I haven’t been blogging much lately.  My apologies.  I’ve found a great new hobby, which has occupied most of my time.  I’ve been solving problems on Project Euler.  Here is a link to my profile there.  Tonight, I just solved my 106th problem, of 222.

“Project Euler exists to encourage, challenge, and develop the skills and enjoyment of anyone with an interest in the fascinating world of mathematics.”

These problems are very interesting and seem tailor-made to someone of my interests.  My major in college was computer science, with a concentration in mathematics.  Actually for Project Euler, a major in math and some comp sci work would probably serve a would-be solver somewhat better.

Most of the problems require some type of computer program.  You’re usually searching for a number with a peculiar property, or adding a bunch of numbers with a similar property.  Many involve prime numbers, which can really stretch a programmer’s technique.  Some are quite unique.  One asks you to evaluate a bunch of poker hands played by two players, asking how many times the first player wins.  Another asks you to programmatically solve a bunch of sudoku puzzles.  One even asks which squares on the monopoly board game are the most popular, statistically.

A new problem has been posted every week since I joined.  I hope they keep this up.  I could be solving these problems for quite a while.  My progress was very quick (several solutions per day) at the beginning.  A couple of weeks ago I had slowed to one solution per day.  Now I have to think about a problem for a day or two before a good solution occurs to me.  I have quite a few I’m thinking about now.

What makes the site so great is that the problems build upon each other.  Knowing how to solve an earlier problem will help solve a later problem.  Efficiency is king, so if your solution takes a few seconds, perhaps the forum for the problem will have a faster algorithm.  This is where the real learning takes place. Some times I’ll wait a few seconds, or even a few minutes for my program to run, only to find that a technique I hadn’t considered gets the answer in a few milliseconds.

Last week I was proud to have solved my 100th problem, which is somewhat of an achievement.  Of all the users that have signed up at project euler, I’m within the top 3%.  My preferred language is python.  Which is good, because while it is one of the most popular languages on Project Euler, it is also one of the most successful.  Soon I’ll write a blog entry about how much I love python.  I just started learning it a few weeks before I started Project Euler.

Users are allowed to submit problem suggestions.  I just sent my first one.  I hope it will be approved, but I understand the approval process is quite long.

Here’s a sad story of a girl called Polly Nomial

Once upon a time (1/t) pretty little Polly Nomial was strolling across a field of vectors when she came to the boundary of a singularly large matrix. Now Polly was convergent, and her mother had made it an absolute condition that she must never enter such an array without her brackets on. Polly, however, who had changed her variables that morning and was feeling particularly badly behaved, ignored this condition on the basis that it was insufficient and made her way in amongst the complex elements. Rows and columns closed in on her from all sides. Tangents approached her surface. She became tensor and tensor.

Quite suddendly two branches of a hyperbola touched her at a single point. She oscillated violently, lost all sense of directrix, and went completely divergent. As she tripped over a square root that was protruding from the erf and plunged headlong down a steep gradient. When she rounded off once more, she found herself inverted, apparently alone, in a non-Euclidean space.

She was being watched, however. That smooth operator, Curly Pi, was lurking inner product. As his eyes devoured her curvilinear coordinates, a singular expression crossed his face. He wondered, “Was she still convergent?” He decided to integrate properly at once. Hearing a common fraction behind her, Polly rotated and saw Curly Pi approaching with his power series extrapolated. She could see at once by his degenerate conic and dissipative that he was bent on no good.

“Arcsinh,” she gasped.

“Ho, ho,” he said, “What a symmetric little asymptote you have I can see you angles have lots of secs.”

“Oh sir,” she protested, “keep away from me I haven’t got my brackets on.”

“Calm yourself, my dear,” said our suave operator, “your fears are purely imaginary.”

“I, I,” she thought, “perhaps he’s not normal but homologous.”

“What order are you?” the brute demanded.

“Seventeen,” replied Polly.

Curly leered “I suppose you’ve never been operated on.”

“Of course not,” Polly replied quite properly, “I’m absolutely convergent.”

“Come, come,” said Curly, “let’s off to a decimal place I know and I’ll take you to the limit.”

“Never,” gasped Polly.

