Moroha　諸刃

One of the more interesting skills learned as an engineer or a scientist is the art/skill (it’s really both) of being able to make reasonable order-of-magnitude estimates. Using a combination of knowledge, common sense, reasoning, intuition, and some quick hand calculations, a skilled engineer/scientist will often try and estimate the critical parameter or result of some system before going through the tedious calculations to get a more accurate answer. This is useful for several reasons: it gives you an idea of what orders of magnitude you will be dealing with, it gives you a ‘sanity check‘ for when you calculate a more accurate answer, and most importantly, often times an order of magnitude estimate is all that you really need.

This process is sometimes called handwaving, back-of-the-envelope calculation, guesstimate, or a ballpark estimate. They are also sometimes called Fermi problems, since the physicist Enrico Fermi was renowned for being able to perform such simple estimates and get within a factor of 2 or 3 of the actual answer (this is extremely good for a simple estimate). Two famous examples of are his calculation for how many piano tuners there are in Chicago, and his estimation of the energy yield of the first atomic bomb test by dropping some scraps of paper and seeing how far the blast blew them.

Often times when you are making such estimates, you simply round (logarithmically) each number to it’s nearest power of 10. This is because your estimates are off by factors at least that large anyway, so there’s little point in carrying through precise numerical calculations. A personal favorite anecdote on this principal came from my brother Porter when explaining why he replaced with the number 1, “Why did I make equal to 1? Because it’s not 10.” (In actuality though, since and , is right close to the dividing line between assigning it to the value of 1 or 10. You can choose either, or just make it 3, which is what I usually do.)

The reason why I’m talking about this is because today I stumbled upon this page for a class at MIT that deals entirely with this subject. The name of the course? “Lies and Damn Lies: The Art of Approximation in Science”. I skimmed through the first chapter and it was very well-written and interesting. You could learn a lot about this skill just by reading the chapter and working through the problems yourself.

Another closely related but slightly more accurate method of estimation uses dimensional analysis. The wikipedia article I linked to is a bit obtuse for the uninitiated, but the two examples halfway through the article are fairly simple to follow and stand well on their own. For dimensional analysis you use your knowledge of the underlying physics of a system to make reasonable assumptions about what parameters are pertinent in your analysis. A more mathematically formalized version of this is called the Buckingham π theorem, and is an extremely useful and versatile tool for the initial analysis of a system.

My favorite example is at the end of the article, where it shows how Geoffrey I. Taylor used the Buckingham π theorem to estimate, again, the energy output of the first atomic explosion. A summation of his original paper with his analysis can be found online. Basically he was able to determine the following relationship:

,
where R is the radius of the exploding shockwave at some time after detonation, t is the time, p is the atmospheric pressure and E is the energy released upon detonation. He used recently declassified movies of the explosion to get the radius and time values, and that allowed him to estimate the energy. In fact, when he published his results it caused quite a commotion in the US Defense Department because the energy output of the atomic bomb was still classified at the time and Taylor’s result was far too accurate for their liking!

A couple of years ago I wrote a post where I talked about taking the imaginary number i to it’s own power an infinite amount of times, essentially

.

I showed numerically at least that it converges upon a single value in the complex plane, but then I speculated about what would happen if I did the same for other numbers? I thought it would most likely be a fractal like the Mandelbrot set or the Julia set, but I never got around to actually solving it.

Well, for the class that I’m TA’ing this semester (intro to computing for Chemical Engineers), I got hold of some simple code for making the Mandelbrot set in MATLAB to show the students a fun and interesting example of using for loops. After I did that though, I figured that it would be fairly simple to modify the code to see if taking a complex number to it’s own power created a fractal or not. So my algorithm was the following:

For any point c in the complex plane, let

.
In the limit as n goes to infinity, if z remains finite, then c is within the set. Otherwise it is not in the set.

