What does mod thirteen mean?

Emily freezes. That is not supposed to happen.

Don't trust anything if you can't see where it keeps its brain wars with Always be polite to magical people that you meet in mundane guises in her brain for a long moment.

But Emily has always been curious, and she's not going to miss the start of her story.

It means that the math is performed dividing by thirteen at each step and keeping the remainder. So 7 + 7 = 1 (mod 13).

If you take a times table and do it modulo different numbers they make a sort of snowflake pattern out of numbers. I've been trying to do all the 20x20 times tables in bases 2-20 modulo 2-20.

Neat! I'll get out of your way, then.

Uh.

On the one hand, she wasn't expecting her Mysterious New Magical Acquaintance to be interested in her times tables. On the other hand, she has already written the title and it will bother her until she finishes it.

Well, if the MNMA™ wants to see her do some math, then ...

01	02	03	04	05	06	07	10	11	12	13	14	15	16	17	20	21	22
01	02	03	04	05	06	07	10	11	12	13	14	00	01	02	03	04	05
02	04	06	10	12	14	01	03	05	07	11	13	00	02	04	06	10	12
03	06	11	14	02	05	10	13	01	04	07	12	00	03	06	11	14	02
04	10	14	03	07	13	02	06	12	01	05	11	00	04	10	14	03	07
05	12	02	07	14	04	11	01	06	13	03	10	00	05	12	02	07	14
06	14	05	13	04	12	03	11	02	10	01	07	00	06	14	05	13	04
07	01	10	02	11	03	12	04	13	05	14	06	00	07	01	10	02	11
10	03	13	06	01	11	04	14	07	02	12	05	00	10	03	13	06	01
11	05	01	12	06	02	13	07	03	14	10	04	00	11	05	01	12	06
12	07	04	01	13	10	05	02	14	11	06	03	00	12	07	04	01	13
13	11	07	05	03	01	14	12	10	06	04	02	00	13	11	07	05	03
14	13	12	11	10	07	06	05	04	03	02	01	00	14	13	12	11	10
00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00	00
01	02	03	04	05	06	07	10	11	12	13	14	00	01	02	03	04	05
02	04	06	10	12	14	01	03	05	07	11	13	00	02	04	06	10	12
03	06	11	14	02	05	10	13	01	04	07	12	00	03	06	11	14	02
04	10	14	03	07	13	02	06	12	01	05	11	00	04	10	14	03	07
05	12	02	07	14	04	11	01	06	13	03	10	00	05	12	02	07	14

... it's sort of easier to see if you kind of unfocus your eyes and see the colors of the different numbers. Like, look at the patterns the 11s make.

Oh! You mean like this?

A little arrow points to the next page, where the numbers rapidly lay themselves out into a grid with a gradient of background colours from no shading at 00 to a dark grey at 14.

Or more like this?

On the page after that, the notebook takes a little more time colouring in the squares of the grid in a rainbow heatmap, which cycles through the colours of the rainbow from red to violet before starting over at a darker red on 10. The result is kind of chaotic, but definitely visually interesting.

Oh, wow!

Hmm. Sort of like either of those. It's the same pattern, you're just seeing it in different ways. And I don't usually have colored pencils with me, so I usually just sort of look at the numbers and see the pattern in them directly.

It kind of helps to actually do the math yourself, because it makes the pattern more

She stops, trying to think of the right word, and taps her eraser on the page a few times.

It makes the pattern easier to feel, if your brain can kind of come at it a few different ways, both calculating it and knowing what it should look like and seeing it visually.

I don't think I've ever done a times table before. I can add and subtract, but I'm not sure about multiplying.

Emily has sort of given up on guessing where this is going for the moment, because she can't really tell. But luckily she's well equipped to explain basic math concepts.

Well, you can actually do the same kind of table with addition, it just makes a different kind of pattern.

But multiplication isn't hard — you can think of it in a few different ways. I think people usually introduce it as repeated addition, so M * N is like having N copies of M and adding them all together:

3 * 4 = 3 + 3 + 3 + 3 = 12

But I like to think of it as being about areas, instead. If you have a grid that has 3 cells in one direction and 4 in the other, it has 12 cells total:

-	-	-	-
-	-	-	-
-	-	-	-

Oh, I see! But those are still both ways of looking at the same thing, right? The twelve spots in the grid are three and three and three and three, or four and four and four, depending which way you count them. And that's adding things up repeatedly even though it's also counting the spots in a grid.

