B
The standard deviation of Set Q

(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... but first, I really need to review standard deviations!

(Some help can be found here.)


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last week's MGRE Math Beast Challenge: correct!

Here's what MGRE has to say about last week's Math Beast Challenge problem:

Let’s start by translating this into algebra.

Abe has k ketchup packets, Beata has m mustard packets, Cruz has s soy sauce packets, and Dion has b barbecue sauce packets. We are trying to find the smallest possible value for k + m + s + b.

We know that 2k = 9m = 7s = 15b. In other words: 2k, 9m, 7s and 15b each equal the same integer number, but what is that number? It would have to be cleanly divisible by 2, 9, 7, and 15. In order to minimize the number of packets owned by the group as a whole, we would want to find the smallest such number. That would be least common multiple of 2, 9, 7, and 15.

First, we find the factors of each number:
2 = 2
9 = 3 * 3
7 = 7
15 = 3 * 5

Then we multiply only the necessary factors together:
2 * 3 * 3 * 7 * 5 = 630.

Note that we leave out a 3, compared to the list of all factors above. Remember that we only include the factors needed to build each of the starting numbers individually. With two 3’s and a 5, we could make either 9 or 15. That’s good enough for a least common multiple.

Now we know that 2k, 9m, 7s, and 15b each equals 630. Let’s find how many packets each person has.

2k = 630, so k = 315 9m = 630, so m = 70 7s = 630, so s = 90 15b = 630, so b = 42

To finish, just add together the individual number of packets (k + m + s + b) = 315 + 70 + 90 + 42 = 517.

It is worth analyzing how the other choices all represent a possible mistake, a quality that makes this question harder than it would be with different answer choices:

(A) is a trap. It’s just the number in the problem added together (2 + 9 + 7 + 15).

(B) CORRECT.

(C) is a trap for those who stop at 630 and don’t remember that it’s just a step on the way to finding the individual number of packets each person has.

(D) is the sum of the packets if starting with 1890 instead of 630 as the common multiple. 2k = 1890 (k = 945). 9m = 1890 (m = 210). 7s = 1890 (s = 270). 15b = 1890 (b = 126). Sum: 945 + 210 + 270 + 126 = 1551.

(E) is another possible multiple of 2, 9, 7 and 15. (2)(9)(7)(15) = 1890. But it’s not the least common multiple. It’s the trap for those who don’t omit a redundant 3, and who also forget to finish the solving. (The value of 2k = 9m = 7s = 15b is not the answer, k + m + s + b is the answer.)

The correct answer is B.

Woo-hoo! But I actually think my own method, with the compound ratio, is quicker.


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going over your errors in reading comprehension

At the tutoring center where I work, I have plenty of SAT students. When it's time to score their performance in Critical Reading, I usually work it this way:

1. For the Sentence Completion portion of Critical Reading, I tell the students whether or not they've gotten a question right, but I don't tell them what the answer is. We go over the question and reason our way, together, to the correct answer.

2. For the Reading Comprehension section, I ask the students to perform what I call "the line-number exercise." I give them the correct answers to the RC questions they got wrong, then I ask them to go back into the reading passages and find the line numbers that provide evidence for why a given answer is correct. If the question already has line numbers in it, I ask the students to write a short sentence explaining why the answer is correct ("B is correct because the passage says X...").

The line-number exercise gets the students to train themselves in cognitive skills that can't be taught explicitly: skills like scanning and inference. I prefer this indirect method-- which places the burden of learning directly on the students-- to easier methods that involve leading students by the nose. Double-plus ungood, that.


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metaphysical froth

[NB: This post is actually a repost of an essay I wrote back in 2008-- here.]

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Philip Pullman's His Dark Materials trilogy (The Golden Compass, The Subtle Knife, and The Amber Spyglass) is a wild metaphysical ride-- imagine Tom Robbins for kids-- that takes the reader through multiple alternate universes, many of which appear to be variations on our own, but at least one of which features an earth on which no human life evolved (the mulefa of that world are sentient, but not human: imagine elephants on motorcycles). We encounter only an infinitesimal fraction of the universes out there; more are born every moment.

The manner in which Pullman's universes are born is boilerplate sci-fi (for a classic example, see Larry Niven's short story, "All the Myriad Ways," in his collection of the same title): as sentient beings are faced with choices, each choice results in a mitotic split by which new universes are born, each universe containing an alternate version of the sentient being who has passed beyond the moment of choice. If a certain Being X has twenty possible choices at a given moment, then twenty different universes will be born, each one instantiating one of those twenty choices. Of course, assuming the existence of libertarian free will, each sentient individual actually faces an infinity of possible actions each moment, so each individual is "producing" infinities upon infinities of universes every moment. If you think that's complicated, apply that scenario to every sentient being.