Since you can’t really evaluate it to infinity, what you do instead is implement an algorithm where you perform the iteration a large but finite number if times, and see if the iteration stays inside a set radius. So here is my fractal. The first plot shows the the complex plane from -20 to +20 on both the real and imaginary axis:

Here the blue football-shaped region in the center is centered at the origin. These points are considered to be ‘within’ the set, since a large number of iterations did not give numbers that were outside of the convergence radius (I set the number of iterations to 50, and the convergence radius to 1000). Any points that are not that deep blue are considered to be ‘outside’ the set, with the color being an indication of how many iterations it took for that point to leave the convergence radius. The dark red points took the least iterations, with the orange, yellow, green, and light blue areas taking progressively more iterations.

I think the feathery wing structures going in the positive and negative imaginary directions are really interesting. They continue up and down seemingly without end, or at least I didn’t see any end for even large plots that I made. This is in sharp contrast to the Mandelbrot and Julia sets which are confined to a finite area.

But the center looks the most interesting, so let’s get a closeup of that.

There’s a lot of interesting stuff going on in this area, with some really beautiful feathery wings, and a jumbled area that just looks like ‘static’. Here’s a zoom-in that let’s us see a little bit of both better.

Wow. Those feathers really are quite beautiful if I do say so myself. That random static-looking area bugs me though, since random noise really isn’t a feature of fractals. Instead they exhibit more of an ‘ordered chaos’ showing infinite complexity. So for my last image here is a much closer zoom on the static-looking area.

This last image has a width and height of just 0.005. Here we can see that it really is infinitely complex and not just random noise. I did another with a width and height of 0.0005, and it looks very similar, so there is some degree of self-similarity, much like we see in other fractals.

So the only thing this set is missing is a name. In reality probably someone has calculated this set and given it a name, but on the slight chance that no one has, I hereby name it the Bassett set. (How’s that for hubris? I bet Benoit Mandelbrot didn’t name the set after himself, but that someone else named it after him in honor of his discovery)

I was chatting with my brother Porter the other day and he told me how he made a Tower of Hanoi game for his children to play with. It’s a fairly simple game with a well-known binary sequence to solving it, in fact it’s one of the few puzzle games where the God’s Algorithm (a shortest-number-of-moves algorithm that can be mathematically proven) exists and is known. The shortest-path solution takes exactly moves to move the disks from one post to another. Porter mentioned in his next blog post that him and his son then calculated how high of a stack The Flash could do if he could move 1 million disks per second, and concluded that it would still take him longer than the age of the universe to complete the 100-disk version.

That’s a good start, but any self-respecting geek has to take things a notch further. First I needed to derive a formula that let’s me calculate how high of a stack I can complete in a given time given a rate of how many disks per second I can move. The original equation we can rearrange as:

,
where N is the number of disks, and n is the number of moves. Then we can replace the n with , where is the frequency of number of moves per unit of time, and t is the total time. So now our equation is:
.
You may notice that I have ignored the . This is because the will be so large that we can ignore the the , as it is inconsequential.

This formula let’s us do some basic calculations. If we take t to be the age of the universe in seconds (about ) then we have the following number of disks we can expect to complete within that time for the given number of disks moved per second:

So we can see that if an immortal Flash were to move disks for the entire age of the current universe at the rate of 1000 disks/s, then he could only expect to complete a stack of 68 disks. Increasing it to 1 billion disks/s only increases it to 88 disks! Still a far cry from completing 100 disks.

So the next question is what kind of energy/power requirement would we need to move the disks at these kind of rates? First we’ll need to make some assumptions on mass and distance. One typical move is shown in the animated gif below (if someone knows how to set it so that it will repeat more than once, please let me know):

In terms of mechanics, to move the disk from column one to column two the disk has to do six steps: 1) accelerate up until its halfway up the column, 2) decelerate until it comes to a stop having just cleared the column, 3) accelerate horizontally until its halfway to the next column, 4) decelerate until it comes to a stop above the new column, 5) accelerate down until it’s halfway down the new column, and finally 6) decelerate until it comes to a stop in the new position.

