Knights and knaves

This kind of puzzle makes use of knights and knaves. A knight is a person who always tells the truth. A knave is a person who always lies. This sort of puzzle was invented, or at least popularized by Raymond Smullyan. They are fun times.

Classic #1:

You are trying to reach the legendary Knight's Village, where everyone is a knight. But you've reached a fork in the road, and you know that if you take a wrong turn, you will instead end up at the Knave's Village, where everyone is a knave. The fork is unmarked, but you see a villager, and resolve to ask him for directions. What single question should you ask him to get to the Knight's Village?

Classic #2:

You are in a labyrinth, and have reached a fork. One of the paths lead out, and the other will certainly lead to your doom (it's that kind of labyrinth). Two guards stand nearby, identical twin sisters. You have heard of these guards, and know that one is a knight, while the other is a knave. What single question can you ask the guards to get out of the labyrinth?

Comic comes from xkcd. Phrasings of the classics are mine. Some of my readers have already seen the classics, so I also wrote a bonus problem just for you.

Bonus problem:

Twenty-five knights and knaves sit around a round table. They all know each other well, but I don't know any of them. But I know that there's at least one knight at the table. I ask them all to point to the next knight on their left. Each one of them points to another person at the table. To my disappointment, even after carefully considering their answers, there is not a single person at the table who is certainly a knight, or certainly a knave. However, I now know how many knights are at the table in total!

How many knights are at the table?

As usual, solutions are posted. Hesitating to look is highly encouraged.

Solution to "Find the fake coin"

See the original puzzle

Reader DeralterChemiker was able to solve the first problem:

Take any three coins and weigh them against any other three coins. If they weigh the same, the heavy coin is one of the other two; weigh those two coins against each other, and the heavier coin will be identified.

If one set of three coins is heavier than the other, take two coins from the heavier set and weigh them against each other. If one of those is heavier than the other, the heavy coin has been identified. If those two coins weigh the same, the heavy coin is the third coin from that set of three.

And Secret Squïrrel solved the second problem:

Ok, the bonus problem requires a bit more work and a more elaborate decision tree but I'll try to be as succinct as I can. I'll label the coins with letters (A to L) so they don't get confused with my numbering of the steps.

Step 1. Weigh any four coins against any other four, say ABCD - EFGH. The three possible results are: (a) balance, (b) ABCD is heavier, (c) EFGH is heavier.

(a) If they balance, then the fake coin must be one of the others (IJKL). Go to Step 2.1.

OR

(b) If ABCD is heavier, then either one of ABCD is fake and it's heavy OR one of EFGH is fake and it's light. Go to Step 2.2.

OR

(c) If EFGH is heavier, then either one of ABCD is fake and it's light OR one of EFGH is fake and it's heavy. Go to Step 2.3.

Step 2.1. (ABCD balanced EFGH).
Now weigh three known genuine coins (say ABC) against IJK. The three possible results are (a) balance, (b) ABC is heavier, (c) IJK is heavier.

(a) If they balance then L must be the fake. For your 3rd weighing, weigh it against any other coin to see whether it's heavy or light. DONE.

OR

(b) If ABC is heavier then one of IJK is the fake and it's light. For your 3rd weighing, weigh two of them against each other (say I - J). If one of them is lighter then that is the fake, otherwise it is the other one (K) and it's lighter. DONE.

OR

(c) Very similar to (a), if IJK is heavier then one of IJK is the fake and it's heavy. For your 3rd weighing, weigh two of them against each other (say I - J). If one of them is heavier then that is the fake, otherwise it is the other one (K) and it's heavier. DONE.

Step 2.2. (ABCD heavier than EFGH).
Weigh two coins from each group against one from each group plus two known genuine coins; eg IJKL are known to be not fake so weigh ABEF against CGIJ. The three possible results are (a) balance, (b) ABEF is heavier, (c) CGIJ is heavier.

(a) If they balance then the fake is one of the others, either D (and it's heavy) or H (light). For your 3rd weighing, weigh DH against IJ. If DH is heavier then D is the fake, if lighter then H is the fake. DONE.

OR

(b) If ABEF is heavier then either A or B are fake and heavy or G is fake and light. For your 3rd weighing, weigh A against B to see if one of them is the fake. If they balance then G is fake. DONE.