The idea that we live in an ever-burgeoning froth of universes is evocative, but is also, in my opinion, unworkable. I want to talk first about the narrative problems it poses for Pullman's plot (this will require explaining the story a bit), and later on about the philosophical problems inherent in a frothy metaphysic.

1. Narrative Problems

Pullman obviously can't lead us through every single universe; for his story to have any coherence, he must confine his narrative to just a few universes. The ones we encounter are:

1. Lyra Belacqua's world
2. Will Parry's world (which is also our world)
3. The world of Cittàgazze (characterized by the predominance of Italian culture, the presence of Coca Cola, and the general lack of adults in the big cities)
4. The world of Lord Asriel's fortress
5. The world of the mulefa, where Mary Malone constructs her amber spyglass

Beings from other universes appear in the story, but we never visit those places.

All the parallel/alternate universes are connected, however, by the existence of Dust, which is particles of consciousness. When matter evolves to the point where sentience appears, there Dust is found. The universes are also connected to the Abyss: interdimensional explosions that rip the fabric of space-time can create holes in many alternate worlds at once, and the Dust from those worlds will begin to drain into that singular Abyss.

It is possible that the universes are also connected "at the top": the idea that all the universes are the products of a single creator God is alluded to in the books, although God is never actually seen, and God's existence is never confirmed. Much of the story focuses instead on The Authority, the first and greatest angel to be formed from the coalescence of Dust. The Authority crowned himself God and told all who followed that he was the creator.

Angels can pass easily between alternate universes without disturbing the overall frothy structure of the Great Metaphysic (my term for the sum total of all universes, not Pullman's). It seems that angels, despite being the most highly sentient of sentient beings, do not produce new universes with their choices. Pullman never directly addresses this issue. Humans, too, can pass from one universe to another; in fact, many doors between worlds remain open because the humans who created them have forgotten to reclose them.

If I've read Pullman correctly, the human ability to travel between worlds began about three hundred years ago in the world of Cittàgazze, where someone or a group of someones created a tool called the "subtle knife." The blade is of modest size and two-edged; one edge cuts through any material (reminiscent of a lightsaber); the other edge, when the proper owner of the knife achieves the correct state of mind, cuts through the fabric of one's universe and, depending on the direction of one's concentration, can slice a window or hole into an alternate world. Shifts in cuts and concentration are what allow the knife wielder to open doors to different worlds. A conscientious user of the knife can step through the threshold and reseal the tear, if he so wishes.

But over the course of three centuries, the various users of the subtle knife have secretly entered different universes, pilfering items and technologies found in them, often leaving the doors between worlds open. Each tear allows a little of the Abyss to peek through, and soul-eating Specters, the children of the Abyss, are created every time a cut with the subtle knife is made. As a result, a good part of the trilogy is devoted to the question of how to repair the open doors, stanch the flow of Dust into the Abyss, and stop the spread of the Specters.

It's all quite complicated, and I'm afraid it's also unworkable from a narrative point of view. The problem is this: if there's one Cittàgazze world, there are many-- an infinity of them, in fact. The moment the subtle knife was created, there wouldn't have been only one such knife, as Pullman's story implies: there would have been an infinity of them, too, with an infinity of people doing an infinite amount of damage to the Great Metaphysic. The plot of the trilogy wraps itself up far too neatly (and happily), and this is problematic because Pullman obviously wants to write a smart story for smart kids, a story that works on many levels. Astute young readers will catch on to the same problems I'm talking about here, and will have the same doubts about the conclusion of Pullman's trilogy.

With a blossoming infinity of subtle knives out there, a simple resolution is quite impossible. How would you track down and stop the wielders when each wielder is producing an infinity of new wielders at every moment? I conclude that Pullman bit off more than he could chew when he decided on such a freewheeling many-worlds scenario for his story. He could have avoided the chaos by hewing to a more modest alternate-universe paradigm, such as can be found in CS Lewis's Narnia series, where God's anteroom is a forest filled with still pools of water, each pool a gateway to a self-contained universe, and little to no interpenetration between universes except whatever God allows. Pullman could also have gone for an even more restricted scenario, such as the one in Stephen R. Donaldson's Thomas Covenant series, which deals with only one alternate world created by a being who, in our world, appears to be a Hindu monk. In terms of narrative, neatness counts, and the more I think about Pullman's story, the less I like this aspect of it. What a contrast with that other well-known series, the Harry Potter heptalogy! JK Rowling offers us only one world, one with quite enough action to keep us occupied, thank you very much. When put next to Pullman's trilogy, Rowling's series looks relentlessly linear.

2. Philosophical Problems

Now let's turn to the matter of the frothy metaphysic itself.