Now you may say that we don’t need to decelerate the disk as it reaches its resting point and just let it smack into the resultant stack, but when the speeds become really fast the energy will be enough to obliterate any disk, so we’ll include the final deceleration. We’ll also assume this is done in a vacuum so we can ignore wind resistance, otherwise the heat would be more than enough to ionize the disks (those that have read the well known Physics of Santa will be familiar with this). At some point the acceleration alone will become greater than the structural integrity of the disks, as well as relativistic effects at some point coming into play. We’ll get to those in a moment. For now though, we’ll simplify the 6 steps above by making them all the same length for every step. We can do this by putting the three columns in a triangular arrangement and then saying the disks are thin enough that we can neglect the change in the height of the stack as we make progress. We’ll say the columns are 10 cm apart and 10 cm high, so each of the 6 steps will be 5 cm. Also we’ll say each disk weighs 100 g.

Using classical mechanics (ignoring relativistic effects), the time to complete one step is simply 1/6 of a move, so that is:

.
The maximum velocity attained (so that we can know if we’re getting close to relativistic speeds) is:
.
The acceleration the disk undergoes (to see of we are overcoming structural integrity) is:
.
The energy required to complete one step is:
.
And finally the power required to keep the disks moving is:
.
Putting all these together in a handy chart we can see the results.



And the results are pretty interesting. We can determine what the trends are with the formulas above, and we can see how the various values scale with the frequency by replacing t in the equations with . That gives us the following scaling arguments: , , , and . In other words, if we double the frequency of the steps we will double the maximum velocity, but the acceleration and the energy will change by 4x, and the required power will change by 8x! This comes out to be a huge power requirement for the higher frequencies.

To get an idea of how huge the power requirements become, here are some power equivalents on the same order of magnitude:
To get , you would need the power consumption of about 10 DVD drive lasers.
To get , you would need the electrical output of a 1×1 m solar panel in full sunlight.
To get , you would need the power output of a typical automobile.
To get , you would need the power output of a Boeing 777.
To get , you would need the total power received by the earth from the sun.
And to get , you would need about 1/3 the total power output from the sun.
(All order of magnitude power estimates are from this Wikipedia page.)

How does this highest frequency look with our other limits on velocity and acceleration? We should be OK on velocity, since is still only 1/5 the speed of light, slow enough that we can still ignore relativistic effects. As for the stress that the object undergoes due to acceleration, that’s fairly simple to calculate too. Stress is simply force over area, given in this simple formula:

.
Where F is the force and a is the cross-sectional area. We can get the force from Newton’s 2nd law: . For the area we will need to assume a cross-sectional area of the disks themselves. Assuming the disks are about 3 cm in diameter, that gives about 7 in area. Calculating the stress that each disk receives we get the following:

Looking up the strength of various materials, the highest two were for high-impact steel and diamond, both being around 1 GPa . So from the table, we could expect to move 1000 disks per second, but the disks would fail due to the acceleration if we tried to go much faster than that.

Assuming that we could make disks out of impossibilium that can take any amount of stress, then for The Flash to complete a Tower of Hanoi with 88 disks, he would need to move 1 billion disks per second at a speed of about 1/5 the speed of light for 10 trillion years, and he would require the constant power output of about 1/3 of the sun (or a series of equivalent-sized stars, since a single star won’t last that long).

That’s it for this post, in the next post I’ll try to include relativistic effects for even faster speeds.

Except for my quest to find an internet connection, the first few days here in Imazu were pretty uneventful. Usually we’re too jet-lagged to do much anyway, so it’s fine with us. Yesterday though, we took a trip to Fukui Prefecture with Ryoko’s aunt and uncle.