OR

(c) If CGIJ is heavier then either E or F are fake and light or C is fake and heavy. For your 3rd weighing, weigh E against F to see if one of them is the fake. If they balance then C is fake. DONE.

Step 2.3. (EFGH heavier than ABCD).
Perform the same weighing as for step 2.2 - weigh ABEF against CGIJ (IJ known to be genuine). The same three possible results are (a) balance, (b) ABEF is heavier, (c) CGIJ is heavier (however, the implications of these results are different from 2.2).

(a) If they balance then the fake is one of the others, either D (and it's light) or H (heavy). For your 3rd weighing, weigh DH against IJ. If DH is heavier then H is the fake, if lighter then D is the fake. DONE.

OR

(b) If ABEF is heavier then either E or F are fake and heavy or C is fake and light. For your 3rd weighing, weigh E against F to see if one of them is the fake. If they balance then C is fake. DONE.

OR

(c) If CGIJ is heavier then either A or B are fake and light or G is fake and heavy. For your 3rd weighing, weigh A against B to see if one of them is the fake. If they balance then G is fake. DONE.

If I may offer a more general tip relating to this sort of weighing problem...

It is useful to think of the problem in terms of information and possibilities. Think about how many possibilities you need to distinguish. In the first problem, there are 8 possibilities, because the fake coin can be any one of 8. In the second problem, there are 24 possibilities, since there are 12 coins which could be fake, and the fake coin could either be lighter or heavier. Every time you use the scale, there are three possible results: tilt left, tilt right, or balance. By observing the result, you have received a piece of information, basically a 0, 1, or 2.

With two weighings, you can distinguish between up to 9 possibilities. With three, you can distinguish between up to 27 possibilities.

What would have happened if, in the second problem, you tried weighing three against three instead of four against four? If the scale tilted left or right, then 6 possibilities would remain. If the scale balanced, then 12 possibilities would remain. Since you can only use the scale two more times, it is impossible to distinguish between all 12 of these possibilities. So you know you were on the wrong track.

Trial and error should get you the rest of the way.

A dance...

My humble readers, let us take a break from skepticism for a moment (it is my blog, so I am always on topic) and talk about one of life's little lessons. Or two, I don't know. The problem with life's little lessons is that they're all contradictory and lack universality.

I am a guy who does not mind being dead serious. I do not need to be quirky all the time. I do not need to be a fun-loving person.

Whenever I see someone describing themselves as "fun-loving" on facebook or the like, it strikes me as subtly ironic. Is it really meaningful to describe yourself as fun-loving?* For it to be meaningful, it has to tell us something we didn't already know. But we already know everyone is fun-loving. What sort of person would not love fun?

*I suppose it does not have to be meaningful. Meaningless self-descriptions are not out of place on facebook. My profile has one too.

But I take pity on these people who idly describe themselves as "fun-loving", and I wish to give their lives more meaning. Therefore, I would describe myself as not fun-loving. I still like joking around and parties and stuff. But the parties I enjoy the most are the ones which are no fun. If that doesn't make any sense, that is okay. You may assume I am being ironic, rather than dead serious.

One fun thing which I definitively do not enjoy is dancing. My dad enjoys dancing. He was into the disco scene before it was popular. Or so he tells me.

I am like one of those characters in one of those stories. One of those characters who is afraid to relax, afraid to dance, afraid to ask girls out. By the end of the story, he has overcome his social anxiety, and is now enjoying life like every healthy person should. I'm a bit like that person, except without the anxiety, fear, or character development. Also, I'm not fictional. I like to think that last bit gives me an edge.

But--true story--that changed! Some nights ago, I actually enjoyed dancing, for reasons which may or may not involve alcohol (I am 21, I am allowed to do this). It was a life-changing realization. Okay, not really. But now I know what it feels like to want to dance. And now I know for sure that this is definitively not how I feel most of the time. So now I am resolved. I'm not ever again going to dance in an attempt to overcome some imagined social anxiety. I'm only going to dance if I want to. Things are better this way, when we do what want, without worrying about social expectations.

In some ways, real-life character developments are so much deeper than their fictional counterparts.

Incidentally, I have been reading Timequake, by Vonnegut. I have been reading several Vonnegut books this summer. Timequake has many real characters, and fictional. Timequake is filled with non sequitur jumps from one subject to another, much like this paragraph, and the next. I wish I could write a bit like Vonnegut.