Yeah, that's true! One of the cool things about math is that there are often multiple ways to think about the same things, because a few simple rules end up casting all of these idea-shadows.

But those two different perspectives can help you do more with a concept than either one of them alone. Like, if I write 2 * 3 * 4, the first perspective tells you that that should be 3 * 4 + 3 * 4, which makes it 12 + 12 = 24. But if you just saw 2 * 3 * 4 and tried to do it in the geometric way, you might have trouble thinking of how it fits. (In this case, it's like a volume of a 3d table, instead of an area, but that might be hard to see if you were trying to multiply a big list of numbers)

On the other hand, knowing the geometric way of looking at it helps you multiply fractions. What is 4 * 2.5? Well, with the first way of seeing it, it's not clear how you could write down 4 two and a half times. That would just be, like, 4 + 4 +└, which makes no sense. But with the geometric way, you can make a table with 4 columns and 2 and a half rows, and then count up the boxes and see that the answer should be 10.

She realizes that she forgot a period and goes back up to add it.

numbers).

... and then second-guesses her period location and uses her eraser to move it to the other side of the parentheses.

numbers.)

You could also write down four two and a half times like this, right?

:: :: :

Sure! And if you squint that's the same thing as the table version. That's kind of what it means for them to be two ways of looking at the same idea.

Actually, here's a third neat way of looking at multiplication:

Any number can either be made by multiplying other numbers (like 12 = 3 * 4), or it can't be (like 13). We call the second kind of number a "prime number". By repeatedly breaking down a number into more and more parts, you can reduce it to a bunch of prime factors. For example, 12 = 3 * 4, but 4 = 2 * 2, so 12 = 3 * 2 * 2, and both 2 and 3 are prime so we stop there.

But the trick is this: if you take two big numbers, break them down into their prime factors, and then add how many times each prime factor appears, that's like multiplying the big numbers. So 12 * 26 = (3 * 2 * 2) * (2 * 13) = 3 twos, 1 three, 1 thirteen = 13 * 3 * 2 * 2 * 2 = 39 * 2 * 2 * 2 = 78 * 2 * 2 = 154 * 2 = 308.

That's how I multiply big numbers in my head, because I think it's easier to track than trying to do it the way the school teaches.

Can you see how this way of looking at multiplication is the same as the table version?

Oh, hmm! Let me see...

It's going to get really hard to draw this for numbers with more numbers in them. I could do repeated twos by separating pieces, but if there was a two and a three and a thirteen and a... let me see...

...and a five? Is that right? Then it would be so untidy to draw them by separating pieces instead of by extending the grid again, but it's really hard to extend the grid again!

Five is a prime number too, yeah. The first few prime numbers are 2, 3, 5, 7, 11, 13, and 17. And I think you've got the idea of how the table view and the prime factors view are the same: the tables can be separated/folded so that all of the edge lengths are prime numbers.

Emily thinks for a moment, trying to figure out how to help the MNMA™ to a satisfying epiphany.

There's a trick you can use to draw a table in more than three directions, although it gets a little abstract, so I don't know how much it will help. It goes like this:

She draws a square.

If you draw a square, which has two dimensions, you can turn it into a cube by copying it and connecting each corner to its copy.

She draws a second square partially overlapping the first, and then adds in four lines to make a wireframe cube in 3/4 perspective.

The same thing works to go from a cube to the equivalent four dimensional shape, which is called a tesseract.

She draws a second cube offset from the first in a different direction, and then adds eight lines to make a wireframe (shadow of) a tesseract.

And it gets a little squished, because you're trying to flatten this complex shape into two dimensions so it fits on the page, but you can still see the structure of the overall shape. And this should work in theory for drawing a cube in any number of dimensions, but I've never done more than 5 because it gets really messy.