I’ve blogged a little about Fukui Prefecture before a few years ago, when we went to visit the nuclear power plant located on one of the many peninsulas there. Interestingly enough, that is right next to the city of Obama, which has gained some notoriety due to the name of the U.S. President. We didn’t go to Obama (I’ve been there before though when my father came to meet Ryoko’s parents. There isn’t much to the city, to tell you the truth), but I did manage to take the following picture as we were driving through Tsuruga, a nearby town:

That’s an advertisement for Obama Ramen. Unfortunately my timing was a bit off and the right edge is cut off by a telephone pole.

The first place we went to was the Fukui Prefectural Dinosaur Museum. It turns out the only place where dinosaur fossils have been found in Japan are in Fukui Prefecture, and to commemorate it the prefecture has built what I can only call a truly, truly excellent museum. I like to go to natural history museums anywhere I go, and this is one of the best I have ever seen. It’s built inside of a giant metallic sphere, which makes the museum easy to spot from several miles away, and makes for a really impressive ceiling once you get into the main body of the museum.
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Two posts ago I showed how many coin flips it would take in order to have a 98% confidence of getting 92 heads in a row (à la Rosencratz and Guildenstern Are Dead). The answer turns out to be coin flips, which if you tried to do by yourself, it would take 100 billion times longer than the current age of the universe. Since I mentioned that only Wowbagger, the Infinitely Prolonged could pull off such a stunt, I think it’s only fair that any other solution also be Adamsian, at least in practicality if nothing else.

My friend Spencer proposed a Dyson sphere to power a huge number of coin-flipping robots. I think he’s on the right track and I had similar thoughts, however my ideas are a bit larger in scale and less detailed. I won’t go into detail on what kind of Dyson Sphere is best or such, since even simplistic models are fraught with difficulties and instabilities (a nice page talking about Dyson Spheres and some simple analysis is here). Instead, let’s just say that we can build some kind of large Dyson network in order to capture a significant portion of the Sun’s energy. We’ll be conservative and say that after light capture, conversion to useful energy, and then maintenance, etc. we can use 10% of the Sun’s radiant energy to power an array of coin-flipping robots.

Spencer also mentioned the concern that once you have good enough robots, that coin flipping is no longer random: exactly precise robots flipping exactly precise coins in the exactly precise way will give the exact same result every time. That may be the case, but we’ll assume that the robots and coins are made imprecise enough that there will be enough random variance in between all the robots to make the system truly random and fair (this is in all reality probably impossible, but we’re in Adam’s universe so we’ll assume it can be done anyway).

Since the robots don’t have to do anything but flip coins and report the outcome, we’ll say each robot consumes about as much power as a toaster oven, or 1000 W. The sun’s luminosity is , so assuming we can use 10% of the sun’s energy we have:

This many robots would give us the same number of flips every second, so that will give us the required number of flips in:
.
Now that is a considerable improvement.

This potential solution does have some problems though, the most obvious being whether there is enough useful material in the entire solar system to build robots, plus the Dyson power grid to run the whole thing, plus a maintenance system to keep it all in good working order, etc. If we limit ourselves to just the easy to use material, like just the asteroid belt, that limits us to about kg of mass. Assuming a total of 10 kg for each robot (including Dyson network power generation, infrastructure, maintenance, etc.), that limits us to just robots. This number of coin-flipping robots would then take 6.14 years to get the required coin flips, which still isn’t bad at all. It might take several millenia to build the coin-flipping robot Dyson network, but once it was up and running you’d have your 92 heads in a row in just a few short years!

So lets say we let our coin flipping Dyson array keep running, say, until the Sun becomes a red giant in about 5 billion years, destroying our Dyson array. We would have

From this can we calculate how many coin heads in a row we can expect to get during this time? Our initial equation is

where F the confidence probability we we desire (we’ve been using 0.98, or 98%), n is the number of heads in a row, and f is the number of coin flips. Rearranging this for n we have:

For some various confidence probabilities we have these results:

So what does this mean? You have a 99.99% chance of getting at least 120 heads in a row, pretty much guaranteed. However, you only have a 50% chance of getting up to 124 heads in a row. What gives? We go from flipping coins for 6 years to 5 billion years, and the only improvement we get is an additional 28 heads in a row? That’s because each additional head in a row has half the probability of occuring, so the probability decreases exponentially with a linear increase in number of heads required. Conversely, for an exponential increase in the number of coin flips, we see only a modest linear increase in number of expected heads in a row.