One of the nice things about modern minimalistic art is its participatory nature. If you see a painting with just a bunch of squares or bars arranged in a random fashion, you might find yourself thinking, I could have made that! Why would a museum bother putting up something I could have made? I contend that this is not such a bad feeling after all. You, too, can make art... doesn't that actually feel pretty good? Blogs are a bit similar to modern art in this regard.

The Fine-Tuning argument (part 2)

See part 1

Here is the Fine-Tuning argument, restated... in limerick form?

Many constants of nature, I'm told,
If changed, would leave the world cold,
But a being divine
Would tune them all fine,
Without God, they'd be uncontrolled.

Don't ever expect me to write poetry again.

If you asked me why I disagree with the Fine-Tuning argument, and if I had enough time and space to explain it (ie here and now), I would begin by pointing to this article by Michael Ikeda and Bill Jefferys. They show, using bayesian reasoning and mathematics, that the Fine-Tuning cannot possibly succeed. If anything, the Fine-Tuning argument shows precisely the opposite of what it attempts to show. You might find this quite shocking. I know I did. Though I thought it was a poor argument, even back in my Catholic high school days, I never imagined that it was entirely backwards.

The first key point is the Weak Anthropic Principle. Ikeda and Jefferys state the WAP as follows: If life exists, and if the world is governed by natural law, then those natural laws are "life-friendly", meaning that they allow for life to possibly exist.

The second key point: What happens if we observe that the natural laws are in fact not life-friendly at all? This implies something supernatural, perhaps a god, or perhaps something else. You might say, in a way, that this is the argument of Intelligent Design. They argue that natural laws, such as natural selection, could not possibly lead to the diversity of life as we know it. That is, they argue that our world is not life-friendly to us. Therefore, the supernatural must exist.

And finally, what happens if we observe that the natural laws are life-friendly? The Fine-Tuning argument would have you believe that this is strong evidence for a god. A life-friendly world is unlikely to exist given a world governed by natural laws, so goes the argument. Therefore, the world is unlikely to be governed by natural laws, given that it is life-friendly.

But if you've been paying close attention, you'd realize that we're falling for the fallacy of having it both ways. The claim now suffers the problem of being unfalsifiable. If the natural laws are not life-friendly, the supernatural must exist. If the natural laws are life-friendly, the supernatural probably exists. No matter what we observe, life-friendly or not, you could conclude that the supernatural probably exists.

That is to say, given your new observation, God is more likely than before your observation. But you already know, before you have made any observations at all, that the world is either life-friendly or not. Therefore, before you observe anything, God is more likely than before you observe anything. It's a logical contradiction. When we encounter contradiction, we have to back up, and question our original assumptions. Our most questionable assumption was that the Fine-Tuning argument is valid. This assumption was incorrect. The only way to resolve the contradiction is by saying that the Fine-Tuning argument is either evidence against the supernatural, or it is neutral.

And that's the basic idea of Ikeda and Jefferys argument. Except they use more bayesian reasoning and math, which is always cool. I would have liked to explain it mathematically myself, but they do it better. It is worth a read.

The Fine-tuning argument (part 1)

One of my favorite apologetic arguments is the Fine-Tuning argument. It goes like this:

"If God exists, there is a high probability that he would create sentient beings such as ourselves. You know, because we humans are so cool, and any old god would want us to exist. If God does not exist, then there is an extremely low probability that sentient beings like ourselves would come to exist. We know this because even if the laws of physics are necessarily configured in the way they are, there are still a bunch of fundamental physical constants whose values are arbitrary. A small change in any of these physical constants is unlikely to produce humans.

"So, given that humans exist, the probability of God is high."

Let me explain first why I like this argument, and then, later, why I disagree with it.

I like this argument, because it involves physics, and gets people interested in physics. Now everyone is asking what a "fundamental physical constant" is. Well, I'll tell you! Most people think of things like "the speed of light" or "the planck constant" as fundamental physical constants, but in fact they're not. Those constants depend on our choice of units; they depend on our definition of a "meter", or a "second", or a "kilogram". So they are not really fundamental. All fundamental physical constants are essentially a ratio between two things. For example, the ratio of the mass of an up quark to the mass of a down quark. Or another example, the ratio of an electron's classical orbit speed to the speed of light (this is known as the fine structure constant). By John Baez's count, there are 26 such fundamental physical constants. Most of them are related to mass in some way. Incidentally, gravity is the least-understood physical force in quantum field theory--perhaps some of the physical constants will be eliminated when we figure quantum gravity out?