“Abscissa,” he swore, using the vilest oath he knew.

His patience was gone. Coshing her over the coefficient with a log until she was powerless, Curly removed her discontinuities. He stared at her significant places, and began smoothing out her points of inflection. Poor Polly. The algorithmic method was now her only hope. She felt his digits tending to her asymptotic limit. Her convergence would soon be gone forever.

There was no mercy, for Curly was a heavyside operator. Curly’s radius squared itself; Polly’s loci quivered. He integrated by parts. He integrated by partial fractions. After he cofactored, he performed runge – kutta on her. The complex beast even went all the way around and did a contour integration. What an indignity – to be multiply connected on her first integration. Curly went on operating until he completely satisfied her hypothesis, then he exponentiated and became completely orthogonal.

When Polly got home that night, her mother noticed that she was no longer piecewise continuous, but had been truncated in several places But it was to late to differentiate now. As the months went by, Polly’s denominator increased monotonically. Finally she went to L’Hopital and generated a small but pathological function which left surds all over the place and drove Polly to deviation.

The moral of our sad story is this: “If you want to keep your expressions convergent, never allow them a single degree of freedom.”

So true! I laughed so hard at this. The innuendo, the mathematical accuracy, the genius, the humanity! I’ll never look at a polynomial the same way again.

It’s the year 2118, and I walk to work like any other day. The computer simulation research facility at the university is an interesting place to work. Mostly, we try to show what would happen in experiments that are too expensive otherwise. The computer systems have learned to simulate reality to such a high degree of accuracy that the value of real-world experimentation has diminished greatly.

I had taken Friday off, so first on my agenda was to discover what had been done while I was absent.

“Hey Jim,” Barbara said.

“Hello,” I replied.

“You’re going to want to look at this.”

“What’s up?”  I ask.

“Well, we were thinking about the 20th century research of physicists such as Stephen Hawking about what happened in the first moments of the Big Bang. We did some digging, and there is actually enough data and math out there on the subject to start the simulation. We spent the majority of Friday collecting the data and setting up the sim.”

“Okay… what happened?”

“Take a look.”

The computer displayed what appeared to be clusters of galaxies, and zooming in, individual stars within the galaxies. “How long has the sim been running?”

“Oh, we’ve been here all weekend. As you know, time is a variable here, so we advanced the clock about 8 billion years, and you have this. 5 billion years later, check out what happens.” She started to advance the clock and individual stars began to form planetary satellites. “Out of curiosity, we cataloged the stars with 8 planets, and came up with a surprisingly short list… only 350 million or so. Of these, maybe 10 million or so had planets with masses similar to our own solar system. And after some more digging, we found this planet.”

A blue-green globe appeared that looked shockingly familiar. “Is that…?” I asked, taking a closer look.

“It gets better. I zoomed in further, and the continental shelfs appeared very similar to our own. We’d have to get some geologists in here to date it, but we figured we could just advance the clock until we began to recognize our own planet more and more. It turns out it wasn’t as hard as you think. Civilization really changes a planet.”

“You’re not saying you can see us, I mean, humans, can you?”

“Yes.” Giggling, she advanced the clock further. The clock was now using the Gregorian calendar, today’s date, only an hour or so ago.

I saw myself taking a shower. “What is this, some kind of joke?”

“Sorry, but no, It’s not a joke. In fact, we’ve discovered we can’t advance the clock past the current time. Let me show you.” She changed view and pushed the clock as far as it could go. There was an image of the two of us, huddling over the computer monitor.

I jumped around, looking for the camera.

“There’s no camera. We can change the angle, if you want proof.” She rotated the view again, and the angle of the view of ourselves changed back and forth, so we could see ourselves from all angles.

Looking around, I saw no movement in the room.  Nothing was recording us.  I was dumbfounded, because  there I was, looking at the computer.  “How is this possible?” I finally managed to blurt out.

“We’re not sure. But it gives rise to some really interesting questions,” she said mysteriously.

“I’ll say. Turn it off… this is starting to freak me out.”

“I’m not sure we want to do that, actually.”

“Why not?”

“Well, think about it. In this simulation is a simulation of us, looking at another simulation of us, and so on, ad infinitum.” She explained.

