Video from the Global Math Project.

And here are some additional answers.

Ask your kids the question: “What Is Math?”

I’d love to hear what they say!

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The monthly *Math Teachers at Play* (MTaP) math education blog carnival is almost here. If you’ve written a blog post about math, we’d love to have you join us! Each of us can help others learn, so in a sense we are all teachers.

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up to first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Click here to submit your blog post

Have you noticed a new math blogger on your block that you’d like to introduce to the rest of us? Feel free to submit another blogger’s post in addition to your own. Beginning bloggers are often shy about sharing, but like all of us, they love finding new readers.

**Don’t procrastinate:** *The deadline for entries is this Friday, January 20th.* The carnival will be posted next week at Travels in a Mathematical World blog.

Help! I can’t keep the carnival going on my own. Hosting the blog carnival can be a lot of work, but it’s fun to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival usually brings a nice little spike in traffic to your blog.

If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

While you’re waiting for next week’s *Math Teachers at Play* carnival, you may enjoy:

- Browse past editions of the
*Math Teachers at Play*blog carnival - Carnival of Mathematics
- Carnaval de Matemáticas

Want to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.

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What teacher hasn’t heard a student complain, “When am I ever going to have to use this?” Didn’t most of us ask it ourselves, once upon a time?

And unless we choose a math-intensive career like engineering, the truth is that after we leave school, most of us will never again use most of the math we learned.

But if math beyond arithmetic isn’t all that useful, then what’s the point?

If you or your student is singing the “Higher Math Blues,” here are some quotations that may cheer you up — or at least give you the strength of vision to keep on slogging.

I don’t want to convince you that mathematics is useful. It is, but utility is not the only criterion for value to humanity. Above all, I want to convince you that mathematics is beautiful, surprising, enjoyable, and interesting. In fact, mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that — by some mysterious agency — capture patterns of the universe around us? Mathematics connects ideas that otherwise seem totally unrelated, revealing deep similarities that subsequently show up in nature.

— Ian Stewart

The Magical Maze

That vast book which stands forever open before our eyes, the universe, cannot be read until we have learnt the language in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

— Galileo Galilei

quoted by Clifford Pickover, A Passion for Mathematics

The investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings.

— George Washington

quoted by William Dunham, The Mathematical Universe

I told myself, “Lincoln, you can never make a lawyer if you do not understand what demonstrate means.” So I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what “demonstrate” means, and went back to my law studies.

— Abraham Lincoln

quoted by William Dunham, The Mathematical Universe

In most sciences, one generation tears down what another has built, and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure.

— Herman Henkel

quoted by Noah benShea, Great Quotes to Inspire Great Teachers

Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals — the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.

— Martin Gardner

quoted by G. Simmons, Calculus Gems

I will not go so far as to say that constructing a history of thought without profound study of mathematical ideas is like omitting Hamlet from the play named after him. But it is certainly analogous to cutting out the part of Ophelia. For Ophelia is quite essential to the play, she is very charming. . . and a little mad.

— Alfred North Whitehead

quoted in The Viking Book of Aphorisms

The mathematician does not study pure mathematics because it is useful, he studies it because he delights in it, and he delights in it because it is beautiful.

— Henri Poincaré

quoted by Theoni Pappas, More Joy of Mathematics

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. The ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

— Godfrey H. Hardy

A Mathematician’s Apology

Mathematics is a world created by the mind of men, and mathematicians are people who devote their lives to what seems to me a wonderful kind of play!

At age eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world.

— Bertrand Russell

The Autobiography of Bertrand Russell

I love mathematics … principally because it is beautiful, because man has breathed his spirit of play into it, and because it has given him his greatest game — the encompassing of the infinite.

— Rózsa Péter

quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians

Did you enjoy these? You can find plenty more on my Math & Education Quotations page.

**I would LOVE to hear YOUR favorite mathematics, education, or inspirational quote. Please share in the Comments section below!**

Never Ending Math Problem photo (above) by Danny via Flickr (CC BY 2.0).This post is part of the #MTBoS #MtbosBlogsplosion blogging challenge.