Many of the counterarguments, too, discuss physics. For instance, it can be argued that many of the fundamental constants are absolutely irrelevant to life. For instance, the mass of the top quark has absolutely no effect on us. The top quark has a lifetime of 5 x 10^-25 seconds, and is not a component of any known object. Creating and detecting a top quark was a task that required some giant particle accelerators and many very clever physicists. Other arguably "useless" constants include the masses of neutrinos, muons, tauons, bottom quarks, charm and strange quarks, the CKM and MNS matrices, and the cosmological constant. But that's not all of them. In any case, even if all the fundamental constants were completely irrelevant, there's still the matter of the equations which relate the constants to physical laws.

Another physics counterargument is, how can we know that there would be no life without these particular fundamental constants? If you think we can just take a new set of fundamental constants and predict the prospects of life, then you have sorely overestimated the power of physicists. A prediction so complex must be informed by experiments and empirical observations. But we don't have any of those; we can only observe and experiment on our own universe. What we can do, however, is look at a specific process which led to life. For instance, the triple-alpha process allows helium nuclei to fuse, forming carbon, which is, as far as we know, essential to life. The triple alpha process seemingly depends on a lucky coincidence: carbon-12 has a quantum energy level which happens to be at 7.65 MeV. If we changed the fundamental constants, would this lucky coincidence disappear? Would there appear any new lucky coincidences to circumvent the process? I wouldn't claim to know.

A third physics counterargument is the idea of a multiverse. If we live in a multiverse, then our universe is simply one of many universes with different fundamental physical constant. We can restate the Fine-Tuning argument as follows:

"If there is a multiverse, there is a high probability that at least one of the universes would allow for our existence. If there is only one universe, then there is an extremely low probability that sentient beings like ourselves would come to exist. We know this because even if the laws of physics are necessarily configured in the way they are, there are still a bunch of fundamental physical constants whose values are arbitrary. A small change in any of these physical constants is unlikely to produce humans.

"So, given that humans exist, the probability of a multiverse is high."

We can use the same argument for the existence of a god, or for the existence of a multiverse. So why should we prefer god over the multiverse?

I must say, my physics intuition tells me that this is a terrible argument for the multiverse. The reason physicists are entertaining the possibility isn't because they just thought up the multiverse and said, "Wouldn't that be philosophically satisfying?" At least, I hope that's not how it happened. I would hope that physicists thought it up because serious cosmological theories required or suggested it. And if we wish to support the idea, we must support it with real scientific experiments or deductions, not just some rhetorical argument. A lot of this cosmological theory is still up in the air, so I wouldn't jump the gun at this time.

If I find the Fine-Tuning argument for the multiverse so unconvincing, one wonders if the Fine-Tuning argument for a god is really any better.

However, none of these are my primary disagreement with the Fine-Tuning argument. I'm a physicist in training, so I would prefer not to commit to the idea that most constants are irrelevant, or that life could exist given different constants, or that the multiverse does or does not exist. Therefore, my primary response to the Fine-Tuning argument has nearly nothing to do with physics, and instead relates to logic.

...To be continued!

Physicists are dreamin'

I will neither confirm nor deny having participated in the making of this video.

And I'm off to Louisiana

... where I'll be visiting some giant lasers.

I don't want to leave you all with absolutely nothing, so here's a video about LIGO.

Einstein's Messengers (courtesy of the National Science Foundation)

(not to be confused with "Einstein's Cosmic Messengers", the musical project inspired by LIGO)

If you don't like the .ram format, it's also on Hulu.

Experiments and Enneagrams

In one of my internet adventures, I came across what is known as the Enneagram. It's yet another personality test, one I had never heard of before. And there's a funny little diagram to go with it.
I'm generally pretty skeptical about personality tests. I accept that people have different personalities. But what is a personality type? It's a bunch of different character traits which tend to come together. How do we know which character traits come together, and which are independent? How are they really organized? These are difficult questions, which would require some critical analysis to get reliable answers. I'm not confident that sufficient critical analysis went into the making of these personality tests.

