Alex Strinka

What Does the Equals Sign Mean?

When kids are taught arithmetic, they're usually given problems that look like "2 + 3 = __", and they're supposed to write "5" in the blank space. So it's not particularly surprising that many kids (and even some adults) don't actually understand what it means. They think it means something like, "Here's the answer" or "Evaluate this expression".

What it actually means is that whatever is on the left of the equals sign is the same as whatever is on the right.

There's some nuance about what exactly it means for something to be the same as something else, which can vary somewhat depending on the context, but generally it just means that the two things have the same value.

So, 2 + 3 = 5 is a true equation because the expression 2 + 3 has the same value as 5.

But that equation could also be written as 5 = 2 + 3. Since thing on the left is the same as the thing on the right, the thing on the right must be the same as the thing on the left too.

You can even write 2 + 3 = 1 + 4. It might not be immediately obvious why you would want to, but being able to manipulate equations like that is the basis of algebra, which is a very useful thing

Sometimes you'll see someone write something like "2 + 3 = 5 + 4 = 9". This is incorrect. What's meant is "2 + 3 = 5 and 5 + 4 = 9", but the correct interpretation is "2 + 3 = 5 + 4 and 5 + 4 = 9". 2 + 3 doesn't have the same value as 5 + 4, so that's false.


Consider this dialogue:

Alice gets a phone call from her friend Bob.
Bob: My car is broken. Can you tell me how to fix it?
Alice: I guess. What's wrong with it?
Bob: I don't know, it doesn't start.
Alice: Well, you could try jumping it.
Bob: And that will definitely fix it?
Alice: It will if the only problem is a dead battery, but there might be something else wrong with it.

You can't give good advice on how to fix a car, if you don't know what the problem with the car is. You can give advice for the most common problems, and you can give advice about how to troubleshoot it, but whether that advice helps depends on whether the problem with the car is something your advice covers. Unless you can give out a detailed technical manual of every part of the car, some problems won't be covered.

Humans are more complicated than cars, and we haven't figured out the fully detailed technical manuals for them yet.

Therefore, all advice is highly situational. Advice that can help one person could hurt someone else.

Which is not to say that advice is bad or worthless. Advice from someone who knows the situation can be very helpful. And even without context, there are many issues that many people share, and it can be helpful to point out common problems. But, any advice that's given without context (for example, self-help books, or random blog posts) should be carefully considered before being put into action.

That especially applies when it's advice that you agree with, advice that makes you say, "Yeah, that's a great idea!". If you feel like that about some piece of advice, then you probably already believe it, and that belief probably already influences your actions. It won't be in the exact same way that the advice says (otherwise your reaction would have been, "Well, duh.") but it's probably more or less in the same direction. Which implies that the advice is less likely to be the bottleneck to improve whatever it is you're trying to improve.

Stack Of Doom
A screenshot of Civilization IV

This post is about Civilization IV, a computer game that's over fifteen years old, and has had two director successors. As always, I strive to maintain my reputation for tackling topical issues.

First, a recap of the relevant game mechanics. In Civ IV, different types of military units get various bonuses. For example, a spearman gets a bonus against mounted units, so in a battle between a spearman and a chariot, the spearman will usually win. An axeman gets a bonus against melee units, so in a battle between an axeman and a spearman, the axeman will usually win. And a chariot gets a bonus when attacking axemen, forming a nice rock-paper-scissors setup.

Furthermore, you can stack multiple units on the same tile. Every unit still moves and fights individually though. When you attack a stack of units, the unit that has the highest chance of victory will defend. So, say there's a stack consisting of one spearman and one axeman. If you attack it with a chariot, the spearman will defend. If you attack it with a spearman or an axeman, the axeman will defend. As you can see, well-composed stacks of units are difficult to attack.

And there's no limit to the number of units you can have in a stack, so generally, it's most effective to put all of your troops together in one giant stack, making what's called a "Stack of Doom". Unless there's a significant tech difference, the only way to reliably counter a stack of doom is with another stack of doom. This is understandably frustrating, and it can make combat a little bit uninteresting.

Civ V dealt with this problem by introducting "One Unit Per Tile". You can only have one single military unit in a tile. In my opinion, One Unit Per Tile is far worse than Stacks of Doom, both from a gameplay perspective and from a realism perspective. Here's an article that talks about some of the problems with One Unit Per Tile.

Here are my ideas for how to fix Civ IV's Stack of Doom:

