The 1's digit of the second-to-last half-adder routes into the A input of the last half-adder. So when she is using only 0-marbles, the effect is the same as putting a 0-marble into the A input of the last half-adder directly.

Since the half-adders wait for the A input before processing the B input, this means that the second-to-last half-adder always finishes its work before the last half-adder does.

Next she tries each of the three arrangements of two zeroes and a one, watching to see the differences between how the marbles make their journey in each case.

In every case, the 1-marble ends up coming out of the 1's digit of the last half-adder. One of the 0-marbles ends up at the 2's digit output output, and the other 0-marble gets dumped out the back into the 'extra marbles' line.

When the 1-marble is input directly to the last half-adder, it follows the same path that it did when put into the B input of the standalone half-adder. When the 1-marble is put into the second-to-last half-adder, it invariably makes its way to the 1's digit output of the second-to-last half-adder, and then into the A input of the last half-adder, where it proceeds as normal.

Hmm. Two ones and a zero, all three ways?

In every case, a 1-marble ends up coming out of the 2's digit of the last half adder and 0-marble comes out of the 1's digit output. Sometimes it's the 0-marble she put in, and sometimes it comes from one of the internal reservoirs of 0-marbles. The spare 1-marble and sometimes a 0-marble roll out the back as extras.

She is, as ever, conscientious about refilling internal reservoirs wherever she depletes one.

Now how about three ones?

That causes both the 1's digit and the 2's digit to disgorge one of the 1-marbles, with the third returned via the 'extra marbles' rail.

"I think I'm starting to see how it works," she says. "Though I'm not sure... how did this system of numbers with only two numbers come about? I think usually there's more numbers than that."

Sandalwood closes her messages.

"Yes! Normally, we use ten different digits," she agrees. "But machines which use ten types of marbles are much more complicated. And even though our current computers work on different principles, it's still true that building them to only handle two different digits makes them a lot simpler, and therefore easier to design, than using ten digits would. So our modern computers also use the two-digit system, which is usually called binary."

"It's a little hard to wrap my head around but I can mostly manage it if I think it through one piece at a time."

Sandalwood nods.

"Yes. Computers can behave in very complex ways. Usually, we work out a small, simple, repeatable piece -- like the half-adder -- and then use it to build larger components without needing to remember the specific internal details of each one. But building systems like this is a difficult skill that many people work hard to get better at," she agrees.

"I think I'm still not sure how adding two numbers together turns into... so much else."

"Entirely reasonable. First of all, computers don't just do addition. You can make similar mechanical systems for subtraction, multiplication, etc. But more than that, letting computers do something involves figuring out how to represent the problem in terms of math," she explains. "For example, to display a private window to someone like I mentioned, you have a system that enters details of the situation into a computer -- where their eyes are, where the window should be, what the window should contain, which photons currently in their eyes took which paths -- and then the computer solves the geometry problem of what angle each photon should be created at over and over again."

She waves a hand.

"A full explanation of how computers work is something that would take many years of education," she claims. "For now, it's sufficient to understand that we can build machines out of simple repeated components to do more complicated tasks. Figuring out how to turn a new task into math is hard, and setting up the system to feed the math problem in the right way is hard, but once you've done those, now the computer can do the calculations repeatedly without additional effort."

"And you have... so many little devices... that you don't have to be careful at all about saving their time and effort?"

"Yes," she agrees. She pulls up a visualization of the current system load. "Right now, about 3% of our computing power is being spent on physical system tracking and manipulation. That includes interface interactions, simulated physics, alternate gravity, traffic routing, atmosphere regularization for teleportation, and a bunch of other stuff. 4% is being spent on purely informational workloads -- financial transactions, sending messages, automatic scheduling, vote tracking, etc. The remaining 93% is being auctioned off, mostly to uploads."

"If the system ever uses more than 10% capacity at once, that's a sign that we need to grow it, and we invest in building more computers," she continues. "So we always have enough."

"A tenth?!" she exclaims. "But—but what—where do you get it all???"

Sandalwood conjures an illustrative ball of stone with swirling gold inlays.

"The same way we make most things. We have technology for moving things around, as you've seen. Fixity crystals, we call them. But 'moving things around' is way more powerful than it initially sounds. 'Moving things around' can mean 'rearranging the smallest components of matter from the middle of the sun to make any material object'. In this case, a computer compares the positions of all the particles in the pattern for the target item to the positions of compatible particles in the sun, and then executes a series of teleports to put all the particles in the right location," she replies.

