“Using mathematics to tell stories and using stories to explain mathematics are two sides of the same coin. They join what should never have separated: the scientist’s and the artist’s ways of uncovering truths about the world.”
—William Frucht
It began with some 4th-8th grade friends who met in my dining room to work math puzzles and play games. At first, a few of the kids wondered how anyone could have fun with math. But we did enjoy ourselves, and Math Club grew until we couldn’t fit anyone else around the table.
When one girl had to move away, I thought, “Why not send Math Club with her?” Thus was born Math Club by Mail newsletter (later renamed Mathematical Adventures), a 4-year snail-mail newsletter that brought puzzles, strategy games, art projects, historical tidbits, and more to readers as far away as Hawaii.
Never very many readers, I admit—certainly never enough to make a profit—but it was a blast while it lasted.
So I thought you might like to read Alex’s story from the beginning, as my original math club kids did. The stories will appear sporadically, scattered between my more typical math-activity and teaching tip posts.
I hope you will have as much fun reading these adventures as I had writing them.
Since I’ve been posting new Alexandria Jones stories this week (beginning here), I’ve gone back and re-read the old Christmas posts. I noticed that the original Graph-It Game included a religious design, but nothing for those who don’t celebrate Christmas.
So I updated the post with a new, non-religious puzzle. Here it is, if you want to play…
Graph-It Game Design
For this design, you will need graph paper with coordinates from −8 to +8 on both the x- and y-axis. Connect the points in each line. Stop at the periods, and then start a new line at the next point.
First, they made a sturdy box with circle, square, and triangle shapes cut in the lid.
To make the blocks large and baby-safe, Alex and Leon bought a 4-foot 2×2 board. Then they asked Uncle Will to help them create a set of special blocks to fit through the holes.
Each block was round and square and triangular, so it could fit exactly through any of the three holes.
She drew the dark outline of each polygon with a ballpoint pen, pressing hard to score the cardboard so the tabs would bend easily.
She cut out shapes until her fingers felt bruised: 20 each of the pentagon and hexagon, 40 each of the triangle and square.
Alex bought a bag of small rubber bands for holding the tabs together. Each rubber band can hold two tabs, forming an edge of the polyhedron. So, for instance, it takes six squares and twelve rubber bands to make a cube.
Finally, she stuffed the whole kit in a plastic zipper bag, along with the following instructions.
Polyhedra Have “Many Faces”
Poly means many, and hedron means face, so a polyhedron is a 3-D shape with many faces.
The plural of polyhedron is polyhedra, thanks to the ancient Greeks, who didn’t know that the proper way to make a plural was to use the letter s.
Each corner of a polyhedron is called a vertex, and to make it more confusing, the plural of vertex is vertices.
Regular Polyhedra
Regular polyhedra have exactly the same faces and corners all around. If one side is a square, then all the sides will be squares. And if three squares meet to make one vertex, then all the other vertices will be made of three squares, just like that first one.
There are only five possible regular polyhedra. Can you figure out why?
Here are the five regular polyhedra, also called the Platonic solids. Try to build each of them with your construction kit.
Tetrahedron: three equilateral triangles meeting at each vertex.
Hexahedron: three squares meeting at each vertex. Do you know its common name?
Octahedron: four triangles at each vertex.
Icosahedron: five triangles at each vertex.
Dodecahedron: three pentagons per vertex.
You can find pictures of these online, but it’s more challenging to build them without peeking at the finished product. Just repeat the vertex pattern at every corner until the polygons connect together to make a complete 3-D shape.
Semi-Regular Polyhedra
Semi-regular polyhedra have each face a regular polygon, although not all the same. Each corner is still the same all around. These are often called the Archimedean polyhedra.
For example, on the cuboctahedron, every vertex consists of a square-triangle-square-triangle combination.
Here are a few semi-regular polyhedra you might try to build, described by the faces in the order they meet at each corner:
Here’s how Alex created tetra-tetraflexagon Christmas cards to send to her friends:
1. Buy a pack of heavy paper at the office supply store. Regular construction paper tears too easily.
2. Measure and divide the paper into fourths one direction and thirds the other way. Fold each line backward and forward a few times.
3. Number the front and back of the paper in pencil, lightly, as shown. Then carefully cut a center flap along the dotted lines.
4. Fold the paper along the dark lines as shown, so the center flap sticks out from underneath and the right-hand column shows all 2’s.
5. Fold the flap the rest of the way around to the front and fold the right-hand column under again. (Shown as dark lines on the diagram.) This makes the front of the flexagon show 1’s in every square.
6. Carefully, tape the flap to its neighbor on the folded column. Don’t let the tape stick to any but these two squares.
7. Gently erase your pencil marks.
Find All the Faces
A tetra-tetraflexagon has four faces: front, back, and two hidden. It is shaped like a tetragon — better known as a rectangle.
Here’s how to flex your tetra-tetraflexagon card:
Face 1 is easy to find. It’s on top when you make the card.
