Math for an MBA, Part 2

The first part of the story is here, along with the questions that took about an hour to consider and another hour to conceptualize. After some deliberation, I settled on four areas that seemed to fit into the "MBA Entrance Exam" mold: Percentages, Averages, Multiple-Variable problems, and Logical Reasoning.

Percentages are simple problems involving percentage calculations: "What is 75% of 40?" belongs in this mold. They have a very practical real-life application in bank lending and interest rates, occasionally make guest appearances in Accounting problems, and tend to show up on project status reports. As a result, I felt that a refresher on these would not be amiss.

Averages are similarly easy, although I wanted to approach them from a roundabout direction. Most problems in this area involve calculating the average of a set of values; I wanted to turn this on its head and ask a question where you needed to get a specific value in a series, given the average as one of your clues.

Multiple-variable problems, on the other hand, were certainties. I had given a friend some help on one of these for her MBA class before, so if there were any problems that were likely to show up, it would be these. You've probably seen them before; these are the tedious word problems where you have to figure out the values of more than one item.

Logical reasoning is harder to describe. This is not a subset of problems, mind you, as much as it is the ability to organize given information, identify an unknown value, and then use one to work towards the other. It's obviously used extensively in business, although in a less mathematical sense. But I'm convinced that it's the mathematical training that helps us apply it to non-math outlets.

Given this outline, all that I had left to do was to write the questions:

1. Three chickens can lay three eggs in three days. In how many days can you expect 18 chickens to lay 18 eggs?

This was a question of logical reasoning, plain and simple. I see it appear in a lot of IQ tests and other cognitive exams, and I've seen some very smart people give some very wrong answers as well. Most people, in fact, will trust in their faulty pattern-recognition senses and say "18 days!" right off the bat.

The answer is a lot more mundane. If three chickens lay three eggs, with all other factors being equal, then it stands to reason that each chicken laid one egg. If these three chickens laid those three eggs in a matter of three days, then it follows that each chicken needs three days to lay one egg. Therefore eighteen chickens would come up with eighteen eggs in those same three days.

2. You have exactly Php 35,000.00 in a bank account that gains 2% compound interest per annum. Assuming that you neither deposit nor withdraw any money from that account, how much will the account contain after two years?

I wanted to throw in at least question that implied a real-world application of a mathematical principle, and this was that question. The easy way to solve this is to just get ((35,000 x 102%) x 102%) for a total of Php 36,414.00. That said, I only realized afterwards that banks hardly use the term "compound interest" anymore, which puts up the wall between theory and practice again.

3. I need an average score of 93 among my exams in order to pass one of my courses. So far, the grades that I got in five earlier exams were 90, 97, 87, 100, and 86. What is the minimum grade that I should get on the sixth (and final) exam in order to pass?

This is a reverse-average problem that is commonly known as "the Student's Dillemma", and I'm sure that a lot of people out there learned to put these calculations together at some point in their academic lives. I won't cover the answer here, as it's really just a throwback to what we were all probably doing around our final exam weeks.

4. A bicyclist travels at a steady rate of 8 kilometers per hour. She leaves her house at 2:00pm and rides her bike to the supermarket. Halfway there, she realizes that she's forgotten her shopping list and returns home to get it, then sets out for the supermarket again. She arrives there at 4:30pm. What is the distance from her house to the supermarket?

I put this problem here for one basic purpose: It encourages the solver to draw a chart. I feel that visualization is an important part of logical reasoning — if you can envision the cyclist's journey in your mind, then so much the better, but if not, you can just doodle something that lets you conclude that she travelled a total of twice her original intended distance from 2:00 to 4:30. That means that she normally travels the path to the supermarket in 75 minutes at 8 km/h... which makes the distance 10 kilometers.

5. A 200-liter mixture is comprised of 20% water, 30% salt, 10% sugar, 15% sand, and 25% gold. This mixture is left out in the sun for a few hours, after which all the water is found to have evaporated. What percentage of the resulting mixture is made up of gold?

And now the problems get a whole lot harder. I set up this percentage problem to illustrate the fact that percentages are non-constant values, and that they change with the introduction or removal of new factors. The easiest way to get the answer here is to realize that you're just looking for an equivalent of 25 parts out of the remaining 80 units, which makes 31.25%.