I'm a big fan of Occam's Razor, which states that we should "not multiply entities beyond necessity." This is normally interpreted to mean "the simplest, most elegant explanation for a given state of affairs is probably the correct one," but in the case of Pullman's Great Metaphysic, there's no need to reinterpret Occam: Pullman's story quite literally multiplies entities beyond necessity!

But let's think for a moment in terms of simple, elegant explanations. Which explanation for the current state of affairs strikes you as simpler and more elegant?

1. There is only one universe.

2. There is an infinity of universes, with new ones being created all the time as sentient beings make choices.

The idea that this one reality (and there can only ever be one reality, as I explained back in this post) contains one universe strikes me intuitively as correct. Parallel universes seem to me to feed an anthropocentric need to spread our egos as far and wide as possible: what a nice fantasy to think that somewhere out there is an alternate Kevin who is at this very moment sipping Mai Tais and surrounded by gorgeous women!

So the froth model seems to fail the test of Occam's Razor, a truly subtle knife if ever there was one. I also think the notion of a frothing reality presents us with a problem only vaguely alluded to earlier: the problem of freedom.

Freedom, conventionally defined, is the ability to do otherwise than what one has done. This suggests that, at a given choice-moment, there is the actual choice made and, potentially, an infinity of counterfactuals, the ghosts of alternatives unexplored. In the froth model, however, there are no counterfactuals: all possibilities are actualized! Stepping back to the God's-eye view, we can see that this means there is no freedom, no shadowy "otherwise." Those "otherwises" actually exist in-- as-- other worlds.

Let's simplify the situation and pretend that at moment M, when Kevin makes a choice, reality suddenly switches to the froth model, and that only Kevin is the generator of universes. What this means, from the God's-eye perspective, is that Kevin is a being whose true shape spreads across a multiverse and resembles a great, branching structure. That structure contains no potentiality, because every single one of Kevin's choices is actualized in some universe somewhere. The shape of this structure is therefore fixed: the branch-Kevin, taken as a whole, is not free. If we follow Kevin along only one world-line, we can see how he might think of himself as free-- how, from his limited perspective, he might come to regret the would-haves and could-haves in his life. But Kevin in his entirety, the infinitely ramified Kevin, isn't free at all: his plural existences cover all possibilities, leaving no counterfactuals.

I somehow doubt that reality is this complicated. I may be wrong, but Occam's Razor is quite persuasive: it's more fruitful to think we all inhabit a single, non-frothing reality, and that counterfactuals, whatever they are and whatever their ontological status, drop away as we pass through each moment of choice.* It also makes little sense, thermodynamically speaking, to say that we, or that our decisions, somehow create whole universes. Easier to adopt the creaturely view that we arise out of a universal matrix, retain some coherence for a time, and then slough back into the cosmic churning-- scattered and dissipated, and never to return exactly as we were.

In conclusion, then: while I found Pullman's trilogy to be a great read, it may have failed in the exploration of one of its most central ideas-- the notion of a ramifying multiverse. The neat conclusion of the trilogy did not take the metaphysic seriously enough, and as a result, the conclusion rang false.

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*The same could be said for quantum-level fluctuations in the structure of abiotic matter. Why should sentience be the sole producer of universes?


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le futur proche

The near-future tense in French is a way of saying "going to," and corresponds, for the most part, to the way we use "going to" in English. The basic construction in French involves the conjugated form of the verb aller plus the infinitive form of the required verb.

sujet + (ne/n') aller [conjugué] (pas) + infinitif

PRÉSENT: Je vais au ciné. (I'm going to the cinema.)
FUTUR PROCHE (affirmatif): Je vais aller au ciné. (I'm going to go to the cinema.)
FUTUR PROCHE (négatif): Je ne vais pas aller au cinéma. (I'm not going to go to the cinema)

Other examples:

1. Elle va faire la vaisselle après le souper.
2. Va-t-il comprendre?
3. Où vont-ils le trouver?

Example (3), above, brings in the idea of direct and indirect objects, and how to place them inside a futur proche construction. The grammatical rule is simple enough: place all objects, direct and indirect, right before the infinitive, and never inside the ne/pas negation. For instance:

1. Où va-t-elle le trouver? (Where's she going to find him/it?)
2. Je vais le faire dans un instant. (I'm going to do it in a moment.)
3. Elle va lui en parler demain. (She's going to talk to him about it tomorrow.)
4. Je te promets que je vais y réfléchir. (I promise you that I'm going to think about it.)
5. Nous allons leur proposer une solution. (We're going to propose a solution to them.)
6. Ils vont nous traquer. (They're going to track/hunt us.)
7. L'écureuil ne va pas les partager. (The squirrel's not going to share them.)
8. Tu ne vas pas le lire? (You're not going to read it?)
9. Ne vas-tu pas le faire? (Aren't you going to do it?)

