Codes and Secret Messages

A barcodeOne of the fun things you can do with numbers is create codes. Codes are ways of organizing information in different, useful ways. In one sense, the Fibonacci sequence is a code that nature uses to create the best spacing between things. If you flip over An Elf’s Equations, you’ll find a barcode on the back. Bookstores use barcodes as a quick and easy way for their computers to identify books. And codes can be used to keep information secret and safe.

But a code doesn’t have to be very complicated to work well. Here’s a very simple code that borrows its complexity from other objects. If you and a friend want to send each other secret messages, all you need is two copies of the same edition of a book. Write your message, then look through the book for all the words of the message. Then write down just three numbers to encode the message: the page number, the line number, and the number of the word within that line. For example, the word “Cretacia” appears on page 75, line 5, word 6. Go check it out!

This kind of code is called a book cipher. To make the code even harder to crack, people write the location as just the three numbers. So “Cretacia” above would be (75, 5, 6). The word “Author” appears on page 311, line 1, word 3, or (311, 1, 3).

Got it? Okay, then I have a secret message for you, and you can only decode it if you have a copy of An Elf’s Equations. Ready?

(74, 4, 7) (151, 21, 2) (261, 12, 2) (108, 11, 4) (245, 11, 8): (48, 28, 6) (48, 9, 4) (176, 14, 4) (61, 20, 7) (79, 4, 1) (63, 23, 9) (212, 17, 1) (287, 13, 5) (207, 7, 8) (214, 4, 5)? “(56, 24, 1) (65, 7, 8).”

Did you figure it out? Don’t post it in the comments! Let other people figure it out for themselves. However, feel free to create your own secret messages and post those, either here or on my Facebook post about secret messages. And you can use this method to send secret messages to your friends, as long as you each have an identical copy of the same book. I recommend that you choose a book with a broad vocabulary. That way, you’ll have lots of different words to choose from.

One thing I’m curious about. I actually don’t have an eBook copy of An Elf’s Equations, so I don’t know if the pages, lines, and words fall in the same places. Could someone with an eBook version check? If it turns out to be complete gibberish, go ahead and post it in the comments, just to show how horribly wrong a code can go!

Tomorrow, I’ll be doing a live question and answer session. It’s much easier for me to answer questions if I have them in advance, so please post your questions in the comments either here or on Facebook, and I’ll do my best to answer them. Thanks!

Make Your Own Thea

Drawing of cacao treeDo you know where chocolate comes from? Do you really?

You might have heard that chocolate comes from cacao beans, and that these grow on cacao trees, but did you know that:

* The scientific name for cacao trees is Theobroma cacao? That’s where Thea gets her name.
* There are four varieties of cacao tree: forestero, criollo, trinitario, and nacional. The nacional variety was found in Peru just nine years ago!
* The genus Theobroma is probably millions of years old, but modern cacao trees first appeared around 10,000 years ago in the Amazon basin, and they’ve been cultivated for 3,000 to 5,000 years.

* Cacao trees only grow in tropical climates very close to the Equator. Most cacao trees grow in West Africa today, but they are also grown in Central and South America, Indonesia, the Caribbean, and Hawaii. So unless you live in one of those places, you can’t have a cacao tree in your backyard.
* Cacao trees can grow to be 30 feet tall, even if they grow in the shade of much larger trees.* Cacao trees can bloom and bear fruit all year around, but they are usually harvested twice per year.
* A single cacao tree can have as many as 6,000 blossoms, but not all of them turn into fruit.
* Cacao fruit, called pods, can be many different colors, from bright yellow to green, brown, red, or even purple!
* Cacao pods grow on the trunks of cacao trees.
* Cacao trees are very vulnerable to fungal infection, and cacao habitats are slowly shrinking due to deforestation and climate change.
* Chocolate was first brought to Europe in 1502, but it wasn’t made into solid bars until 1848.
* The first chocolate house in the United States in 1765, owned by John Hannon and Dr. James Baker, after whom Baker’s Chocolate is named.
* When I was a student at MIT in Cambridge, MA, I used to walk past a chocolate factory every day on my way to class.

If you’re interested, you can read more about cacao trees here:

I’m fascinated by cacao and chocolate, which is why one of my characters is Thea, a baby magical intelligent cacao tree.

Thea was no ordinary tree. The Enchanted Forest School that Petunia and Millie attended took place in the branches of an enormous, intelligent, and magical oak tree, a dodonos named Quercius. Dodonoi were extremely rare — there were only four in the whole Enchanted Forest Realm before Thea came along.  …At school, Millie had accidentally transformed a green bean into a cacao bean, and because it had sprouted so close to Quercius, it became a dodonas, an intelligent, magical tree: Thea. One day, Thea would walk and talk and even attend school, learning to use her magic. But for now, she was still a baby and needed care and protection.A Pixie’s Promise, p. 17.