One interesting quirk about this problem was the fact that you technically don't need the volume of the original mixture to solve it. In fact, you can give the original mixture and quantity you want and the answer will still be the same... but I wasn't about to introduce that to people who had spent years away from their high school math classes.

6. A motorboat needs three hours to travel upstream, but it only needs one hour to travel downstream. When there is no current, the motorboat moves at a constant four kilometers per hour. What is the rate at which the river's current flows?

This took things a little further; It's actually rather difficult to solve if you slept through most of your math classes. In fact, it's another problem that encourages you to draw... although a chart instead of a diagram is needed in this case.

Rate-Time-Distance problems like these usually need a bit of background. You need to know that Rate x Time = Distance, of course, but you also need to know that an opposing force will lower an object's effective rate of travel (and vice-versa). Ergo, the river will slow you down by its own rate when you go upstream, but it'll make you go faster by the same rate when you go downstream.

Assuming that the river's rate is R, we get:

(4 + R) x 1 = distance travelled downstream = distance travelled upstream = (4 - R) x 3

From there, it just boils down to:

(4 + R) = (4 - R) x 3
4 + R = 12 - 3R
4R = 8
R = 2 km/h

7. Three bowling balls and four frying pans weigh 54 pounds in total. Four bowling balls and one telephone weigh 54 pounds in total. Three telephones and eight frying pans also weigh 54 pounds in total. What is the total weight of one bowling ball, one frying pan, and one telephone?

This is the classic three-variable problem: Three unknown quantities, and three equations. I chose the objects completely at random only because I like choosing objects completely at random.

The interesting part is that I deliberately screwed around with the numbers here — while a bowling ball weighs 12 pounds and a telephone weighs 6 pounds, a frying pan weighs 4.5 pounds. I find that some solvers normally get thrown off by the decimal value for some reason, perhaps because it makes them think that they're on the wrong track.

8. Anthony, Beatrice and Charles win the lottery on a single ticket. They decide that they will each take 30% of the total, and then set aside the remaining 10% for future needs. After the money is deposited in their bank, however, each of the three friends arrives separately to claim their share. Anthony arrives first and withdraws 30% of the money. Beatrice arrives a few hours later, and withdraws 30% of what's left. Finally, Charles arrives some time later and withdraws 30% of what's left. At this point, only Php 205,800.00 is left in the account. How much did the three friends originally win in the lottery?

This is actually a problem that gets featured in a lot of puzzle books. While it's possible to solve this by means of basic algebra, the circumstances of the problem tend to leave people confused on where to start. Yes, this actually centers more on logical reasoning than percentages. Yes, I threw the two of them together to try and confuse my solvers further.

Problems like these encourage logical thinking — they force the solver to stop, think, and determine their battle plan before trying to tackle the problem. In this case, the way the logic should go is that that 205,800 represents 70% of the money that Charles saw (before he took his 30%). This amount represents 70% of the money that Beatrice originally found in the account, which is 70% of the money that Anthony found in the unblemished account. Therefore:

205,800 is 70% of the money that Charles saw...
— Charles originally saw 294,000 in the account.

294,400 is 70% of the money that Beatrice saw...
— Beatrice originally saw 420,000 in the account.

420,000 is 70% of the money that Anthony saw...
— Anthony originally saw 600,000 in the account.

*

And now I must admit that it all turned out to be for nothing, because we both ended up so busy the night before the exam that we never got around to the problems. I managed to fire off a quick question about chickens and eggs (which my friend got wrong), but we otherwise weren't able to go through the eight items above.

The next morning, my friend called to tell me that the exam was much easier than he expected, and that math only played a very small role in this regard. There were plenty of real-world logical scenarios and a few questions of general knowledge, but nothing beyond the simplified "What is 75% of 40?" percentage problems that I mentioned at the start of this article.

I suppose it's too much to ask. Math is more a tool for mental stimulation in schools, something that acts as a precursor to the logical thought that we use when we're older. You can't expect business professionals to maintain passion (much less practical use) for these theoretical concepts... especially when it comes to a post-graduate program that concentrates on cooperation and networking.

The test still sits on my desk, however, waiting for the next time that another MBA applicant asks for my assistance. On top of that, I have more word problems where that came from. It's only a matter of time before somebody asks me for another favor...

Math for an MBA, Part 1

A few days ago, someone asked me to give him a hand with the MBA entrance exam he was taking on Saturday morning. "You still remember what we learned in Math class," he told me, "and you obviously practice it a lot."

