The Odd Number Rule

Hey, Vsauce, Michael here. Why though? Why are any of us here? What's the purpose? What does it all mean? Well, sometimes if we listen closely enough when we ask why, we can hear an answer, and it's another question: Why?

Why? What? Our journey begins here. Checking the rulebook, but you won't find anything in there. It says a dog can't play. He's right—ain't no rule said the dog can't play basketball. Is that true? Is there really no rule against a dog playing middle school basketball? Vsauce, magic 8 ball, is there really no rule against a dog playing basketball? The answer is yes, or is it?

We have begun. The actor who played Air Bud, Buddy the dog, passed away in 1998 at the age of nine. Now, take a look at this. Let's talk about World War I. On June 28, 1914, Franz Ferdinand, the heir to the throne of Austria-Hungary, and his wife Sophie were assassinated in Sarajevo while riding in a car. The car's license plate was A III III118.

They were killed by Gavrilo Princip, who said during his trial that he wanted the unification of all Slavs free from Austria. The event sparked a series of escalations in Europe, the July Crisis, which ultimately caused World War I, a war that ended on Armistice Day, November 11, 1918. Pretty odd, huh? A spooky coincidence, if you will.

But look, odd things happen all the time. Oddness is all around us, especially in the form of odd numbers. The grip they have on our reality is revealed when we release our grip. Notice anything strange? Under the influence of gravity, things move in an odd way. If something starts moving because of gravity, after say one second, it will have traveled some distance.

Now, interestingly, during the next second, the distance it will travel will be three times as far as it moved during the first second. During the third second, it will travel five times as far, then seven times, then nine times, eleven, thirteen, fifteen. We used seconds, but it doesn't matter what time interval you use. You will always find this pattern: the odd number rule hiding right there within the very phenomenon that keeps us on Earth.

Are the odd numbers more like God numbers, right? Is the odd number rule the face of the universe looking back at us? Well, here is something that Galileo wrote: "Philosophy is written in this grand book, I mean the universe, which stands continuously open to our gaze. But it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it. Without these, one is wandering about in a dark labyrinth."

Before this pattern in the grand book of the universe was read and appreciated, people wondered why things fell. Maybe stuff just has a desire to be in its own proper place, right? Rocks with the Earth, fire with the air. That makes sense, right? It was fun to imagine causes and write our own stories, but underneath our desire for explanations, a story was already being told, one that is written in the language of mathematics: measure and relation.

Now many of these stories are being told all around us all the time, one on top of each other. But if we can look past those disturbances, each isolated event is narrating a quiet, simple story—not of why the universe is, but of what the universe is. Galileo recognized what this pattern spelled mathematically: it spells "Hello, I am speeding up at a constant rate."

And that was huge! All this time, for millennia, we had been wondering about the causes of gravity, its cosmic purpose, why it existed. But now here was someone saying, "Yeah, yeah, cool, but like what does it do? It makes things fall. Great. Well, what does that mean? What does it mean to fall? Fall? How?"

Now, the human mind alone couldn't answer that, but the universe was telling us; we just had to speak its language and read it. This is "I am gaining velocity at a constant rate" spelled with odd numbers.

How? Well, what's an odd number? The concept of oddness comes from the observation that sometimes when you have a bunch of things, everything you have can be paired up, but sometimes that can't happen. Now, when it's possible, when you can pair everything up that you have, it is said that the number of things you have is even. If you can't do that, then the number of things you have is odd.

Now, what this means is that the number of objects in an even group is equal to 2 times n, where n is just the number of whole pairs that you have. So n is any whole number—n could be, for example, 50, giving us the even number of 100, which is also the number of legs centipedes have. Or is it? Centi-hundred.

"Peed," feet having 100 legs is right there in the name, but it's only approximate. Centipedes have been observed to have anywhere from 30 to 345 legs. Now 100 falls pretty nicely in that range, but every pair of legs on a centipede is attached to a body segment, which means that in order to have 100 legs, a centipede would need to have 50 body segments. But not a single species of centipede has an even number of body segments.

50 segments is nowhere to be found in the library of centipede genomes. So while centipedes on average have a number of legs quite close to 100, they never naturally have exactly 100. The point is, 2n is the mathematical way to spell even. If I add just one more thing, the amount I have is no longer even; it's odd.