Try changing the following sentences from the present to the near-future tense.

1. Je fais du jogging chaque jour.
2. Tu y penses.
3. Le fait-il?
4. Nous jouons au foot.
5. Visitez-vous Paris?
6. Elles prennent le train.
7. Pourquoi n'étudies-tu pas le chinois?
8. Pourquoi ne l'étudies-tu pas?

Answers (highlight the space between the brackets to see):

[
1. Je vais faire du jogging chaque jour.
2. Tu vas y penser.
3. Va-t-il le faire?
4. Nous allons jouer au foot.
5. Allez-vous visiter Paris?
6. Vont-elles prendre le train?
7. Pourquoi ne vas-tu pas étudier le chinois?
8. Pourquoi ne vas-tu pas l'étudier?
]


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"like" vs. "as (if)"

Like I said...

Have you ever said the above? It's common (and perfectly tolerable) in informally spoken and written English; I'd expect to see the above construction in blog posts and emails, and to hear it in conversation. But in terms of proper grammar, as would be required in a formal research paper, the above is incorrect. The correct construction should be:

As I said...

The same problem occurs here:

You look like you've seen a ghost.

That should actually be:

You look as if you've seen a ghost.

[NB: Purists may see the above and argue that "as though" is more appropriate, claiming that "as if" is counterfactual. A question for another blog post, perhaps?]

"But, WHY?" I hear you screech. The general rule is this:

Use as (if) before a clause (or verbal construction); otherwise, use like.

CORRECT: You look like my sister. ("my sister" is a noun phrase, not a clause)

CORRECT: You look as if you're hungry. ("you're hungry" is a clause)

CORRECT: As I mentioned earlier... ("I mentioned" is a clause)

CORRECT: But in terms of proper grammar, as would be required in a formal research paper, the above is incorrect. (though not a clause, the phrase "would be required" is a verbal construction)

Simple? Clear? Try your hand at the following. Answers will be between the brackets following the quiz; highlight that space to see them.

LIKE/AS QUIZ

1. Do _____ I do, not _____ I say.

2. Why can't you be more _____ your brother?

3. You've failed, Your Highness, for I am a Jedi, _____ my father before me.

4. _____ I was saying before I was rudely interrupted...

5. _____ sands through the hourglass, so are the days of our lives.

6. This case isn't _____ that case at all!

7. _____ you wish.

8. Please make corrections _____ necessary. (Watch out! This one's tricky.)

9. I am, _____ you now clearly see, a vampire.

10. All the same, I am not _____ other vampires, for I am a vegetarian.


ANSWERS (highlight the space between the brackets to see)

[

1. as, as ("I do" and "I say" are both clauses)
2. like ("your brother" is not a clause)
3. like (from "Return of the Jedi"; "my father" is not a clause)
4. As ("I was saying" is a clause)
5. Like (from an old daytime soap opera)
6. like ("that case" is not a clause)
7. As ("you wish" is a clause)
8. as (the sentence can be written out more fully as "Please make corrections AS THEY ARE necessary," or "...AS YOU DEEM necessary.")
9. as
10. like ("other vampires" is not a clause)

]



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this week's MGRE Math Beast Challenge

From here:

Abe, Beata, Cruz, and Dion each collect a different type of condiment packet, and each person only collects one type. Twice the number of ketchup packets in Abe’s collection is 9 times the number of mustard packets Beata has, 7 times the number of soy sauce packets Cruz has, and 15 times the number of barbecue sauce packets possessed by Dion. If each collector owns at least one packet and only whole packets, what is the fewest possible number of packets owned by all four people?

(select only one)

A. 33
B. 517
C. 630
D. 1551
E. 1890

Go to it! My own answer will appear in the comments. Hint: the easiest way to solve this problem is probably via compound ratios. Ever worked with those? I started using them only a few months ago myself! Compound ratios basically look like multi-tiered fractions, because that's exactly what they are. Here's an example of such ratios in action:

In Arkansas, for every 2 snakes there are 7 gerbils; for every gerbil there are 5 mice. On Farmer Brown's many-acred property, there are 6125 mice. How many snakes are on Farmer Brown's property?

Start building a compound ratio by first assembling the data you have:

Let snakes = s; let gerbils = g; let mice = m.

s : g : m

2 : 7 : x

y : 1 : 5

Solve for x by making a proportion: 7/x = 1/5. X therefore equals 35.

s : g : m

2 : 7 : 35

No need to solve for y! We now have our compound ratio, and we can derive other ratios from the above information. To wit:

s/g = 2/7 (given)
g/m = 7/35 = 1/5 (given)
s/m = 2/35 (we'll need this info)

Now that we know the basic ratio of snakes to gerbils to mice, we can apply the compound ratio to the problem.

s : g : 6125

2 : 7 : 35

The snakes-to-mice ratio is 2 : 35. Set up a proportion:

2/35 = s/6125

Solve for s:

35s = 6125•2

s = (6125•2)/35 = 350

Farmer Brown's looking at 350 snakes on his property.