"Yeah, well... the math in an MBA course isn't that tough. I've helped out at least one other person with his MBA homework, and it didn't go any farther than algebra and number theory. It's high school stuff."

"I don't remember most of what we studied in high school. And it's the entrance exam... who knows what questions will come out? You know this better than I do, so give me a hand here."

I admit that I couldn't deny him the favor, and I suppose that part of me was itching to do some heavy-duty tutoring, so I said yes. Since we only had about one night's worth of time to go through the basics (us being working stiffs and all), I needed to gather my thoughts and imagine what kind of math was likely to come out in a bunch of MBA classes.

Eventually I came up with a list of eight questions for discussion, which I'll post here. I've come up with Math tests before (mostly for my siblings when they were in school), but this is the first time I've come up with a series of questions for an MBA applicant. If there was any point to this entire exercise, it lay in the matter of what was likely to come out, and what was not.

I eliminated such irritants as Calculus and Trigonometry right off the bat, for example. I felt that these were highly unlikely to come up in a Business Administration course, because they obviously require a lot of theoretical background and advanced thinking. Geometry was the next to go, because as basic as the math is, the concepts didn't apply to management principles. Probability and Combinatorics remained by the wayside as well, although those were among my favorites to explore.

After an hour's worth of thought, I had pared down my ideas into a limited set of concepts that felt as though they belonged in an MBA environment, and were fundamental enough to appear in an entrance exam. From these, I pulled together eight questions that my friend and I could discuss, and I now post them below for your viewing pleasure:

1. Three chickens can lay three eggs in three days. In how many days can you expect 18 chickens to lay 18 eggs?

2. You have exactly Php 35,000.00 in a bank account that gains 2% compound interest per annum. Assuming that you neither deposit nor withdraw any money from that account, how much will the account contain after two years?

3. I need an average score of 93 among my exams in order to pass one of my courses. So far, the grades that I got in five earlier exams were 90, 97, 87, 100, and 86. What is the minimum grade that I should get on the sixth (and final) exam in order to pass?

4. A bicyclist travels at a steady rate of 8 kilometers per hour. She leaves her house at 2:00pm and rides her bike to the supermarket. Halfway there, she realizes that she's forgotten her shopping list and returns home to get it, then sets out for the supermarket again. She arrives there at 4:30pm. What is the distance from her house to the supermarket?

5. A 200-liter mixture is comprised of 20% water, 30% salt, 10% sugar, 15% sand, and 25% gold. This mixture is left out in the sun for a few hours, after which all the water is found to have evaporated. What percentage of the resulting mixture is made up of gold?

6. A motorboat needs three hours to travel upstream, but it only needs one hour to travel downstream. When there is no current, the motorboat moves at a constant four kilometers per hour. What is the rate at which the river's current flows?

7. Three bowling balls and four frying pans weigh 54 pounds in total. Four bowling balls and one telephone weigh 54 pounds in total. Three telephones and eight frying pans also weigh 54 pounds in total. What is the total weight of one bowling ball, one frying pan, and one telephone?

8. Anthony, Beatrice and Charles win the lottery on a single ticket. They decide that they will each take 30% of the total, and then set aside the remaining 10% for future needs. After the money is deposited in their bank, however, each of the three friends arrives separately to claim their share. Anthony arrives first and withdraws 30% of the money. Beatrice arrives a few hours later, and withdraws 30% of what's left. Finally, Charles arrives some time later and withdraws 30% of what's left. At this point, only Php 205,800.00 is left in the account. How much did the three friends originally win in the lottery?

While it would be easy for me to just put up the answers to each of these and call it a day, that wouldn't be enough for a second article on this subject. I feel that there's a specific reason why I prepared each one of these questions... and that these reasons are worthwhile to discuss.

That said, those words will have to wait till tomorrow, I think. It's almost two in the morning at this time, and in the intervening hours, you're welcome to have a crack at these. I know that it's no Sudoku, but wouldn't you be interested to see how much you remember from your high school math?

Export to Hard Drive

I spent a few minutes doing a backup of this blog a little while ago, based on Dominique's revelation that Blogger now has an Export feature. That's not to say that I'm paranoid and constantly afraid that Big Brother is going to delete all my records one day, but I have had a lot of words posted here for the past four or five years.