“Right, the chain of simulations continues on forever.”

“Exactly. Now ask yourself, what is the probability that we are at the top of this chain of infinite simulations, and we are the real people and everyone below us is the simulation.”

“Huh? But you built this simulation yourself.”

“Well, I might be a simulation of the real me, in another universe above us. There is no way to tell. The fact that we can’t advance the date beyond the current time seems to indicate something. If we turn this computer off, we destroy the simulations, the universes below us. And if we’re just a simulation, we do it because the universes above us will also turn their computer off, destroying us, and everything we know.  So we can’t ethically shut down the simulation.”

“Yeah… I agree.  Don’t turn it off.”

I noticed that this severely crippled Internet Explorer’s estimation algorithm for the length of the download, which works in a very specific way, and was thrown off given the circumstances of the download as described above. It would give a very fast estimated time to completion, then gradually go up as it realizes the rest of the download will not go as fast as the beginning. At some point, the download time will start to give a more realistic estimate, and will start to count down almost as if the download speed had been constant the whole time.

So I came up with an interesting set of math problems: some basic, some not so basic. I’ll present the answers to each question after I state the problems.

A download of a 50MB (51200KB) file starts at a constant 120KB/S transfer rate for the first minute. After this, it continues at 40KB/S for the duration of the download.

1) Calculate the time needed to complete the download.

A program computes the estimated time remaining for the completion of the download by taking the average past download speed (total amount downloaded divided so far by total time taken so far) and projecting this speed forward, to the end of the download.

2) Give the value of the estimation after 1 minute.

3) Give the value of the estimation after 5 minutes.

4) Give the value of the estimation after 10 minutes.

5) Give a function f(t) whose input is the number of seconds from the start of the download and whose output is the value of the estimation algorithm, described above.

6) Give the maximum value for this estimation throughout the length of the download.

7) How much time into the download is the estimation clock running backward exactly half as slow as real time?

Answers:

1) The answer to the first few questions are easy, and require only simple arithmetic. After the first minute, 60*120 = 7200 KB will have been downloaded, leaving 44000 KB left. At 40KB/s, the rest of the file will take 1100 seconds, for a total of 1160 seconds, or 18 minutes, 20 seconds.

2) After 1 minute, 7200 KB will have downloaded in 60 seconds, giving 120KB/S. Projecting 120KB/s forward gives 44000/120 = 367 seconds or 6 minutes, 7 seconds.

3) After 5 minutes: 1*60*120 + 4*60*40 = 16800 KB in 300 seconds, or 56 KB/s. Projecting forward gives 34400/56 = about 614 seconds or 10 minutes and 14 seconds.

4) After 10 minutes, 1*60*120 + 9*60*40 = 28800 KB in 600 seconds, or 48 KB/s. Projecting this forward gives 22400/48 = about 467 seconds or 7 minutes, 47 seconds.

You can see from the values here that the estimated time grows after the first minute, and begins to decline again at some point before the tenth minute.

5) The formula will need to be piecewise. The time, in general, is the amount not downloaded divided by the average speed. f(t) for t > 60, it is (51200-(120*60+40*(t-60))/((120*60+40*(t-60))/t). This last formula is the interesting one, and simplifies to 1280*t/(120 + t)-t.

6) local mins and maxes are where the derrivative equals zero. The derrivative of the formula where t > 60 is 153600/(t^2 + 240*t + 14400) – 1. This is zero when t is approximately 272, or 4 minutes and 32 seconds. The value of the estimation function at this point is about 616, or 10 minutes, 16 seconds.

It is interesting that over the course of the 18 minutes and 20 seconds taken to complete the download that the value of the estimation function never goes above 10 minutes and 16 seconds.

7) The question is asking, in other words, “what is the value of the function where its derrivative is -.5?” So we have -.5 = 153600/(t^2 + 240*t + 14400)-1… or t^2 + 240t + 14400 = 307200. Solving for t we get approximately 434 seconds, or 7 minutes, 14 seconds. The estimation at this time is roughly 569 seconds, or 9 minutes, 29 seconds. So basically, two seconds later (436), it will say there is 9 minutes and 28 seconds to left… running backwards half as fast as real time.

Math is so much fun.