A couple of posts ago I talked about how laughably improbable it would be to get 92 heads in a row on a fair coin. To sum up, the probability is:

The probability of this happening is so abysmally low that you could flip coins your entire life and never expect to see this happen. Or could you? How many times would you need to flip a coin to see a reasonable chance of this happening?

A pointless question? Most certainly. But trying to answer pointless questions that can be solved by math is one of the trademarks of a geek. So a quick review of probability and statistics lead me to the Geometric Distribution, which gives the probability of an event occurring after a given number of trials.

Here p is the probability of the event occurring once in one trial, i.e. . k is the number of trials, and P is the probability of the event occurring once within k trials. However, this equation doesn’t quite give us the probability distribution we need. This function will give us the probability of getting exactly one event (heads 92 times in a row) out of k trials (flipping a coin 92 times in a row k times). We’re not interested in the probability of exactly one success, we’re interested in the probability of one or more successes. For that, we need the related Cumulative Distribution Function. Basically it’s the sum of the probabilities of 1, 2, 3,… up to k successes out of k trials. It’s actually pretty easy to derive without performing sums for arbitrarily large values of k. Since we want one or more successes, that means the only thing we don’t want is a failure for every trial. The probability of a single failure is simply , so the probability of k failures is . Since we want every possible combination except every trial a failure, we just subtract this from 1 (the sum of all possible combinations is of course equal to one). This function comes out to be:

Choosing a reasonable number for F, we’ll select 0.98. Solving the above equation for k we get:

This equation is exact, but it has a big problem. No normal calculator or computer is going to be able to calculate the answer because of the denominator. The logarithm of one is zero, so the logarithm of a number very very close to one is a very very small number. But no normal calculator is going to be able to handle
(Note I said no normal calculator. I used Mathematica for this and it works fine. But anything limited to double-precision arithmetic isn’t going to get you there.) Fortunately there is a convenient Taylor series expansion for . It is

The first term in the series will give more than sufficient precision in this case, so we have

This tells us how many trials we we will need to to have a 98% confidence of getting 92 heads in a row, but it doesn’t tell us how many coin flips we will need. Now a trial is defined as 92 coin flips, and if all 92 are heads it is a success, otherwise it is a failure. However, we don’t need to do all 92 flips each time, as soon as we get our first tails, we already know that the trial is a failure and we can start over. Since a fair coin is going to result in tails half of the time, then that means on average we will have two flips for every trial. So if f is the total number of coin flips we will need to get a 98% confidence, then:

This is a very very big number. If we flipped a coin once a second, how long would it take us to get this number of flips?

And how long is this? This is really, really long. Astronomers estimate the current age of the universe to be years old. That puts it as 100 billion times longer than the current age of the universe. This is a feat that could only be pulled off by, say, Wowbagger, the Infinitely Prolonged. According to this fascinating article on the eventual heat death of the universe, at this point there will be no matter left in the universe but white and black dwarfs (and black holes, but I don’t know if they’re considered to be matter within our universe or not), and they will be flung from their orbits due to gravitational radiation.

So is all lost? Is there no way to get 92 heads in a row? For all practical purposes, yes, there is no way. As for impractical purposes though, in a subsequent post I’ll detail a way that we could accomplish it long before the death of the universe using a scheme that in all reality would only make sense in a Douglas Adams’ universe.