If I add yet another thing, I’ve added another pair, and it has gone up by one, and the number is even again. This means that an even number plus just one more will always be odd.

Now that we know how to mathematically spell odd and even, we can move on to moving. If you and me hugged, well, that would be wonderful; we would both get so much out of it. But if during the hug you suddenly ran away from me while yelling, "I'm leaving you" at a velocity of two meters per second, well, I would be devastated. But I would know that deep down you still cared for me because you told me your velocity.

And if I know your velocity, I can know just how far away you are from me at all times. All I have to do is take your velocity and multiply it by the number of seconds that have elapsed since you left.

Now take a look at what a graph of your velocity would look like. Here we have velocity in meters away from me per second, and we have time in seconds since the hug ended. Now, if your velocity is two meters per second and that persists, then here's a graph of your velocity. But look at this: after a number of seconds have elapsed, like after one second, the distance you are away is just the rate times the time elapsed: two times one.

Well, that also happens to be the area of the rectangle bounded by a line coming up from the time of your velocity graph and our axes. Well, that's pretty convenient! After two seconds, you are two times two, four meters away from me, which is also the area of this shape we've created. After three seconds, you are three times two, one, two, three, four, five, six meters away. It's always the area.

Well, that's pretty convenient! But let's change up the story. What if we hugged and it was great? I mean, just almost not even a hug. Like something new is being invented here. But then during that, you suddenly leave; you run away from me, and while you run, you yell, "I am leaving you" at an ever-changing velocity.

Well, I would say, you know what? That is a lot less helpful! Now let's say that this changing velocity you have that I don't know happens to be an acceleration of two meters per second per second. What does that mean?

Well, that means that your velocity is getting bigger by two meters per second for every second that you run. It would look like this on a graph: you begin hugging me with no velocity, and then one second later, your velocity is two meters per second larger. So, two larger than zero is two. That's your velocity after one second. After two seconds, your velocity is two larger still, so four, and then six. Your velocity graph would look something like this.

But now here's a question: if you're moving like this, is the distance you are away from me at any given time still just the area of the shape bounded? It's harder to see that because you don't maintain any single velocity for any amount of time; your velocity is always changing. So we don't have nice rectangles.

But we can see that it is true by putting ourselves in my shoes—abandoned, alone. I don't even know your emotion; I don't even know how you're accelerating. It's very scary! But I can decide to take matters into my own hands and measure your velocity as often as I like.

Let's say that I measure it just once. I measure it after, let's say, two seconds. Alright, wonderful! Now, the question I want to answer is how far away you are after three seconds. Alright, so I have this measurement. I know that you were traveling at four meters per second. Well, I mean, look, if you're going four meters per second for one second, then that's a rectangle like this, and it's got an area of four.

But you're not four meters away, are you? You're a lot further than that because you weren't traveling four meters per second for this entire second. You were at all times actually traveling a bit faster, and I'm totally forgetting all the distance you covered before my measurement, which is not zero, okay? So clearly, I need to measure your velocity more frequently.

So let's say that I measure your velocity twice during the three seconds I'm curious about. Let's say I measure it after one second and then after two seconds. Well, now I can imagine rectangles here. This one is two by one, so it has an area of two. This one is four by one, so it has an area of four. Two plus four is six, but you're a lot more than six meters away as well because you weren't traveling two meters per second this entire time; you were actually speeding up, covering more and more distance.

I'm totally forgetting all the distance you covered over here—basically, this method will always leave me short, but the more frequently I check your velocity, the better things get. Take a look at this. Oh, yeah! Oh, wow! As you can see, the more frequently I check your velocity, the closer my calculation gets to being based off of what you really did—your actual motion.

But also, the more frequently I check your velocity, the closer the combined area of all of my rectangles gets to simply being equal to the actual area bounded by your velocity curve, which means that the area under your velocity curve, even when your curve is like this, is still totally the distance you are away from me.

Now let's take this knowledge and apply it to the odd number rule. Before people knew how things fell, there were some pretty good guesses. For example, if the reason things fall is that they have within them a desire to be in their own proper place—in this example, the table—well then, maybe the closer this object gets to the table, the more excited it is to be there and the faster it goes, in which case its velocity would change over time like this, getting steeper and steeper and steeper. But Galileo didn't think that.