For instance, look at the enneagram picture. Each of the numbers represents a different personality type. Do those lines really represent mysterious connections that we've discovered between different personalities? Or is it just guesswork based on the decimal expansion of 1/7 (0.142857142857...)? Various sources suggest the latter...

That's not to say that there's no point to taking such a personality test. I thought it was fun to play around with.

The first thing I did was look at the Wikipedia article. Glancing through it, I was reminded of the Forer effect. The Forer effect causes people to think "Hey, that's totally me!" whenever they read a vague personality description. Looking at the first few personality types, they all gave me at least a bit of a feeling of, "That's totally me!" But is any one of the personality types more me than the others? This calls for an experiment!*

So here's what I did. I copied the section of the wiki with the nine personality types. Using search and replace, I removed all references to "ones", "twos", "threes", and so forth. Then I jumbled the order of all the sentences. I couldn't tell which sentences correspond to which personality type, because I'm just not that familiar with the enneagram personalities. So I went through the sentences one by one, and decided whether they described me or not. Afterwards, I resorted the sentences so I could see which personality types came out on top.
Here's a picture of the end result. Green means that I felt it totally matched me. Red means that it didn't really match at all. Yellow means something in between. It's sort of hard to tell, but I suppose the picture indicates that my personality type is 5 ("the investigator"), followed by 4 ("the individualist") and ... maybe 3 ("the achiever")? That totally sounds like me. I mean, I'm doing investigation right now, of course I would be the investigator. 3 and 4 also had some resonance with me.

Are these results consistent? I tried taking a five minute test to find out. Lo and behold, it said I am most likely to be 3, 4, or 5. That's pretty neat, huh? So I tried taking yet another test, this one a bit longer. This time it said I was mostly type 1 ("the reformer"), followed by 9 ("the peacemaker"), and then 5. Actually, now that I think about it, 1 and 9 match me much better than 3 or 4. The reformer is all about having high standards and working for change. Why, I do have rather high standards; that's why I'm so critical! The peacemaker wants everything to go smoothly and without conflict. That's so true, I do tend to avoid confrontations, and I have an extremely slow temper.

The more I look into it, the more it seems like all of the personalities match, to varying degrees. But if I look carefully, all of them also have things which I think are totally wrong.

Take, for instance, number 5, which seems the best match for me overall. The description of 5 seems to get my motivations totally wrong. The Wikipedia article says that fives are often either neglected or intruded upon early in life. It's funny how they put both these opposing possibilities there, so that no matter what kind of childhood you had, you're likely to think, "yes, YES, that's me!" In any case, I feel like I was neither neglected nor intruded upon, but somewhere in the happy middle. The article also says that fives often withdraw because they feel they have limited energy. I think I hardly ever feel that way. I have all the time in the world.

Another odd thing about the descriptions, is that several of them talk about feelings which we are unaware of. Gee, if I'm unable to discern my own feelings, I hardly expect a personality test based on numerology to know any better.

I wonder. What would happen if I just took all the green text from my experiment, and created a new personality number 10? That would totally fit me much better than any of the other types!

*I hope no one takes this to be a rigorous experiment. I mostly did it because designing experiments is a fun process in itself.

The thief's assistant

Two college students on vacation were walking in through a small town, absorbed in an incomprehensible debate.

"...doubt that the soul operator would commute with the Hamiltonian."
"Why should--hey what's that over there?"

There had been a single figure walking up ahead of them, but now there was a second figure which looked much more threatening. A little old lady, the kind that always gets her purse stolen in comic books, was getting her purse stolen.

"HEY! What are you doing!" The thief looked up, and then ran away with the purse. One student stayed to make sure the lady was okay, while the other one chased after the thief into an alley.

But he ran straight into a big guy who had stepped in his way. "Whoa there. What's the rush?" The college student looked him up and down. He looked quite friendly, not threatening at all despite his size.

"I'm chasing a thief, he stole a woman's purse back there! Come on, help me!" The student tried to go around the big man, but it was a narrow alley, and the guy didn't seem to want him to go past. The student suddenly realized how odd it was that there could be such a narrow alley in such a small town.