And that's how compound ratios work!



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rock and roll! got last week's MGRE problem correct!

I admit I felt shaky about my solution to last week's MGRE Math Beast Challenge problem, but I turned out to be correct. Here's MGRE's explanation:

Let’s put the words into equation form:

(Total Income – Exclusion)(Tax Rate) = Income Tax.

The question asks about total income, so we’ll solve the equation for Total Income:

Total Income – Exclusion = [Income Tax / Tax Rate]
Total Income = [Income Tax / Tax Rate] + Exclusion

To maximize total income, we’ll minimize Tax Rate (smaller denominator→larger value) and maximize the Exclusion:

Maximum Total Income = [$8700/0.15] + $9800 = $58,000 + $9,800 = $67,800.

To minimize total income, we’ll maximize Tax Rate (larger denominator→smaller value) and minimize the Exclusion:

Minimum Total Income = [$8700/0.35] + $5200 = $24,857.14 + $5,200 = $30,057.14.

The correct answers are C, D, E, and F.

The letters C, D, E, and F correspond to the values I had selected. MGRE arrived at the exact same range that I had arrived at, too: roughly $30,057 at the bottom end, and $67,800 at the top end.

Triomphe!


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Hindu/Western parallels

A very interesting article concerning "overlaps" between Hindu and Western thought.


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belle recette

Chaussons aux champignons, brie, et persil (Jasmine Cuisine).

A vous de déchiffrer et de cuisiner!

les chaussons = slippers, turnovers


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Americans and the media

You might enjoy this long piece from a 1996 issue of The Atlantic titled "Why Americans Hate the Media." Excerpt:

As with medieval doctors who applied leeches and trepanned skulls, the practitioners cannot be blamed for the limits of their profession. But we can ask why reporters spend so much time directing our attention toward what is not much more than guesswork on their part. It builds the impression that journalism is about what's entertaining—guessing what might or might not happen next month—rather than what's useful, such as extracting lessons of success and failure from events that have already occurred. Competing predictions add almost nothing to our ability to solve public problems or to make sensible choices among complex alternatives. Yet such useless distractions have become a specialty of the political press. They are easy to produce, they allow reporters to act as if they possessed special inside knowledge, and there are no consequences for being wrong.

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this week's MGRE Math Beast Challenge

From here:

For Jack, income tax is between 15 and 35 percent of total income after an “exclusion” amount has been subtracted (that is, Jack does not have to pay any income tax on the exclusion amount, only on the remainder of his total income). If the exclusion amount is between $5200 and $9800, and Jack’s income tax was $8700, which of the following could have been Jack’s total income?

(Choose all that are appropriate)

$13,000

$23,100

$33,200

$43,300

$53,400

$63,500

$73,600

Go to it! I'll leave my response in the comments.


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solution to last week's MGRE Math Beast Challenge

From MGRE, this explanation for why the answer to last week's Math Beast Challenge is indeed (C), 36:

In situations in which two people, animals, cars, etc. are traveling on a straight line:

ADD the rates when moving in opposite directions.
SUBTRACT the rates when moving in the same direction.

In this case, if the prey is running at 40 kph and the predator is chasing at 48 kph, then the predator is catching up at a rate of 8 kph.

Since the distance between the two is 80 meters, we can simply use the formula Rate × Time = Distance.

However, note that the distance is in METERS and the rate is in KILOMETERS per hour. We will have to convert. Let’s use 1 kilometer = 1,000 meters to make a proportion:

80 = 1,000x
0.08 = x

Thus, the distance = 0.08 kilometers.

From Rate × Time = Distance,
8t = 0.08
t = 0.01

Thus, the time is 0.01 or 1/100 HOURS, but we need our answer in SECONDS. 1/100 of an hour is 60/100 of a minute, which is 3/5 of a minute, which is 36 seconds.

Or, make a proportion:

x = (0.01)(3600)
x = 36

The correct answer is C.

This was exactly the solution as my friend Charles described it. Bravo!


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high and low christology

The term christology is analogous to theology: it refers to ordered discourse about the Christ.* Who was Jesus? Was he truly the son of God, or just a particularly talented teacher? Did Jesus even exist? What does it mean when Christians declare that Jesus is the Christ? What is the significance of the crucifixion and resurrection events? What scriptural, historical, philosophical, and psychological arguments can be made regarding who this Christ was? Christology deals with all these matters.