"There are also tricks for generating matter from nothing, but that's more expensive. Anyway, once we have the pattern for something, we can make many copies of it cheaply. So producing more computers to keep up with demand is not a very large expense."

"...that's made of the sun??? Isn't the sun not made of the right things for that???"

"So it's true that the sun is mostly Hydrogen, which is not the right element to make stone out of," she agrees, tossing the ball up and down a little. "But for one thing, the sun is very big, so even trace elements are present there in large quantities. And for another, elements are actually made up of even smaller parts called protons and neutrons, which are again made up of even smaller parts called quarks. So if you can rearrange individual quarks, you can turn one element into another element."

"I think," she says, consideringly, "that that explanation would make more sense if I knew what hydrogen is."

"Oh! I'm sorry," Sandalwood replies. "When you said that the sun was not made of the right things, I assumed that meant you knew what it was made of."

This first contact managed to skip all the math and science they were expecting to be a necessary prerequisite to communication, but now it sure is circling back around to math and science.

Their visitor has responded well to having a tactile thing to experience, so she summons a sample-platter periodic table. The gasses are in little clear canisters, as is the mercury, and anything that reacts explosively with air. Radioactive elements have been not had samples included at all, but the periodic table is otherwise complete.

"So obviously there are a great many materials. And I already told you that they're all fundamentally made up of quarks," she begins, ignoring leptons for the moment. "But we usually think of there being a difference between the parts they can be broken down into mechanically, chemically, and nuclearly."

She summons a bowl, and crushes the stone orb in her hand, letting the pieces rain down into the bowl. "Mechanically, you can grind this orb up and see that it is made of bits of stone and of gold. You can see them here."

She takes a scoop of the stone and drops it into a beaker of acid, swirls that while waiting for it to dissolve, and then pours another beaker into it to precipitate the silicon.

"Chemically, you can dissolve the stone and eventually figure out that the stone is made of silicon, which is this part falling out of the liquid like snow, and oxygen. Silicon and oxygen are what we call 'elements'. Elements are these things," she says, gesturing at the periodic table. "They are the pieces things are made of which you can't break down just by using chemistry. Hydrogen is also an element."

She points out Hydrogen on the periodic table.

"The sun is mostly Hydrogen, with trace amounts of other elements."

She is prepared to keep lecturing if that seems warranted, but she pauses to see if this has spurred any new questions, or if their guest wants to play around with the element samples for a bit.

"Hmm," she says. "I think I... don't expect changing something into something else to tell you very much about what the first thing was made of, necessarily? But I guess you're saying that the way you're doing it, it does do that?"

Sandalwood thinks about this for a moment.

"It's not so much that changing one thing into another tells you about what the thing was originally made of," she replies. "So much as that our method requires you to already know, very precisely, what it is made of. But our technology can tell where particles are, and what kind of particles they are, in the same way that it moves them, so this isn't a problem in practice. Technically, the fixity field simply moves a particle to where it already is, and determines the kind of particle thus moved by looking at how much mass it just moved, even though the movement itself is infinitely short."

She lets that explanation hang for a moment and then says "I can explain how the sensing works in more detail if it's important, but it requires a decent amount of calculus knowledge. Does the first part of that explanation make sense?"

—she shakes her head. "I mean—" She gestures at the beakers. "Changing one thing into another, like you're doing there."

"Oh! Chemistry," she exclaims. "Uh ..."

She stares for a moment at the accumulated chemistry detritus, and then clears it away, replacing it with a taper in a jar on an old-fashioned balance scale.

"So we used to think that when things changed in certain ways, like being burned, that there wasn't much -- or any -- relationship between the things they were made of before and the things they were made of afterwards. But a few hundred years ago, a scientist studying what things were made of noticed something strange."

She snaps her fingers, lighting the taper on fire.

"If you burn something in an enclosed container, how heavy it is doesn't change. That is, even once the taper burns down, the whole enclosed jar will weigh the same amount. Which is a bit curious -- the burnt taper is clearly smaller. And if you weigh it alone, without the rest of the jar, it is lighter. So burning the taper made the air inside the jar heavier."

She waves a hand.

"Now, that's not conclusive proof that what something is afterwards is based on what it was beforehand," she continues. "But it is evidence that there is something which is conserved when you burn things. Same thing for other operations, like mixing flasks of chemicals, or melting things. If you do it in a sealed container, none of these operations change how much the whole system together weighs. Which suggests that one rule of Chemistry -- the art of changing substances into other substances by mixing and burning and so on -- is that you can't end up with something which weighs a different amount than what you started with. Does that make sense so far, or no? I promise I'm going somewhere with this."