Turn the card over to find Face 2.
Face 3 is hidden behind Face 2. Fold your flexagon card in half (vertically) so that Face 1 disappears. Unfold Face 2 at the middle, like opening a book. Face 3 should appear like magic.
Face 4 is hidden behind Face 3. Fold the card (vertically) to hide Face 2, then open the middle of Face 3. Face 2 vanishes, and Face 4 is finally revealed.
When Faces 2 and 3 are folded to the back, you will notice that any pictures you drew on them will look scrambled. What happened?
Add Your Designs
Alex wrote a holiday greeting on Face 1. Then she drew Christmas pictures on the other three faces of her card.
Alex dropped her pen and shook out her tired fingers.
“I’m stumped,” she said. “I’d like to send a special Christmas card to some of my friends from camp last summer. But I can’t think of anything that seems good enough.”
Leon leaned his chair back in thought.
Then he snapped his fingers. “I’ve got it! We’ll throw a handful of sand in each of their envelopes. You know, to make them remember all the fun you guys had digging up old stuff.”
Alex humphed. “How would you like to get sand in your Christmas present?” she asked. “Besides, it wasn’t stuff. It was artifacts.”
“You should not make such a display of your ignorance, young man,” Dr. Jones said. “Stuff, indeed!”
Mrs. Jones put her hand to her forehead and sighed dramatically. Then she turned to Alex. “Have you considered doing a jigsaw puzzle card? They sell them at the hobby store.”
“I’ve tried those before,” Alex said, “but the ones I had always warped. The puzzles didn’t go back together very well.”
Dad Gets an Idea
Dr. Jones got an out-of-focus, “I’m thinking” look in his eyes. He stood up, tapped his chin with his pen, and walked away. He almost ran into the wall, but he caught himself. Shaking his head, he disappeared into his study.
“Why don’t you two address those envelopes while we wait for your dad’s inspiration to reveal itself? I need to put a little one down to S-L-E-E-P.”
Alexandria Jones and her family are fictional characters from my old Mathematical Adventures newsletter. Their stories appear sporadically as I find time to transcribe them from the back-issues. You can find them all on this blog page.
Here are all the Alexandria Jones stories Christmas stories, with activity and craft ideas…
Alexandria Jones and the Christmas Present Quandary
Alex designs tessellation wrapping paper, hunts for the perfect Christmas tree, and comes up with a lively present for her brother. We meet the rest of Alex’s family — her father was introduced in an earlier issue — along with historical figures Maria Agnesi and Leonhard Euler, and we take a brief glance at mathematics from China.
And New This Year: Alexandria Jones and the Magic Christmas Cards
Dr. Jones suggests a way to make the “best Christmas cards ever” (according to Alex), and the Jones children create geometric gifts to celebrate the holiday.
Alexandria Jones collapsed onto the couch with a dramatic sigh. Her father, the world-famous archaeologist Dr. Fibonacci Jones, glanced up from his newspaper and rolled his eyes.
“I don’t even want to hear about it,” he said.
Alex’s brother Leonhard was playing on the floor, making faces at the baby. He looked up at Alex and grinned.
“I’ll take the bait,” he said. “What happened?”
“Mom called my bedroom a Strange Attractor.”
“Oh? What does it attract?”
“I don’t know. Mostly books and model horses. But what’s so strange about that?”
The Mathematics of Chaos
Animation of a double compound pendulum showing chaotic behaviour.
Dr. Jones laughed and put down his paper. “Strange attractor is a technical term from the branch of mathematics called dynamical systems analysis — often called chaos theory.”
“So my bedroom is a math problem?”
“No. I think Mom meant your bedroom was chaos.”
“Oh.” Alex looked like she might pout, then she shrugged. “I guess she’s right, at that. So what is a strange attractor, really?”
“Well, when scientists first drew graphs of classical, non-chaotic systems — like a planet’s orbit or the flight of a football — it was surprising how often they got an ellipse or parabola or some similar curve,” Dr. Jones explained. “For some reason, nature seemed to be attracted to the shapes of classical geometry.”
Alex and Leon enjoyed their baby sister, but they were amazed at how much work taking care of a baby could be. One particularly colicky night, everyone in the family took turns holding the baby, rocking the baby, patting her back, and walking her around before she finally succumbed to sleep.
Then Alex collapsed on the couch, and Leon sank into the recliner. They teased each other with these story problems.
For Leon’s Christmas gift, Alex made the Graph-It game. She wrapped a pad of graph paper and wrote up the instructions:
To play Graph-It, one person designs a picture made by connecting points on a coordinate graph. He reads the points to the other player, who tries to reproduce the picture.
For toddler Renée’s Christmas gift, Alex and Leon crafted a puzzle set of wooden blocks. 
Here’s how Alex created tetra-tetraflexagon Christmas cards to send to her friends: 
Denise Gaskins' Let's Play Math 