"A thief? Here?" The big man looked the college student up and down. He looked like the kind of young adult who was always trying to be a hero, always imagining up dragons to fight. Idealistic. Arrogant. "There are no thieves in Townsville. Townsville is a friendly community."

"But he's right there! Hurry, before we lose him!"

The man turned his head towards the escaping thief. He doesn't react. "There are no thieves in Townsville. Don't you dare insult us."

The student sighed. There was no way he'd catch up anymore, not even to get a good look at him. He started to get a little angry at the big man blocking him. "Well then what do you call that guy who ran away?"

"If that were really a thief, he couldn't be a citizen of Townsville, much less a good citizen. He must be a traveler, much like yourself."
"But that's beside the point! You should at least have let me chase after him, if you weren't willing to help!"
"Why are you getting so angry at me? I am not a thief! How dare you accuse me."
"I'm not accusing you. It's not your fault if there is a thief in Towns--"
"I am a citizen! Citizens of Townsville never steal because they always ask politely first."
"I think you're misunder--wait, what was that?"
"I said that no proper Townsville citizen is a thief. I think you are misunderstanding me."

The student was quite exasperated and disturbed at this point. He was about to give up. "And yet there are still thieves."

"No one here is a--"

The student felt he had heard enough from this guy. "Okay, okay, no one in Townsville is a thief. Could you just direct me to the police already, so I can report this?" The man blocking him, satisfied that he had defended the dignity of his town, directed him to a dusty old building in the furthest corner of town.

Yes, I am asexual

Some months ago, I wrote a rather confused post about my sexual orientation. But now that I have the benefit of hindsight, I can present the issue more clearly and accurately. So consider this my second try.

What does asexual mean?

I consider myself to be asexual. An asexual is a person who does not experience sexual attraction. At least, that's what it says at AVEN, the internet nexus of asexuality. It is strongly recommended that you read the FAQs at AVEN for the official presentation.

Of course, it can be a little unclear what "sexual attraction" really means. It's unclear to me, anyways, since I don't have any first-hand experience of it. Different people have described it in different ways, leaving a caricatured image in my mind. It happens you look at someone, or talk to someone, or otherwise interact with someone. You feel aroused, or it leads to sexual fantasies, or it makes you want to talk to them, maybe enter a relationship or have sex. I'm unclear on the details, but it's some instinctive feeling which causes people to, on some level, desire sex with another person.

Asexuals, by definition, are not merely repressing or inhibiting their feelings. They simply don't experience them in the first place. If you tell an asexual that they're just repressing themselves because of the puritanical forces in our culture, they will be A) annoyed by your ignorance, and B) a little confused about what this feeling is that they're supposedly repressing.

Asexuals lack sexual attraction. However, this does not necessarily mean they lack other things. For instance, most people closely associate romance and sex, because they rarely have one without the other. Many asexuals do experience romantic attraction without sexual attraction. In the asexual community, it's common to speak of a romantic orientation (ie heteroromantic, homoromantic, biromantic, aromantic) in addition to a separate sexual orientation (ie heterosexual, homosexual, bisexual, asexual). I've discussed in the past some scientific literature which supports this disassociation.

Another surprise: some asexuals do enjoy sex. Just because they lack that feeling which causes them to seek sex doesn't necessarily mean that they don't enjoy it when it happens. But I think most asexuals do not enjoy it, in the same way that most straight men would not enjoy sex with other men. This can cause problems (which can be overcome) in a asexual/non-asexual relationship, since the partners have different levels of sexual desire.

More surprises: Some asexuals experience crushes of a nonsexual nature. Most asexuals masturbate, just like everyone else. Some asexuals can recognize sexual attractiveness (or "hotness"), but are unaffected by it. And some of them have fetishes which are unrelated to other people.

"How come I've never heard of these people?" you ask. It's rather uncommon, rather difficult to recognize, and is often simply dismissed. At least one survey has indicated that asexuals represent 1% of the population. However, this is probably an underestimate, since asexuals are exactly the type of people who are uninterested in taking a survey about sex. Most asexuals themselves have never heard of the concept. Most researchers have never heard of the concept. The scientific literature on the subject is small enough that you can potentially read all of it.

If asexuality were better-known, I think it would help a lot of asexuals understand their own experiences, and might also provide insight into human sexuality in general.