  1. More collateral damage. Typically, when you attack a stack of units, only the one defending unit takes any damage. But some units, like catapults and cannons, and also the Chinese unique unit do collateral damage, which means they can do damage to other units in the stack. My first suggestion is to simply make more units inflict collateral damage, and make collateral damage more harmful.
  2. More targeting. Targeting is a concept that exists in Civ IV, but is only used by a single unit. The Khmer unique unit is the ballista elephant. Like standard war elephants, it gets a bonus against mounted units, but unlike any other unit in the game it targets mounted units, which means that if it attacks a stack with any mounted units, one of the mounted units will be the defender. My second suggestion is to make more units have this ability.
  3. Flanking. This is a new concept that doesn't exist in Civ IV. (There is a concept with the same name, but that's just a special version of collateral damage.) The idea is that if you attack an enemy unit, and there's a friendly unit directly adjacent to the enemy, but not on the same tile as the attacker, then the attacker gets a bonus.
  4. Attacks of Opportunity. This one I'm less confident about. It's another concept that doesn't exist in Civ, but it is related to Zones of Control, a concept that existed in Civ I and II. In those games, if you had a unit that was adjacent to an enemy unit, you could attack or you could retreat, but you couldn't move your unit to another tile that's also adjacent to an enemy unit. It was really restrictive, which I think it why it was removed in later games. My idea here is to reintroduce that, but less restrictively. You can still move around an enemy unit, but doing so would give the enemy an attack of opportunity. In other words, they would get a free shot at your unit, and your unit wouldn't get a shot at them.

All these suggestions are intended to work with Civ IV's existing combat system, but they would require making other changes for balance. The existing units that do collateral damage and targeting would need to be buffed in some way to distinguish them. And since defending a city requires putting a stack of units in it, there would need to be a way of countering these abilities in a city. Perhaps defensive structures like walls and castles could reduce or nullify these abilities. You would probably also need to change the costs, strengths and abilities of various units, and maybe add and remove some.

Sturgeon's Law

According to Theodore Sturgeon, "Ninety percent of everything is crap". I think that's worded a bit too strongly. I would say something more like, the majority of everything is low quality. Of course, there's no objective way to evaluate quality, but I think that's true in most cases, by most standards.

Why is this? I think it's because there are far more ways for a work to be bad than for a work to be good, and it takes skill to reliably produce quality work. It's like a dart board. Most throws won't hit the bullseye, unless you're really good at darts.

Some individuals are particularly skilled, and produce mostly high quality work. Ninety percent of plays may be crap, but not ninety percent of plays written by Shakespeare. And more generally, while the majority of everything is low quality, a carefully selected subset of everything doesn't have to be.

I think this is part of the reason some people perceive recent works as being worse than older ones. When you look at old books, movies, music, etc., you typically find the ones that we still read, watch and listen to today. When you look at recent books, movies, songs, etc., you're more likely to see a representative sample. The latter will be worse, but only because it hasn't been winnowed by time.

A seemingly paradoxical implication of this law is that by increasing output, you increase the amount of both low quality work and high quality work. The relative proportion doesn't have to stay constant, though. In fact, it could move in either direction. For example, the proportion of high quality work can increase by an artist getting more skillful through practice. Or it could decrease by removing a selection barrier that had been in place before.


There are two kinds of eclipses. A lunar eclipse in when the Earth casts a shadow on the Moon, making the Moon appear dark. A solar eclipse is when the Moon casts a shadow on the Earth. The people who are in the shadow will the see the Moon block the Sun, making the Sun dark. The picture above is one I tookd during a total solar eclipse.

In my previous post, I said that a full moon is when the Earth is between the Sun and the Moon. And as you might expect, a lunar eclipse can only happen during a full moon. But we get a full moon about once a month, and lunar eclipses only happen about twice a year. Why isn't there a lunar eclipse every month?

The explanation requires some 3D geometry, which can be hard to visualize. So, to help, I've made some interactive 3D models. To start, here's a model showing the Earth's orbit around the sun. You can click and drag to rotate it. Note that it's not to scale to make it easier to see. The grid is just to visualize the plane that the orbit lies on.

This interactive 3D model requires javascript to be enabled.

The Earth moves around the Sun in a circle, with the Sun at the center. (Technically, it's not actually a circle and the Sun isn't quite at the center, but that's a close enough approximation for now.) That circle lies on a plane, called the ecliptic. The ecliptic doesn't move, at least not significantly.

The Moon orbits the Earth on a circle too, but the plane of the Moon's orbit doesn't line up with the plane of the Earth's orbit. This is called the inclination of the Moon's orbit. You can adjust the slider to see how different inclinations look. The actual inclination of the Moon's orbit is about 5 degrees.

This interactive 3D model requires javascript to be enabled.

You can see how when the Moon is just slightly inclined, it's possible for it to be on the far side of the Earth, but not in Earth's shadow. But now, you might be wondering, how can there ever be an eclipse?

Well, as the Earth moves around the Sun, the plane of the Moon's orbit changes very little. (A process called nodal precession causes the plane to spin around the Earth, but it happens slowly.) This next model shows what the Moon's orbit looks like as the Earth moves around the Sun.

This interactive 3D model requires javascript to be enabled.

If you adjust both the Earth's and Moon's positions to 90 degrees, you can see the Moon will be in Earth's shadow no matter its inclination. This is why we get about two lunar eclipses per year.

This model also helps demonstrate why solar eclipses are less common than lunar eclipses. Since the Moon is smaller than the Earth, it has to be that much closer to being perfectly lined up. Furthermore, the Moon's shadow can't cover the entire Earth, so only a small portion of the Earth can see a solar eclipse when it does happen.

Because the above models aren't to scale, they might give the wrong impression about the actual sizes and distances of the solar system. So the next model is the same as the last one, except it's to scale. If you can't see anything, zoom in.

This interactive 3D model requires javascript to be enabled.

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