I ran across this page that is an outline for a lecture given by a professor of statistics at Berkeley. The title of his talk is “The top ten things that math probability says about the real world”, but he just glosses over six of them and spends the majority of his lecture discussing the last four. Still, all of the points are valid and important, in fact a lot of his lecture covers subjects that are pet peeves of mine. But the one that never ceases to amaze me is is the title of my post: people are predictably irrational in actions involving uncertainty.

Take for example the opening scene in Rosencrantz and Guildenstern Are Dead, where they are flipping the coin. Rosencratz (or is it Guildenstern?) gets heads something like 92 times in a row. Now assuming a fair coin, the odds of that are laughably improbable: . You’d have a millions of times better chance of winning the lottery than achieving this feat. In fact, given a lottery that has a one in one billion chance of winning, you’d have a better chance of winning said lottery 3 times in a row then you would of getting 92 heads in a row on a fair coin.

Proof: .

But that’s not what’s important here. The issue in question is what people will predict the next coin flip to be. If they see the large number of successive identical coin flips, and you then ask them what the probability of the next flip also being heads is, they will usually give one of two answers: 1) It is most likely to be heads, because the coin is obviously ‘on a roll’ of heads. 2) It is most likely to be tails, because it’s had so many heads in a row that there is a ‘negative balance’ of tails that needs to be met. This is despite any and all assurances that the coin is perfectly fair. So the real answer is of course, 0.5 probability of heads, and 0.5 probability of tails. This is always true, no matter what the previous record of instances may be. The thing that many people fail to realize is this:

In any simple game of pure chance, every turn/round/instance is completely independent of previous turns, and and every single turn has the exact same probability every time. This is how casinos make the majority of their money.

So why are most people so predictably irrational in such situations? Obviously I’m not a psychologist (or other such similar profession, but see this slide from the end of the lecture), but I think it has to do with the fact that as humans, we almost never have to make judgments in situations where the outcome is truly random. Such situations have only arisen quite recently in human history with the advent of gambling. And even then there is only a subset of gambling games that are purely random (like craps or roulette, assuming they are truly fair) while many have a combination of chance and skill (card games fall into this category) and some are flat out not fair (slot machines).

In most everything that we deal with in daily life, even when there are events that seem random when we we observe them, they are almost never random. For example, take my daily bus commute. Even though the bus has a regularly scheduled time to arrive, from my perspective it appears random within a time frame of +/- 10 minutes. Also how long it take to arrive at school or home also appears to be random, with a total time of anywhere from 15 to 45 minutes, depending on traffic. But in reality, both when the bus comes and how long it takes to arrive at my destination are not random at all. The problem is that the number of variables that go into determining these two times are so vast and unpredictable that the end result may as well seem to be random when it isn’t.

Back to my former point though, I think we as humans tend to find pattern and correlation in many things (even when they don’t exist) because finding correlations and patterns is extremely useful. Such thought processes have fueled man’s scientific progress, and help humans navigate the dangerous minefield of social interaction. It has its downsides though. People losing lots of money in gambling is obvious, but also things like finding pictures of Mary or Jesus in just about anything, or the existence of most every pseudoscience out there (numerology, cryptozoology, paranormal phenomena, etc.).

Since it was on major news and aggregate sites yesterday, I can pretty much assume that all of the internet has seen yesterday’s story on the super mouse. Basically, researchers have found a way to genetically alter mice so that they have an astounding 10 times the normal number of mitochondria in their muscles! (Quick review: mitochondria are the fundamental energy plants of cells: they breakdown glucose from which they provide energy for all of the cell processes) These mice eat twice as much, but have half the body weight because they have almost no fat reserves. Recalling the movie Bladerunner you might think, “but they die at a young age, right?” Actually, they live longer than normal mice, and were still producing litters at three years of age – equivalent to over 80 years old in human terms.

So what are some possible ramifications? From the BBC article:

Other research groups have produced similar novel rodents by altering different aspects of their genetics. One criticism of the work is that it could open the door to abuse, with the spectre of athletes resorting to gene therapy to try to improve their performance.