To be honest, the Multiply site that I set up some months ago already acts as a backup — that is, whenever it's picking up my posts properly. However, I feel more comfortable knowing that there's a method that allows me to be responsible for making my own archives, as opposed to worrying about a point in time when I might lose everything.

That, and I've just realized that all of my posts since 2004, all of the code and comments and image links, comes to a mere 7.1mb on my hard drive. I feel so inadequate.

Then I mused on the fact that I have a lot of words posted here, so I opened up the XML file in a browser window, then took advantage of the clipboard in order to copy and paste the resulting mishmash into a Word document. The whole operation took me ten whole minutes at my current level of memory (and thirty more minutes because MS Word insists on annotating every single spelling and grammatical error in its documents), but I eventually got the numbers: The backup file contained about 3.6 million characters in 640,000 words. Given a conservative estimate that I wrote about 70% of those words, that puts my total so far at about 450,000 words scattered throughout this blog. I'm no slouch there, at least.

That also implies that I write an average of five or six letters per word. This is a surprise, considering my habit at wordiness.

I'm now working on exactly what kind of backup schedule and procedure I should use. I mean, I'm not sure if it's a smart move to produce archives only when I feel like doing so, not in an age where our government is debating legal standards for open reporting. Pushing a seven-megabyte file each time I post something new, however, will just eat my time in front of the modern boob tube.

Or I could just forget about my budding paranoia and just post more often. But then that would discourage me from playing with all the lovely numbers.

Ah well.

A Man's Psychology Can Be Inferred by His Choice of Street Fighter

Ryu. You are an intensely competitive person. You exercise yourself both mentally and physically before an endeavor; you don't like walking into situations unprepared. You are incapable of thinking outside of the box, and instead resort to tried-and-tested methods in order to achieve your goals. You use these methods for a reason, mainly the fact that they work... and on top of that, you know exactly how to implement them to the best of your advantage.

Ken. As above, except that you're far more flashy and charismatic. You've got the mad skills — but some time ago, you realized that Number One happens to be a lonely place. You've played the field and are comfortable in the knowledge that you know how to deal with things... enough to go about them with a casual attitude and a sardonic smile on your face. You're really not as good as you think you are, but you're not going to lose as long as people think otherwise.

Chun-Li. You think on your feet. You're a guarded, cautious person who prepares a strategy beforehand and fully expects it to fall apart... which means that you have plenty of practice making last-minute changes. Reaction is almost instinctual in you, and life for you can be reduced to a series of moves and countermoves against the people you play with. You know when to push and you know when to wait; it's all part of the same game.

Guile. It's not about how well you do in the world, really. Instead, you feel that it's all about how you can take advantage of the opportunities that you observe. You are constantly on the lookout for openings — weak spots where you can drive your point home, or at least areas where you can push your own agenda. You're not into self-improvement as much as you are into using your strengths: If someone breaks one of your arms in a fight, you're just going to hit them with the other one. The only way people can get past you is if you let them through.

Blanka. You like to get up close and personal; the best way to solve a problem is to throw yourself into the solution. You have a lot of scars from experiences gone wrong, and often your misadventures outnumber your achievements... but your successes are held in high enough esteem that people don't consider you a slouch. You may not be strong or smart enough to hang with the best of them the first time, but eventually you'll get there. Just expect to get a lot of pain for all your trouble.

E. Honda. People don't expect much from you, and that's where the secret lies. You like being the underdog, the person who doesn't seem to hold much promise until you snatch success out from under everybody's nose. You're not as fast as them — but you're fast enough. You're not as skilled as them — but you're skilled enough. Your plan is to let them settle into their preconceived notions for as long as possible... then pull the gloves off and show them how you really roll.

Zangief. You hit things to make them work. You don't pretend to have an understanding of the basic nature of things like all the others claim to have, and that's because you don't. Why bother gathering every bit of information you can, when you just have to know what to do? That's the way the world works: You just need to point yourself in the right direction and pull the trigger. Everything else is just an irritating buzzing noise in your ears.

Dhalsim. You like surprising people, especially when you can defy their expectations so completely. If they expect you to go left, then you go right. People are at their worst when they're extremely predictable, and that's how you discovered the wonder of creativity and random choice. No one can ever figure out exactly what you're doing until the last moment, and that's when they find out that your eye has been on the prize for a long time. They may be ready, but they certainly won't be ready for you.