Want to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.

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I like to use games as a warm-up with my co-op math circle. Some homeschoolers make every Friday a game day, and some turn gaming into a family lifestyle.

If you’d like to add more play to your family’s day, check out Cait’s 2017 Gameschooling Challenge.

“Playing games with your kids offers a host of educational benefits, plus you build relationships and make memories. I am constantly amazed by the amount of learning that happens when I sit down to play games with my children.”

—Caitlin Fitzpatrick Curley

Gameschool Challenge

- How to Make Math Cards
- Game: Tens Concentration
- Math Club Nim
- Tell Me a (Math) Story
- Math Game: What Number Am I?
- Math Game: Fan Tan (Sevens)
- Horseshoes: A Place Value Game

“Games put children in exactly the right frame of mind for learning difficult things. Children relax when they play — and they concentrate. They don’t mind repeating certain facts or procedures over and over, if repetition is part of the game.”

- Number Bond Games
- Active Math Game: Rock
- Maze Game: Land or Water?
- Math Game: Chopsticks
- Addition Games with Cuisenaire Rods
- Free Multiplication Bingo Game

“Coming back from winter break can be hard. Everyone is sleepy, unfocused, and daydreaming of the holiday gifts that await them at home after school. And that’s just the teachers!”

—Andrew Gael

Beat the Back to School Blues…Play a Math Game

- Game: Times Tac Toe
- Contig Game: Master Your Math Facts
- 30+ Things to Do with a Hundred Chart
- Game: Hundred Chart Nim
- Euclid’s Game on a Hundred Chart
- The Game that Is Worth 1,000 Worksheets
- Math Game: Thirty-One
- Multiplication Models Card Game
- Review Game: Once Through the Deck
- Princess in the Dungeon Game

“If you play these games and your child learns only that hard mental effort can be fun, you will have taught something invaluable.”

- Fraction Game: My Closest Neighbor
- Game: Target Number (or 24)
- 30+ Things to Do with a Hundred Chart
- Hit Me! (A Math Game)
- Alcumus Online Problem-Solving Game
- Math Games with Factors, Multiples, and Prime Numbers
- Math Game: Logarithm War
- The Function Machine Game

“Mathematics is mental play, the essence of creative problem solving. This is the truth we need to impart to our children, more important than fractions or decimals or even the times tables. Math is a game, playing with ideas.”

—Denise Gaskins

Let’s Play Math: How Families Can Learn Math Together—and Enjoy It

They don’t have to be math! Please share in the comment section below!

This post is part of the #MTBoS #MtbosBlogsplosion blogging challenge.

Want to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.

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The first way to make your math blog grow is to write posts. Here’s an #MTBoS blog challenge that seems doable: Only one post a week, so maybe even I can keep up.

With the start of a new year, there is no better time to start a new blog! For those of you who have blogs, it is also the perfect time to get inspired to write again! Please join us to participate in this years blogging initiative…

Once you’ve got your post blogged, please share it with us!

The Math Teachers at Play (MTaP) blog carnival is a monthly collection of tips, tidbits, games, and activities for students and teachers of preschool through pre-college mathematics. We welcome entries from parents, students, teachers, homeschoolers, and just plain folks…

[Spiral fractal photo (above) by Kent Schimke via Flickr (CC BY 2.0).]

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If you know of any other resources, please share in the comments below. And as I find new goodies, I’ll add them to the list here:

- NASA Lesson Toolkits
- Lesson Plan: “When Computers Wore Skirts”
- Girls Build LA Lessons & Resource Materials
- Max’s Hidden Figures Lesson Plan
- Norma’s Resource Folder

- The True Story of “Hidden Figures,” the Forgotten Women Who Helped Win the Space Race (Smithsonian Magazine)
- Hidden Figures: Margot Lee Shetterly’s book about NASA’s black women mathematicians and engineers is timely and eye-opening (Scientific American)
- Official Movie Home Page
- Trailers and Cast Interviews

Before computers were machines, computers were people who computed things. This complicated task often fell to women because it was considered basically clerical. That’s right: computing triple integrals all day long qualified as clerical.