Where do I fit in?

I have absolutely no problem with sex or romance. I think they are great ideas. However, I consider myself asexual and aromantic because I cannot for the life of me understand the concept of "hotness", and I do not ever have crushes of any sort. No person has ever prompted feelings which were recognizable as romantic or sexual desire. I never even think about the idea, except by conscious effort. I'm really not against the idea of a relationship, but it would be quite difficult to start one, since my mind never labels anyone as a candidate. And if I did somehow get into a relationship, I might have difficulty actually falling in love or feeling sexually attracted to them.

Or not. Who knows?

Asexuality is a description, not a commitment. If it changes in the future for me, then that's that. Either I was wrong about my orientation, or my orientation changed. It is thought that asexuality, like other sexual orientations, can sometimes change, but it is rare, and not by conscious choice. Basically, I'm not going to count on it.

How do I feel about this? Frankly, I don't like it, to the point that I want to deny it. I don't really know if I care about the asexual part, but I do care about the aromantic part. I mean, no romantic love ever? That sort of bums me out. At least I still have friendships and non-romantic love. In any case, many asexuals on AVEN are completely comfortable with their asexuality. If you read their FAQs, you might notice that they emphasize how happy asexuals are. I do not doubt it.

But between the internet and me, I think the reason this is officially emphasized is because if they are unhappy with their asexuality, then according to the DSM (the psychologist's "bible"), they have a disorder called Hypoactive Sexual Desire Disorder (HSDD). That's right, asexuality is pathologized in much the same way that homosexuality was in past decades,* but only if the asexual experiences personal distress. In my opinion, if something causes me a little distress, that does not necessarily mean it is a disorder.

*As a matter of fact, HSDD derives from another diagnosis, HSD, which at one point included homosexuals, since they are supposedly inhibiting their natural heterosexuality.

I will say that I am happy about discovering the concept of asexuality. It has opened new doors. Before, I would have reacted to the ideas of romance and sex with disinterest or confusion. Now, it finally makes sense why it doesn't make sense to me. It's like a barrier to my understanding has been lifted, or at least partially lowered. And I'm sort of curious about what's on the other side.

What will I do with my life now? Probably the same thing as I have always done before. I'll go on living it.

Find the fake coin

You have eight coins which look identical. One of them is fake, made of a slightly heavier metal than the others. The difference in weight is too small to feel by hand, but you have a precision scale which can weigh one object (or set of objects) against another.

If you are only allowed to use the scale twice, can you pick out the fake coin from the real ones?

Bonus question: You have twelve coins which look identical. One of them is fake, and has a slightly different weight than the others. You do not know whether it will be heavier or lighter, and you cannot feel the difference by hand. Can you pick out the fake coin if you are only allowed to use the scale three times?

See the solution

Rationalizing the caste system

While looking up karma on Wikipedia, I happened to click on this external link to a despicable article trying to justify the caste system. I don't know enough about Hinduism to know where this website, GitaMrta, fits in, but this is just too terrible to pass up.

We all work. At work we have a boss. Your boss is higher than you and you are lower than your boss. Does this mean that your company is bad for making you lower than your boss? No. You are lower than your boss, because your boss deserves to be higher than you. He is better qualified and experienced and thus he deserves to be higher than you. You are less qualified and experienced and thus you deserve to be lower than your boss.

Ugh! So it's saying that when we work for someone else, we're necessarily beneath that person. And since we're beneath another person, we must deserve it. This is the worst kind of confusing is and ought.

Every organization in the World is following the Hindu caste system. As the employees in every organization are given a role in the organization according to the qualifications of the employee.

The current caste system practiced in India is based on birth and not qualifications and thus it should be rejected. As stated in the Bhagavad-Gita.

It's funny how the one real caste system is so horribly wrong, even they have to admit it. But according to their reasoning, it couldn't possibly be because the caste system itself is wrong. It must be a case of mislabeling. See, all those other more just systems are really caste systems, and the one in India is just some corrupted version.

The Bible openly advocates enslaving human beings. Is this not far worse than the Hindu caste system?

"Slaves, obey your earthly masters with respect and fear, and with sincerity of heart, just as you would obey Christ." (Ephesians 6:5)

They can only make the caste system look good by comparing it to slavery of all things? Gosh, well at least Christians have the decency to ignore, reinterpret, or do whatever they need to do to rebuke slavery. I don't see GitaMrta rebuking the caste system.