But Professor Hanson played this down. “Right now, this is impossible to do – putting a gene into muscle. It’s unethical. And I don’t think you’d want to do this. These animals are rather aggressive, we’ve noticed.”

It makes the animals more aggressive? He thinks that is going to keep unscrupulous athletes from gene-doping? What hole has he been living in? This shows one of the major misgivings I have with medical research in general, and genetic research in specific: no thought for the consequences. The article briefly addresses the issue of athletes trying to gain a genetic advantage, but what about military applications? Who doesn’t think the pentagon or some other nation would pay top dollar for genetic ‘super soldiers’?

The truth of the matter is, no matter how much the public decries it as unethical, as soon as the technology becomes available there will be people trying to genetically engineer a ‘super soldier’ or a ‘super athlete.’ If nothing else, the sheer amount of money put into professional sports and military technology is an indicator that if the technology exists, there will be people willing to take the chance. You just have to genetically engineer a zygote, implant it into a surrogate mother from a 3rd world country who is willing to do such things for money, and have the baby brought to gestation at a facility in a ‘country of convenience’ (somewhere in eastern europe, SE asia, etc.). You now have your genetic superbaby. And what about when gene therapy through retroviruses becomes viable? You can bet there will be athletes willing to give it a chance.

Similarly, as soon as the technology to target and trigger genes for increased intelligence, beauty, immune system, etc. are found, there will be rich parents from all over the world willing to pay an exorbitant sum of money to ensure that their child is a perfect beautiful genius, no matter what laws may be in place to stop such genetic tampering. Even if you pass laws in the U.S. and other 1st-world nations, once the technology exists there will be clinics in India, Bulgaria, Georgia, and such places where the genetic treatment can be done.

I realize this technology has great potential to help cure all sorts of genetic and other health problems, but it really is a Pandora’s box. Professor Hanson’s statement seems incredibly naive to me. This is classic Pandora: he has too narrow of an idea of where and how his technology may be used or abused, and that others – even those with similar technical ability – may not share his sense of morality.

Today on the internet I saw the calculation for i^i, where i is the square root of -1. It can be solved pretty easily using Euler’s formula, where any number z in the complex plane can be expressed as

where is the absolute value of the number, is the base of the natural logarithm, is of course , and is the angle in radians of the number measured counterclockwise from the positive real axis, essentially giving the complex number in polar coordinates. So for example , , , etc. We know that is going to be some number in the complex plane, so we can express that number as , where is itself a complex number.
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Is I mentioned in the earlier post, I would post the answer later. Most people that were kind enough to comment on it also gave the correct answer. I thought the trick was pretty sneaky, so I wrote an email to the author of the website:

Cliff,
It took me a few minutes, but I really had to slap myself on the forehead when I realized the trick. The trick being that *none* of the six cards appear after you click the one of the six eyes. The five that appear after the click are all different. The original six has three red face cards: a king, a queen, and a jack, and three black face cards, also a king, a queen, and a jack. When you click on the eyes, then five face cards are shown. However, for each face card that was there before, the suit of each card changes to the other suit of the same color. A king of diamonds is replaced by a king of hearts, a jack of spades is replaced by a jack of clubs, etc. Then you show five of these cards instead of six, and no matter which of the six they chose, that card will not appear with the five. Since the person playing is only paying attention to the card that they picked, they only notice that the card they chose is no longer there: they don’t notice that all the other cards have changed because they weren’t paying attention to that detail. Quite an ingenius little trick.

I’m curious, what percentage of people figure out the trick to this? I’m sure you won’t post this reply (and I wouldn’t ask you to) on your site since it would ruin the trick for everyone, but I’m certainly not a genius and I highly doubt that I’m the first to figure it out.

Also, I find it hilarious that many people claim it didn’t work and thier card wasn’t removed! I guess they remembered thier own card wrong, or something like that.
–Derek

Alas, Cliff never wrote me back.