— Samantha Schumacher

Hidden Figures Movie Review

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Two of the most popular New Year’s Resolutions are to spend more time with family and friends, and to get more exercise. The 2017 Mathematics Game is a prime opportunity to do both at once.

So grab a partner, slip into your workout clothes, and pump up those mental muscles!

For many years mathematicians, scientists, engineers and others interested in mathematics have played “year games” via e-mail and in newsgroups. We don’t always know whether it is possible to write expressions for all the numbers from 1 to 100 using only the digits in the current year, but it is fun to try to see how many you can find. This year may prove to be a challenge.

**Use the digits in the year 2017 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.**

- You must use all four digits. You may not use any other numbers.
- Solutions that keep the year digits in 2-0-1-7 order are preferred, but not required.
- You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.
- You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
- You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.

- You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
- You MAY use a double factorial,
*n*!! = the product of all integers from 1 to*n*that have the same parity (odd or even) as*n*. I’m including these because Math Forum allows them, but I personally try to avoid the beasts. I feel much more creative when I can wrangle a solution without invoking them.

As usual, we will need every trick in the book to create variety in our numbers. Experiment with decimals, two-digit numbers, and factorials. Remember that dividing (or using a negative exponent) creates the reciprocal of a fraction, which can flip the denominator up where it might be more helpful.

**Use the comments section below to share the numbers you find.**

But please don’t spoil the game by telling us how you made them! You may give relatively cryptic hints, especially for the more difficult numbers, but be careful. Many teachers use this puzzle as a classroom or extra-credit assignment, and there will always be students looking for people to do their homework for them.

**Do not post your solutions. I will delete them.**

There is no authoritative answer key for the year game, so we will rely on our collective wisdom to decide when we’re done. We’ve had some lively discussions in past years. I’m looking forward to this year’s fun!

As players report their game results below, I will keep a running tally of confirmed results (numbers reported found by two or more players). My expat daughter is coming home for a visit this month, however, and we’ll be traveling to see extended family. So this tally will almost certainly lag behind the results posted in the comments.

Percent confirmed: 49%

1–37, 39–43, 45–50, and 79.

Reported but not confirmed: 51%

38, 44, and 51–100.

Numbers we are still missing: 0%

Wow!

Students in 1st-12th grade may wish to submit their answers to the Math Forum, which will begin publishing student solutions after February 1, 2017. Remember, Math Forum allows double factorials but will NOT accept answers with repeating decimals.

Finally, here are a few rules that players have found confusing in past years.

**These things ARE allowed:**

- You must use each of the digits 2, 0, 1, 7 exactly once in each expression.
- 0! = 1. [See Dr. Math’s Why does 0 factorial equal 1?]
- Unary negatives count. That is, you may use a “−” sign to create a negative number.
- You may use (
*n*!)!, a nested factorial, which is a factorial of a factorial. Nested square roots are also allowed. - The double factorial
*n*!! = the product of all integers from 1 to*n*that are equal to*n*mod*2*. If*n*is even, that would be all the even numbers, and if*n*is odd, then use all the odd numbers.

**These things are NOT allowed:**

- You may not write a computer program to do the puzzle for you — or at least, if you do, PLEASE don’t ruin our fun by telling us all the answers!
- You may not use any exponent unless you create it from the digits 2, 0, 1, 7. You may not use a square function, but you may use “^2”. You may not use a cube function, but you may use “^(2+1)”. You may not use a reciprocal function, but you may use “^(−1)”.
- “0!” is not a digit, so it cannot be used to create a base-10 numeral. You cannot use it with a decimal point, for instance, or put it in the tens digit of a number.
- The decimal point is not an operation that can be applied to other mathematical expressions: “.(2+1)” does not make sense.
- You may not use the integer, floor, or ceiling functions. You have to “hit” each number from 1 to 100 exactly, without rounding off or truncating decimals.

- Mathematics Game Worksheet

For keeping track of which numbers you’ve solved.

- Mathematics Game Manipulatives

This may help visual or hands-on thinkers.