* No, this is not a serious psychological study or profile. This is a completely random article with no academic basis whatsoever, so lay off any accusations that I've favored one character over another. Street Fighter™ is the property of Capcom Co. Ltd., and I assure them that this post was written for parodical purposes (more as a curiosity than anything else). Don't sue me, or else I'll go all M. Bison on you. Psycho Crusher!

Six-Word Stories

Here's a few six-word stories for Bahay Talinhaga's Independence Day effort. Because I like challenges, I've tried to do one for each of various genres. Because I have masochistic tendencies, I based these off of the ones available in the Talecraft storytelling game. And because I'm borderline suicidal, I've tried to place all of these in a contemporary Philippine context, something that I usually don't do in the course of my regular writing.

One conclusion I've developed, mind you, is that I'm too wordy to squeeze all of my ideas into a mere six words. That said, that probably just means that the items below are just the tip of the iceberg that happens to be my warped little mind.

Here you go, Mr. Chikiamco. Hopefully these are of good enough quality for you.


Adventure: Intrepid archeologist seeks lost Kho videos.

Fantasy (Bangsian Fantasy): Demon runs day spa for celestials.

Fantasy (Contemporary Fantasy): Secret squatter society battles subdivision monsters.

Fantasy (Fairytale Fantasy): Princesses. Paedophilia. Politicians. Put something together.

Fantasy (High Fantasy): Spellcasting cop shakes down otherworldly citizens.

Gothic: Strange giggling noises in Manila Cathedral.

Historical: Ferdinand Magellan's last words: "Nice... sword..."

Horror: Balut vendors terrorized by giant duck.

Mystery: Imelda's three thousand shoes are missing!

Romance: Is that MMORPG girl really female?

Science Fiction: Solution to Abu Sayyaf: Giant Robots.

Thriller: Must... stop... president... from... running... again!

Sleep Experiment

Nope, this isn't really a post. At least, that's what I keep telling myself.

I just happened to remain in the office for about fifteen hours today — almost double the recommended daily allowance — because a critical web site went down at about five in the afternoon and we had to spend a good part of the evening trying to straighten it out.

And that leaves me tired. Dog tired, in fact.

I think that I have only a few minutes left before I doze off. To be honest, I'm only tapping away at the keyboard because I'm curious to see how long I can last before the dowsiness overwhelms me and I'm off to crawl under the blankets. Or even better, I can drop off with my face on the keyboard and go to work the next morning with QWERTY printed across my forehead.

My eyelids are drooping a bit. I imagine that I'll finally start nodding off when I begin hallucinating, and I'm surprised that I know exactly what it's like. Strangely enough, there's no single bit where your vision blurs — you just start seeing things right off the bat. After a while, it becomes easier to tell which items in your line of sight are real, and which ones are not... or at least, that's what the purple giraffe is telling me right now.

And then there's the alternating urge to speak both French and Latin, in that sequence. And then a period of awkward silence as I wonder why I would invest my time in such an effort when I could be speaking Swahili instead.

That mattress would be good right about now.

I sometimes worry about my teeth. I haven't been grinding them in my sleep for a while now, but I still worry. I haven't had any nightmares in a while, but suffice to say that the idea of waking up with nothing but nerves and gums ranks foremost among them.

Now I begin asking myself nonsensical questions, which is a clear indication that I'm in the home stretch. Are bananas funny, for example? Will magpies steal shiny items that aren't necessarily metal or bits of glass? What pieces of technology finally killed the tape recorder, and why does the LP yet survive? How many pieces of taffy does it take to change a bottle of Spanish vodka?

And now I begin swaying from side to side — literally — and backtracking throughout this post to edit the numerous spelling errors that I'm starting to make. Even when I'm sleepy, I'm an obsessive-compulsive proofreader.

And now I slowly nod off. The very air feels heavier now. The silence feels oppressive, enough to make me want to escape into the noise in my mind. There was a time when I would be using this time to write a thousand straight words in the span of a single hour, but that time is fairly distant. I've gotten old.

Soon it shall be a question of sleeping at eight in the evening and getting up at one in the afternoon. Exist but don't exist. Slumber bluff, they call it — to pull a fast one on the caterpillar and the Sandman. Fred Flintstone would be proud of the bowling ball.