Now you can create your very own Thea to care for and protect! You will need:

  • Paper and/or cardstock
  • Cardboard from a cereal box or shipping box
  • Scissors
  • Tape and/or glue
  • Markers, crayons, colored pencils, and/or paint
  • Optional: pipe cleaners, popsicle sticks, stickers, glitter, googly eyes – whatever is fun for you!


  1. Download MakeYourOwnThea-template and print it out. You might want to print extra copies of page 3. If you can print it on cardstock, you can skip steps 3 and 4 below.
  2. Cut out the two tree shapes. Be sure to cut the slot in each trunk. You might want to ask an adult for help with this.
  3. Use the tree shapes as stencils and draw their outlines onto some cardboard.
  4. Cut out the cardboard tree shapes. You might want to ask an adult for help with this. Adults, for heavy cardboard, you might want to use a box cutter or other sharp implement.
  5. Slide the two tree shapes to form an X. The tree should be able to stand on its own.
  6. Color the leaves, cacao pods, and flowers on page 3.
  7. Cut them out and decorate the tree’s branches and trunk with them. While Thea is actually too young to have flowers or fruit (she’s less than a year old, and cacao trees usually don’t bloom until they’re 3-5 years old), you can use them if you want.
  8. Add other decorations, such as eyes, pipe cleaners, and stickers. Make your tree magical!
  9. Take a photo and post it in the comments on the Dianna Sanchez Facebook page.

Tomorrow, I’ll be posting a secret message! Have your copy of An Elf’s Equations handy to decode it.

Also, on Friday, I’ll be doing a live Question and Answer session. If you post questions in the comments on my Facebook page, I’ll gather them up and answer them.

Happy treemaking!

Fun with Fibonacci

The Fibonacci Series depicted on a nautilus shell.

Image by Karin Henseler from Pixabay

I have long been fascinated by the Fibonacci sequence because it seems to show up everywhere. You might have noticed it, that certain swirl that shows up on things. Here are a few examples:





In An Elf’s Equations, Sagara describes it like this:

Start with 1, then add the previous number to it (in that case, 0), then add the previous number to that number… 1, 1, 2, 3, 5, 8, 13, 21… Sagara could describe this elegantly as an equation: xn= xn-1+ xn-2.  Not that anyone else she knew appreciated that elegance. She could draw this as a series of blocks, then draw a line through their intersections to make a spiral, the same spiral on snail shells and flower petals and pine cones, even faces. Taking the ratio between any two consecutive numbers in the sequence and averaging them gave Sagara the Golden Ratio, 1.618034, which could usually be simplified to 1.6. This number appears over and over and over again in nature. Sagara’s mother told her it even determined the shape of great storms and swirls of stars in the sky. –pp. 46-47

I think Sagara does an okay job of describing the Fibonacci sequence, but I know someone else who does a spectacular job of explaining it: Vi Hart. Her Doodling in Math Class YouTube series shows just how much fun math can be, and how much it influences art. She has three videos on the Fibonacci sequence. The first is the simplest, Part 2 gets a little more complicated, and Part 3 is very complicated, so see how much you can challenge yourself! I recommend you watch them several times because Vi Hart talks REALLY fast.

Want to find some spirals yourself? Ordinarily, I’d suggest that you go shopping in the grocery aisle of your supermarket and look for pineapples, artichokes, cauliflower, and stalks of brussels sprouts to try out, but since we’re all avoiding grocery stores, go outside! Pine cones, leaves on stalks, flowers (if they’re up and growing in your part of the world), and even acorn caps all have Fibonacci spirals on them. Can you find them? Can you find other spirally things in your yard?

And if you can’t, try drawing slug cats and flowers and pine cones, or come up with new creatures made out of spirals, Sagara, for example, draws spiral dragons.

Tomorrow, I’ll give you a template so you can build your own model of Thea!

Decorate Your Room like Sagara’s Room

Hello, everyone! I hope you’re all finding interesting and fun things to do while practicing social distancing. This week, I’m going to be posting activities for you to do while you’re home. Today’s theme is Sagara’s bedroom, which is decorated with MATH. Here’s an excerpt from An Elf’s Equations:

Sagara turned and ran lightly along a branch as broad as the Path by Millie’s house until she reached her bedroom. Pulling aside a curtain of cultivated ivy, Sagara touched a smooth river stone set in a niche beside the door, and it began to glow with soft, golden light. She entered a room made entirely of living woven branches, much like the classrooms in Master Quercius’s branches at the Enchanted Forest School. This room was much smaller and private, decorated with mathematical constructs: fractal patterns, the Fibonacci spiral, graphs and diagrams. A long strip of paper circled the leafy ceiling with the first two hundred digits of pi. — p. 31

What the heck is all this stuff? Well, here are some brief explanations of what they are and why Sagara thinks they’re cool.