Two other terms are in common use among biblical scholars: low christology and high christology, sometimes also called low, ascending christology and high, descending christology. These terms refer, in the main, to the attitudes taken by the New Testament writers toward Jesus's role and significance: low christology emphasizes Jesus' humanity; high christology emphasizes Jesus divinity, his cosmic nature.

Since it's Good Friday, I thought it might be appropriate to focus on these varying christologies for a moment. One of my favorite approaches to this topic is to ask people to compare the four gospel accounts at the moment of Jesus' death on the cross. In each case, what are Jesus' final utterances?

Mark is, according to most biblical scholarship, the first of the four gospels (Matthew was written later). How does Mark portray Jesus' final moments?

Mark 15:33-37

When it was noon, darkness came over the whole land until three in the afternoon. At three o’clock Jesus cried out with a loud voice, “Eloi, Eloi, lema sabachthani?” which means, “My God, my God, why have you forsaken me?” When some of the bystanders heard it, they said, “Listen, he is calling for Elijah.” And someone ran, filled a sponge with sour wine, put it on a stick, and gave it to him to drink, saying, “Wait, let us see whether Elijah will come to take him down.” Then Jesus gave a loud cry and breathed his last.

Let's move on to Matthew.

Matthew 27:45-50

From noon on, darkness came over the whole land until three in the afternoon. And about three o’clock Jesus cried with a loud voice, “Eli, Eli, lema sabachthani?” that is, “My God, my God, why have you forsaken me?” When some of the bystanders heard it, they said, “This man is calling for Elijah.” At once one of them ran and got a sponge, filled it with sour wine, put it on a stick, and gave it to him to drink. But the others said, “Wait, let us see whether Elijah will come to save him.”

Then Jesus cried again with a loud voice and breathed his last.

Now Luke:

Luke 23:44-46

It was now about noon, and darkness came over the whole land until three in the afternoon, while the sun’s light failed; and the curtain of the temple was torn in two. Then Jesus, crying with a loud voice, said, “Father, into your hands I commend my spirit.” Having said this, he breathed his last.

Finally, the Fourth Gospel:

John 19:28-30

After this, when Jesus knew that all was now finished, he said (in order to fulfill the scripture), “I am thirsty.” A jar full of sour wine was standing there. So they put a sponge full of the wine on a branch of hyssop and held it to his mouth. When Jesus had received the wine, he said, “It is finished.” Then he bowed his head and gave up his spirit.

Note that, in Mark and Matthew, Jesus' final utterance is a scream-- this coming after a lorn cry of abandonment. Mark and Matthew's depiction of Jesus constitutes a low christology: the emphasis, here, is on Jesus' humanity. Luke, meanwhile, seems to offer us a slightly loftier perspective, but it's in John's gospel that we see a radical shift to high christology: John's version of Jesus proclaims, "It is finished" (some render this as "It is accomplished."). That's the sort of utterance you'd hear from someone who has orchestrated events to turn out as they did. This gospel moment, then, is an example of high christology: Jesus is exalted in death.

None of this is to say that Mark and Matthew are entirely low-christological works, or that John is entirely high-christological. I chose the above passages because they illustrate, quite clearly, what the terms mean. The exercise for the budding biblical scholar is to see whether he or she can spot high- and low-christological moments throughout each respective gospel, and to form an opinion, after doing such research, as to which gospels lean more one way or the other. (You can probably guess which way John leans!)

For those who are celebrating Easter: a mindful Good Friday and Holy Saturday, and a Happy Easter. For my Jewish friends, who begin their own celebrations at sundown today: a mindful Pesach to you and yours.

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*Another term used in Christian scholarship is pneumatology: ordered discourse about the Holy Spirit. The three terms together form a trinitarian set: theology (theos), christology (christos), and pneumatology (pneuma).


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les verbes avec l'auxiliaire être

I. Verbs conjugated in the present perfect with "to be" as the auxiliary

I am become Death, destroyer of worlds.

--the Bhagavad Gita, as quoted by scientist Robert Oppenheimer, one of the "fathers" of the atomic bomb

You don't hear it much in modern English, but in older English, it was common for some verbs, especially verbs of motion and verbs of change of state, to be paired with "to be" instead of "to have" in the past tense. The above Oppenheimer quote is an example of this: "I am become Death" instead of "I have become Death." In Stephen R. Donaldson's The One Tree, a powerful character named Infelice arrives at a gathering and announces her presence by saying: I am come.

This should alert you to the fact that not all verbs are conjugated in the present perfect with the auxiliary (i.e., helping verb) "to have." There are verbs that take "to be" as their auxiliary.