Is everyone of the same intelligence? Is everyone very clean? Is everyone very honest? Is everyone very religious? Is everyone very peaceful and non-violent?

The simple answer to the above questions is NO. Thus this proves that everyone is not equal. The Hindu caste system states that there are 4 categories to which a person belongs to, according to his/her qualification.

• Peacefulness, self-control, austerity, purity, tolerance, honesty, wisdom, knowledge, and religiousness are the qualities of Brahmans. They should be the leaders of society so that others can follow in their footsteps.
• Heroism, power, determination, resourcefulness, courage in battle, generosity, and leadership are the qualities of Ksatriyas. They should be the administrators of society.
• Those who have diminished qualities from those above, are the Vaisyas. They are the producers of society.
• Those who have further diminished qualities from above, are the Sudras. They are the workers.

For a moment, I considered being charitable. Perhaps they are just arguing for a meritocracy. But no, they advocate a strictly structured system, nearly like the one in India. Hint! Just because people are unequal doesn't mean they make an ordered set! It does not mean that you can or should separate them into four categories from best to worst! It does not mean that jobs should be assigned according to these preposterous groups!

Note that those who eat meat or kill (humans or animals) don't fall within the 4 categories. They are not qualified to be called civilized humans. They are called Yavanas and Mellechas, the lowest among mankind.

This final note is especially ironic considering another thing they had said earlier:

Every second, the unbelievers are born and this is simple proof that the all mighty God very much loves the unbelievers. Also the fact that the unbelievers have existed for millions of years and the believers for only 1400 years, is proof that God wants the unbelievers to exist more so than the believers.

The real God (Lord Krishna) is full of love and mercy, he is not a bully or dictator or hater of any living being.

This is love and mercy? Bullshit.

Solutions to "Dividing a plane"

See the original puzzle

Every time you draw a line, it "splits" all of the regions which it goes through. Every time a region is split, it increases the number of regions by exactly one. Therefore, to maximize the number of regions, you want to have the line go through as many different regions as possible.

For example, here is a picture of the fourth line being drawn.
The fourth line (in color) crosses four regions. This is the best you can do, because there are only three other lines, and it can only cross each one once. Every time it crosses a line, it goes from one region to another. If it crosses three lines, then it crosses four regions. If it splits four regions, then the total number of regions increases by four.

In general, whenever you add the Nth line, the number of regions increases by N. When there are zero lines, we start out with a single region. Therefore, the maximum number of regions is equal to 1 + (1+2+3+...+N). If we simplify this equation,* we get 1+N*(N+1)/2.

The circles are more or less the same idea. Whenever you draw a circle, it splits every region it crosses into two. For example, here is a picture of the third circle being drawn:
The third circle splits at most four regions. This is the best it can do because there are only two other circles, and each circle can only be crossed twice. Just like with the lines, each time the circle crosses a line, it goes from one region to another. So if the circle crosses four lines, then it goes through four different regions, splitting each of them into two.

In general, whenever you add the Nth circle, it can crosses 2*(N-1) lines. Therefore, it goes through 2*(N-1) different regions. Note that there is an exception to this rule, when N=1, and it crosses zero lines. Even if the circle crosses zero lines, it still goes through one region, splitting it. The maximum number of regions which can be created by N circles is therefore 2+(2+4+6+...2*(N-1)). This simplifies to 2+N*(N-1). Note that because of the exception mentioned earlier, this equation fails for N=0, when there is only one region.

The ellipses are the same idea yet again. Except now, each pair of ellipses can cross themselves at most four times, rather than two. So the maximum regions which can be created by N ellipses is 2+(4+8+12+...4*(N-1)). This simplifies to 2+2*N*(N-1). Note that this equation also fails for N=0.

Also see the reader comments for other ways to think of the problem.

*If you are wondering how I simplified the equation, consider the sum (1+2+3+...+N).
Let x = 1+2+3+...+N
2x = (1+2+3+...+N) + (N+(N-1)+(N-2)+...+1)
2x = (1+N) + (2+N-1) + (3+N-2) + ... + (N+1)
2x = (N+1)*N
x = N*(N+1)/2