A representation of the number pi

Pi Goes On Forever

Pi is a strange and unusual number. It’s the number we’ve discovered that describes round things: circles, balls, even the orbits of planets (if the orbits were perfect). Draw a circle and put a dot in the exact center of the circle, then draw a line straight through that dot from one side of the circle to the other. We call that line the diameter. Now put a dot anywhere on the circle. The length of the circle if you go all the way around it is called the circumference. So if you have a piece of string, and you make a circle out of it, the length of the string is the circumference. If you measure the circumference and the diameter very carefully, you can find pi.

For example, let’s say you have a string that’s a foot long. If you make a circle out of it and measure its diameter, you’ll find that it’s a little under four inches, about 3.8 inches. Now, divide 12 inches by 3.8 inches. You can use a calculator, it’s okay. You’ll get a number with a lot of digits after it: 3.157847368 is what my calculator says. That’s pretty close to pi, but not exact because we didn’t measure the radius exactly enough. If you measure it really, really carefully, it’s more like 3.14159. Here’s a little video of how that works:


Many, many digits of pi. Image by Andrew Martin from Pixabay

Now here’s the really cool thing. If you could measure the diameter and the circumference absolutely perfectly, the digits would go on forever. What’s more, they never repeat. Mathematicians call this a transcendental number. They have calculated millions of digits of pi, and they still haven’t found the end. Sagara loves this. She loves thinking about the fact that it goes on and on and on forever. So she glued together pieces of paper and copied out as many digits as she could and put them up all around her room. You can do that, too! How many digits of pi can you write out?

A Mobius Strip has Only One Side

Take a piece of paper. Cut it into three strips and tape or glue them into one long strip. Now give the strip one twist and tape the ends together. Here are some good step-by-step instructions. Now, put a dot in the middle of the strip, anywhere, and start drawing a line down the middle of the strip. Keep going. Eventually, you’ll reach the dot again. HOW IS THAT POSSIBLE???

It’s possible because a Mobius strip actually only has one side. That twist you put in the loop means that you connected on side of the paper to the other, making an infinite loop. Sagara thinks this is better than magic, and she has little mobius strips dangling from the branches of her room. When she gets bored, she takes one down, draws that line, and then cuts along the line to see what happens. Try it! I guarantee you’ll be surprised.

Fractals Can Be Infinite

Fractals are repeating patterns that appear in nature. Be warned! Once you start seeing them, you can’t stop.

Here’s a simple example of a fractal. Draw an equilateral triangle – that’s a triangle whose sides are all the same length and whose angles are all the same 60 degrees. Now, draw another, upside down triangle inside that triangle, with each of its points at the middle of the larger triangle’s sides. Suddenly, you have four triangles. Do that again. And again. And again.

via the Boston University Math page on Serpinski Triangles

If you had a big enough triangle, or a good enough magnifying glass, you could keep on doing this forever. The triangle doesn’t even have to be equilateral. Try drawing some triangles of different sizes and see if you can keep dividing them up the same way.

Tangrams: Many Pictures from the Same Shapes

Sagara loves turning things around and seeing them in new and surprising ways. Tangrams are images that you make from a small set of shapes. By courtesy of my friend Rebecca Rapoport, co-author of Math Games Lab for Kids, you can download and print the basic tangrams set and many different tangrams shapes to make (there are some other cool math downloads on that page, too). I recommend coloring the pieces for the tangrams many different colors, then seeing how the colors fit into the shapes. Sagara enjoys trying to create new shapes for herself and has several unique creations posted around her room.

As for Fibonacci spirals, we’ll get to that tomorrow. Happy mathing!

My Book Birthday is Pi Day!

Screenshot of video of Dianna Sanchez reading from An Elf's Equations

At long last, An Elf’s Equations is available for purchase on and available for order at your local bookseller! Today is also Pi Day (3.14), so to celebrate both, I read from the book live on Facebook. You can also watch me read from Chapter 1 here.

As I discussed in my previous post, I’m going to start providing activities related to An Elf’s Equations on Monday and every weekday at 11am EDT while schools are closed for the next two weeks. Here’s my schedule of activities for next week:

Monday, March 16: Decorate your room like Sagara’s room! Sagara loves math, and her room is decorated with all kinds of fun math puzzles and diagrams. Check this blog for downloadable activities.

Tuesday, March 17: Fun with Fibonacci! Draw your own Fibonacci spiral, watch videos about how the spiral is used in art, and go find Fibonacci spirals in nature.

Wednesday, March 18: Make your own Thea with downloadable patterns. Recipe for something chocolate.

Thursday, March 19: Secret message! Use a code to find a message hidden in An Elf’s Equations.

Friday, March 20: Live Question and Answer session. I’ll answer questions asked in the comments on Facebook.

So check back here on Monday at 11am EDT for more fun stuff!