II. La maison d'être

In modern French (and in German, too, by the way*), a certain set of verbs is still conjugated in this manner. French teachers teach these verbs in several different ways, but one of the more prominent ways (aside from invoking "MRS. D.R. VANDERTRAMPP"**) is to use a visual aid called "La maison d'être," or "the House of être." See below:

(reference found here)


III. Some quirks of être verbs

As in older English, French être verbs (henceforth "E-verbs") are those that involve motion or a change of state. When you conjugate an E-verb au passé composé, the past participle must agree with the gender and number of the subject. To wit:

Je suis monté(e)
Tu es monté(e)
Il est monté
Elle est montée
Nous sommes monté(e)s
Vous êtes monté(e)(s) (remember, with "vous," this could be singular!)
Ils sont montés
Elles sont montées

Another quirk to remember is that certain E-verbs can also be conjugated with avoir if they're transitive (i.e., transferring their action from the subject to an object or objects; see here for more info). Some E-verbs that can also be A-verbs:

passer: J'ai passé le livre à mon frère. (I passed the book to my brother.)
descendre: J'ai descendu les bagages. (I took down the bags.)
         (Or, more sinisterly) J'ai descendu le flic. (I killed the cop.)
sortir: J'ai sorti la poubelle. (I took out the garbage.)
monter: J'ai monté mon sac à dos. (I've put up my backpack.)

There are others, but you get the idea. If the E-verb is being used transitively, i.e., with an object, then it's actually an A-verb in that context. Be very careful with this!


IV. Les verbes pronominaux et les verbes réfléchis: pronominal and reflexive verbs

There is an entire class of verb that is always conjugated with être: le verbe pronominal. A subtype of this verb is le verbe réfléchi, or reflexive verb. In the infinitive form, such verbs have the pronoun "se" in front of them.

The pronoun se normally means something like "(to) oneself/each other," but sometimes there's no good reason for the se to be there. In such cases, the pronoun se is simply a pronoun and implies no reflexivity. A good example of this is the verb s'apercevoir, to notice/realize, which can be both pronominal and reflexive.

PRONOMINAL: Elle s'aperçoit que son mari n'est pas revenu. (She realizes that her husband hasn't come back.)
REFLEXIVE: Elle s'aperçoit dans la glace. (She notices herself in the mirror.)

Elle s'est aperçue que son mari n'était pas revenu. (She realized her husband hadn't come back.)
Elle s'est aperçue dans la glace. (She noticed herself in the mirror.)

As you see above, the se plays no real role in the pronominal sentence.

Now look at the following reflexive verbs:

Ils se parlent. = They talk/speak to each other.
Ils se sont parlé. = They talked/spoke to each other.

Elles se voient. = They (fem.) see each other.
Elles se sont vues. = They saw each other.

Elle se parle. = She talks to herself.
Elle s'est parlé. = She talked to herself.

Did you notice something? In the above se parler examples, the participle parlé did NOT receive an "e" or an "s"! Why? Because in this instance, the pronoun se is an indirect object. The original verb is parler à [quelqu'un], so parler takes an indirect object. You can see this when parler is used non-reflexively:

Je parle à Jeanne. Je lui parle. (lui = indirect object)
J'ai parlé à Jeanne. Je lui ai parlé. (not "Je lui ai parlée.")
Je parle à Jeanne et Hélène. Je leur parle. (leur = indirect object)
J'ai parlé à Jeanne et Hélène. Je leur ai parlé. (not "Je leur ai parlées.")

Compare this to the past-tense rule for direct objects, where there is agreement in gender and number:

J'ai vu Jeanne hier. (Jeanne = direct object)
Je l'ai vue hier. (the participle is inflected as feminine singular)

Let's look at an example with a reflexive verb:

se donner des cadeaux = to give each other gifts

Ils se donnent des cadeaux. (présent)
Ils se sont donné des cadeaux. (passé composé)

Why not "se sont donnés," with an "s" at the end? Because the direct object is cadeaux, while se (each other) is the indirect object. No agreement necessary.

V. L'intérrogation! (The quiz!)

Ecrivez la bonne locution au passé composé.

1. Elle _____ (se voir) dans la glace.

2. Ils _____ (se donner) des compliments.

3. Il _____ (tomber) de la falaise (cliff).

4. Elle _____ (naître) en 1995.

5. Vous _____ (passer) par le même bâtiment trois fois!

6. Je lui _____ (passer) le sel.

7. Nous _____ (monter) les valises.

8. Elles _____ (descendre) de la montagne.

9. Ils _____ (aller) voir leur famille en France.

10. Tu _____ (sortir) le chien?

LES REPONSES (highlight the space between the brackets to see the answers):
[
1. s'est vue
2. se sont donné (no "s"!!)
3. est tombé
4. est née
5. êtes passé/passée/passés/passées ("vous" can be sing., pl., masc., fem.)
6. ai passé
7. avons monté
8. sont descendues
9. sont allés
10. as sorti
]

Vous avez eu 10 sur 10, j'espère...!