The Enter button becomes the focal point for my existence, and I have this desire to begin listing down random dates off the top of my head. The kangaroo reaches the goal post, and wondrs aloud as to why I insist on counting sheep. They have a poker table set up in the next meadow, and the one on front is offering me two-to-one odds on a full house. In the corner, an octopus plays the saxophone.

Truth is a question of black and white, but Bogart calls them all shades of gray. Listen to me now, Humphrey... you should never have switched from scotch to martinis. No wonder the moon smiles; we are all its lunatics.

Disclaimer: June 2009

MANILA, PHILIPPINES — A sixteen-year-old amateur writer was injured this morning in what investigating politicians called "a brazen attack against a controversial Internet personality".

Internet writer Adaspera P. Astra emerged with a fractured wrist and a bruised superego after having been clubbed vehemently with confrontational comments and sharp rhetoric through her own weblog. The incident occurred at about 10:39am, after Astra had participated in an impromptu online debate the previous evening.

"I just turned on my blog and there were, like, three thousand comments there. Like, they wanted to crucify me!" the luckless writer declared.

Astra had previously been identified as the perpetrator of a number of plagiaristic attacks on the Internet, where she would obtain content from other weblogs and personal profiles and post it as her own original writing. The teenage blogger had raised ire for her derogatory responses to the accusations, particularly because the results were easily identified as incompatible with her personal speech patterns.

"It's, like, injust, you know?" Astra commented. "I just want to get, like, more readers kasi eh. How's Google Adsense supposed to, like, pay me money if people don't read my site, ha? It's not like information isn't free on the web."

Online behavioral analyst Sean Uy was among the first to respond to Astra's original statement. "What Astra does not realize," Uy said, "is that while information is often free to obtain on the Internet, this privilege does not exempt users from following standard copyright regulations. Every article has an author and originator regardless of where that article is published. While only some of those articles hold conditions as to how and when they can be used, it must be presumed that standard copyright law applies to all such articles."

The subject of at least three class-action suits, including one from the southwestern United States, Astra had no immediate response to the charges leveled against her. "I'm, like a celebrity!" she said in a UltraMegaPinoyChat forum twelve hours before the attack. "They don't know who I am! Why should they, like, care, ha?"

Uy, who runs a four-year-old blog, clarified the situation further. "Many authors don't like seeing their work used without permission. There is a real risk here — the risk that the author will not be recognized for a project in which he or she has invested some amount of effort, the risk that someone else will profit from the unwarranted theft of such effort, and the risk that the work will be twisted and interpreted out of context.

"I'm proud that I've written and originated all of the works on my blog," Uy said. "While I occasionally use the works of other authors, I try to leave them the correct acknowledgements so that their efforts are recognized. I even remain willing to open dialogues with other writers in case they feel that there's something wrong with my usage. Miss Astra, on the other hand, has originated no such works, has taken no measures to recognize others' efforts, and goes as far as to take offense against her infamy."

News of the incident was a matter of public record within government congressional hearings by late afternoon of the same day, during which Rep. Isagani Batongpuso denounced the incident as an addendum to his speech on Charter Change.

"It is vital to the safety and security of this country that the rights of even incredibly insignificant wrongdoers like Adaspera P. Astra are protected by the Constitution," Batongpuso noted. "It is with this consideration that we must move forward to establishing a constituent assembly before our opponents in the majority voting bloc decide to come back from their foreign junkets."

"Philippine Law holds allowance for copyright law as established even among digital publications," Uy mentioned, "although it is formative in the face of the Internet age. With that said, a lot of copyright disputes could be circumvented if people just asked for the use of articles. People can ask me for the use of my work, for example, and I'm usually willing to let them use it with little more than a byline or link to my site.

Uy also discussed the presence of several online organizations who had begun to help support and enforce authors' rights on the Internet. "I subscribe to a Creative Commons License myself," Uy said. "It's right on the lower area of the sidebar in my main web site."

On Astra's side, the controversial blogger is still unwilling to disclose exactly how she encountered her fractured wrist. "It could have come from, like, all the pangugulit that my enemies do," she said. "But we don't even have to be enemies. They can, like, just keep writing their stuff."

One responding doctor, who declined to be identifed, had a different opinion: "It looks like she just slammed her hand down on her keyboard in frustration," he said. "You can see the letters imprinted on her arm."