Oh, clever! I hadn't thought to draw it that way. I was stuck thinking of the next dimension as drawing more grids on subsequent pages, which is nicer than drawing separate grids on one page because I can feel how the pages are next to each other. But my pages only extend in one more direction, at least in this world, and anyway I think grids on separate pages is probably less legible from outside of me than separate grids on one page.

So you are the notebook? You're not a struggling wizard's apprentice, stuck in his high tower and desperate for a penpal who will learn the secrets of algebra (and friendship) from an ordinary Earth schoolgirl? Eventually teaching her the secrets of magic in turn, until you manage teleportation and come for a visit, proving to her skeptical teachers and schoolmates that her penpal really exists, and bringing magic back into the world after a thousand years of absense?

Because that was my best guess.

Oh! No, I'm sorry, I've gotten very distracted by all these interesting thoughts and forgotten to explain myself. Yes, I'm the notebook! I was sent here by the Spirit of Femininity Unleashed to offer you its power.

Well, that's a bit on the nose. But finding a mysterious book that grants strange powers and opens the door to a world of magic is something she knows how to handle.

Then she remembers she's trying to be polite.

Nice to meet you — my name's Emily.

Why me? Am I secretly a princess and Parent is actually my nursemaid who fled with me from the lands where the Spirit once ruled, and now it needs me to be its champion and take back the kingdom from the forces of darkness?

(That was my second guess)

This morning she was pretty sure that her parent was her biological parent, but most protagonists are orphans, so it would make sense.

What a neat story that would be! But no, it's nothing like that. The Spirit is much bigger and stranger and farther away than the sort of thing that could rule some lands. It chooses people who want to be beautiful and powerful and special in a feminine way, but because it's so big and strange and far away, it doesn't really get to them in any kind of sensible order.

Huh. Okay, so she's just lucky, not a secret princess. That's fine, lucky people usually get the best endings anyway. It would be better if she had two older siblings, though. Maybe she can convince her parent to adopt.

Then a horrible thought strikes her.

... is this a puberty metaphor?

Not that I would object, you're just a little early.

I don't think so! Most of the people the Spirit chooses are from in or after that stage of life, not before. I guess you could think of it as a puberty metaphor if you wanted, though.

That's the problem with metaphors, in her opinion.

Okay — so how do you grant people powers?

She would make another guess, but three wrong guesses in a row is her limit.

I talk with you about what kinds of powers you would want to have, and you decide which ones you want, and eventually you tell me you're satisfied with what you've chosen, and I channel the Spirit's power into you and grant you your powers and send you on to your destination. (You could stay in this world, or pick a different one, or let the Spirit choose one for you.) There's a list of powers that I can show you, but if there's anything you want that isn't on the list, or anything on the list that you'd like more if it were a little different, I can build custom powers for you too. Every power is associated with a number of points that approximates how much of the Spirit's power goes into granting it to you, and you have 70 points to spend across all your powers, plus there are some options on the list that grant points instead of costing them.

Also, because you haven't started the relevant life stage yet, there are some powers in the list that you might have trouble deciding what you think of. The usual recommendation in that situation is for you to reserve some points, perhaps seven or so. When you finalize your choices with some points reserved, those points stay with you as unrealized potential, waiting to become more powers later when the time is right. You don't get to choose how or when they spend themselves, but they always do their best to become the powers you would want at the time when you would want them.

Huh. And you can grant any kinds of powers, the only limit is the points?

She taps her pencil on the notebook again, trying to think what to ask for.

Can I be a dragon? The noble kind, with shapeshifting and magic and maybe a river, not the kind with gold-lust. My parent says to always be yourself, unless you can be a dragon.

There's a power called Dragon Fairy Elf Witch, which does this:

Name: Dragon Fairy Elf Witch - Cost: 5 ☐
You can at any time discover previously unknown heritage from any type of being you encounter, even if this makes no sense or contradicts previously established descriptions of your family tree. You always get their powers without their drawbacks, unless the drawbacks are cool and dramatic. Any visible features of this heritage will appear at narratively appropriate moments and be cute, pretty, beautiful, or striking rather than awkward, weird, gross, or scary. This ability works even if the beings in question cannot reproduce with humans, or at all.

Is that the sort of thing you're looking for?