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.
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.
The Jones family sat around the dining table performing a traditional holiday ritual: the Christmas card assembly line.
First, Dr. Fibonacci Jones (the world-famous mathematical archaeologist) signed for himself and his wife. He handed the card to Alex, who signed for herself and baby Renée. Then Alex’s younger brother Leon added his own flourish. Finally, Mrs. Jones wrote a personal note on the cards going to immediate family and close friends.
One-year-old Renée sat in her high chair, chewing the corners of an extra card.
Alex Poses a Problem
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.
Mrs. Jones put down her pen and picked up Renée.
“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.”
Alex laughed. “If you keep that up, Renée will learn to spell before she’s out of diapers!”
Leon thumbed the stack of envelopes and groaned. “C’mon, sis. Back to work!”
Before long, Mrs. Jones came back and chased the kids away from the table. “I’ll finish this,” she said.
Unfolding the Magic
Alex and Leon ran to the study. They found Dr. Jones at his desk, playing with a piece of paper.
“Ah, there you are,” he said. “Here, Alex. What do you think?”
“Well,” she said, “it looks like a regular piece of paper that’s been folded over on itself.”
Dr. Jones nodded. “Now you know a sheet of paper has two faces—that is, it has a front and a back.”
Leon reached for the paper and flipped it over. “Is that why you put red stripes on one side and blue stripes on the other?”
“Observe,” Dr. Jones said.
He took the piece of paper and folded it in half. Then he unfolded it and handed it to Alex.
“Hey, how’d you do that?” she asked. “Now there are blue polka-dots on this side.”
“Cool! It’s magic,” Leon said.
“It is called a tetra-tetraflexagon,” Dr. Jones said, “and it has one more hidden face. Can you find it?”
Alex folded the paper this way and that. Then she held it up in triumph.
“Look, red dots—I did it!”
She gave her dad a tremendous hug. “Thanks, Dad! I’ll make magic flexagons. They’ll be the best Christmas cards ever!”
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.
The MIT Mathmen got the ball on their own 20-yard line for the last drive of the game. They were down by 2 points, so they needed at least a field goal to win the game.
If quarterback Zeno and his offense advanced the ball halfway to the opposing team’s end zone on each play…
Here’s how Alex created tetra-tetraflexagon Christmas cards to send to her friends: 
Denise Gaskins' Let's Play Math 