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*German examples:
Ich bin gekommen. = I have come. (not Ich habe gekommen.)
Er ist gegangen. = He has gone. (not Er hat gegangen.)

**The mnemonic "MRS. D.R. VANDERTRAMPP" is an acrostic for remembering a cluster of être verbs. To wit:

Monter (motion)
Rester (motion, or lack thereof)
Sortir (motion)

Devenir (change of state)
Retourner (motion)

Venir (motion)
Aller (motion)
Naître (change of state)
Descendre (motion)
Entrer (motion)
Revenir (motion)
Tomber (motion)
Rentrer (motion)
Arriver (motion)
Monter (motion)
Partir (motion)
Passer (par) (motion)

a very interesting story

Thanks to a friend of mine, I've been made aware of a 1961 short story by a gent named Hal Draper. The title of the story is "MS Fnd n a Lbry"-- a title whose meaning the story makes clear. Read it here. An excerpt from the beginning of the story:

From: Report of the Commander, Seventh Expeditionary Force,
Andromedan Paleoanthropological Mission

What puzzled our research teams was the suddenness of collapse and the speed of reversion to barbarism, in this multigalactic
civilization of the biped race. Obvious causes like war, destruction, plague, or invasion were speedily eliminated. Now the outlines of the picture emerge, and the answer makes me apprehensive.

Part of the story is quite similar to ours, according to those who know our own prehistory well.

On the mother planet there are early traces of *books*. This word denotes paleoliterary records of knowledge in representational and
macroscopic form. Of course, these disappeared very early, perhaps 175,000 of our yukals ago, when their increase threatened to leave no place on the planet's surface for anything else.

First they were reduced to *micros*, and then to *supermicros*, which were read with the primeval electronic microscopes then extant.
But in another yukal the old problem was back, aggravated by colonization on most of the other planets of the local Solar System, all of which were producing *books* in torrents. At about this time, too, their cumbersome alphabet was reduced to mainly consonantal elements (thus: thr cmbrsm alfbt w rdsd t mnl cnsntl elmnts) but this was done to facilitate quick reading, and only incidentally did it cut down the mass of Bx (the new spelling) by a full third. A drop out of the bucket.

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this week's MGRE Math Beast Challenge

From here:

A predator is 80 meters behind its prey, which is running away at a rate of 40 kilometers per hour. If the predator chases at 48 kilometers per hour and both animals run along a straight-line path at their respective constant rates, how long will it take, in seconds, for the predator to catch the prey? (1 kilometer = 1,000 meters)

(A) 10

(B) 24

(C) 36

(D) 80

(E) 100

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


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the answer to last week's MGRE Math Beast Challenge

I blogged last week's MGRE Math Beast Challenge here. Was the answer indeed 4√5, as I had guessed?

Yes, it was. MGRE says:

A rhombus has four equal sides. In this figure, each rhombus side lies on a face of the cube. In order for the rhombus side lengths to be equal, the two corners of the rhombus that lie on a cube edge must lie exactly at the mid-point of that cube edge, as shown:



By Pythagorean Theorem, each rhombus side length is thus √(1² + 2²) = √5. If you don’t quite believe that the rhombus corners we placed at the midpoint of the cube edge MUST be at that midpoint, try placing those corners elsewhere on the edge. If the front right rhombus corner lay, say, 1.5 up from the base of the cube and 0.5 down from the top surface of the cube, the front edge of the shaded shape would be √(1.5² + 2²) = √6.25, while the right edge would be √(0.5² + 2²) = √4.25, i.e., the shaded shape would not be a rhombus.

If each rhombus side length is √5, then the perimeter of the rhombus is 4√5.

The correct answer is D.

We could also apply some common sense and use the answer choices. Each rhombus side length is longer than the edge of the cube, but less than the diagonal of a face of the cube. If x is the rhombus side length, then 2 < x < 2√2. The perimeter of the rhombus must be between 8 and 8√2, exclusive. Approximating √2 as 1.4, the perimeter must be between 8 and 11.2, exclusive.

(A) 8. TOO SMALL
(B) 2√6. TOO SMALL
(C) 4√3. TOO SMALL
(D) 4√5 is between 4√4 and 4√6, which is double choice (B). Between 8 and 9.52. OK
(E) 4√2 + 4√3 ≈ 4(1.4) + 4(1.7) = 12.4. TOO LARGE

I find that that second method takes too long without the use of a calculator.


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