He believed that when objects fell, their velocities increased at a constant rate. So, graphed like this, their velocity would just follow a straight line. Let's draw a straight line—beautiful! Now let's divide this graph up into strips of equal time—perfect!

As we know, the areas of the shapes we have bounded here are the distances traveled during their respective time intervals. So what's the area of our first shape here—a little triangle? Well, we don't know what it is; we don't have any actual markings here on our axes, but that's fine; it doesn't really matter.

We'll just call this one. So in the first second or the first time interval, our object traveled a distance of one unit of space—perfect! Now, how far did our object travel during the second time interval? Well, that distance will be equal to that area. Well, what is this area?

Well, one thing we can do to make this easier is to divide the shape like this. Now we have two shapes: a triangle and a rectangle. This triangle has the exact same area as our initial triangle. That's because they are congruent triangles. They both have the same base length, and because this velocity line has a constant slope, it rises the same amount per time elapsed all the time, so their heights are the same and they're both right triangles.

Their areas are the same, so the area of this space is the area of that triangle plus the area of this rectangle. What is the area of this rectangle? Well, one thing we can do is divide the rectangle in half with a diagonal like this, and look what we've created.

This triangle has the exact same area as our initial triangle. It has the same height, it has the same base, they're both right angles, and look, the same goes for this one. It's a right triangle; its base is the same as this space; its height is the same as this one's height. Gosh, this is just the area of the initial triangle, but twice!

So we have two of them, and we have two of them one time, okay! Perfect! Now let's move on to finding the area of that shape. Once again, I can divide this shape like that, and now I have a triangle, which as we know has the same area as our initial triangle right here. So we have that area one, and then we also have, look at that, two rectangles, which as we know we can divide in half like this, giving us now four triangles equal in area to the initial one.

So what we've done is we've taken the initial area and we have doubled it. Well, once and then twice, so two times! Now, as we continue doing this, we find that we're always adding one more of our original areas, and then we're always doubling that original area one more time. Here we've doubled it one, two, three times—doubled three times.

Well, look at this pattern right here. We can think of one as being equal to two times zero plus one, can't we? Well, these numbers are whole numbers, and they're just going up by one: zero, one, two, three. This is all of the general form 2n + 1. That is how you mathematically spell odd.

So, the odd number rule is not some kind of magical coincidence; it's just what happens when an object's velocity increases at a constant rate. The answer was right there in front of us all along, written in the mathematical language of the universe.

The universe also contains Colorado, one of its best rectangles. Or is it? Let's find out after a quick word from our sponsor.

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Welcome back! Now, Colorado looks like a rectangle—a shape with four sides that all meet at the same angle—but Colorado doesn't have four sides. It has 697, making it not a rectangle, but instead a hexa-hecta-indecagon! And this was all by mistake.

The United States Congress initially defined Colorado as a geospherical rectangle, the space bounded by two lines of latitude and two lines of longitude. In the 19th century, surveyors set out to demarcate its shape—to actually mark it on the ground—and they did a phenomenal job with the tools that they had. But mistakes were made, some were large enough that you can spot them on many common maps of the state.

In 1925, the US Supreme Court ruled that the borders, as surveyed, were the correct ones and would legally stand as the official recognized edges of the state, not the originally intended theoretically pure geospherical rectangle. Thus, ending the shaping of Colorado that began with lines drawn by the compromise of 1850.

We have arrived! Article 18, Section 5, Paragraph 0 of the Washington Interscholastic Activities Association 1996-97 Constitution, the set of rules Air Bud would have been subject to at the time and place of the movie, states that in order to participate in an interscholastic athletic activity like basketball, a student must be a "regular member of the school," defined as someone enrolled half-time or more.

The section does refer to RCW 28A 225010, a list of approved exceptions, but not one of these was considered in the case of Air Bud. Had they been, it would have been discovered quite quickly that Air Bud, the dog, qualified under absolutely none of them.

There is no rule that specifically states a dog cannot play basketball, but there are and were rules that said that this dog, Air Bud, should not have been allowed to play basketball.

And as always, thanks for watching!