- Mathematics Game Student Submissions

For elementary through high school students who wish to share their solutions.

For more tips, check out this comment from the 2008 game.

Heiner Marxen has compiled hints and results for past years (and for the related Four 4’s puzzle). Dave Rusin describes a related card game, Krypto, which is much like my Target Number game. And Alexander Bogomolny offers a great collection of similar puzzles on his Make An Identity page.

*2017 Adventures photo by Kitty Terwolbeck and Origami Star by uschi mitzkat via Flickr (CC BY 2.0). New Year’s Resolutions from Wikipedia.*

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John 1:1-5; 8:12 recited in English, Cantonese, Japanese, Spanish, Croatian, Turkish, and French.

In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God. All things were made through him, and without him was not any thing made that was made. In him was life, and the life was the light of men. The light shines in the darkness, and the darkness has not overcome it.

… Again Jesus spoke to them, saying, “I am the light of the world. Whoever follows me will not walk in darkness, but will have the light of life.”

Scripture quotations are from the ESV® Bible (The Holy Bible, English Standard Version®), copyright © 2001 by Crossway, a publishing ministry of Good News Publishers. Used by permission. All rights reserved.

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The carnival features prime numbers, self-referential logic, calculus puns, word problems, Pythagorean triples, arithmetic games, geometric coloring designs, and more.

Click here to go read the carnival blog!

Past carnivals are still full of mathy treasure. Check them out:

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Counting all the fractional variations, my massive blog post 30+ Things to Do with a Hundred Chart now offers nearly forty ideas for playing around with numbers from preschool to prealgebra.

Here is the newest entry:

**(34****)** The Number Puzzle Game: Rachel created this fun cross between the hundred-chart jigsaw puzzle (#7) and Gomoku (#23). You can download the free 120-board version here or buy the complete set at Teachers Pay Teachers.

*[Photo by geishaboy500 (CC BY 2.0).]*

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So I updated the post with a new, non-religious puzzle. Here it is, if you want to play…

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.

(-8,8) – (-8,0) – (0,8) – (-8,8) – (-4,4) – (0,4) – (0,8) – (8,8) – (4,4) – (0,8).

(8,8) – (8,0) – (4,0) – (4,-4) – (8,0) – (8,-8) – (0,-8) – (4,-4) – (0,-4) – (0,-8) – (-8,0) – (-8, -8) – (0,-8).

(-8,-8) – (4,4) – (0,4) – (4,0) – (4,4) – (8,0).

(8,-8) – (-4,4) – (-4,-4) – (0,-4) – (-4,0) – (-8,0).

(0,-2) – (0,-4) – (4,0) – (2,0) – (2,-2) – (-2,-2) – (-2,2) – (2,2) – (2,0) – (1,1) – (1,0) – (2,0) – (0,-2) – (-2,0) – (0,2) – (1,1) – (-1,1) – (-1,-1) – (1,-1) – (1,0) – (-4,0) – (0,4) – (0,-1) – (-1,0) – (0,1) – (1,0) – (0,-1) – (0,-2).

Color in your design and hang it up for the whole family to enjoy!

Of course, the fun of the Graph-It Game is to make up your own graphing puzzle. Can you create a coordinate design for your friends to draw?

You can see all the Alexandria Jones Christmas posts at a glance here:

“Love Christmas Lights” photo by Kristen Brasil via Flickr (CC BY 2.0).

Want to help your kids learn math? Claim your free problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.

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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.

How can that be?

Read all the posts from the December 2000/January 2001 issue of my *Mathematical Adventures of Alexandria Jones* newsletter.

“Christmas Tree Closeup” photo by Zechariah Judy via Flickr (CC BY 2.0).

Want to help your kids learn math? Claim your free problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.

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She cut the boxes open, which gave her several big sheets of thin cardboard.

Then she carefully traced the templates for a regular triangle, square, pentagon, and hexagon, as shown below.

Click here to download the polygon templates

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.

*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* 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* 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:

**Icosidodecahedron:** triangle, pentagon, triangle, pentagon.

**Truncated octahedron:** square, hexagon, hexagon.

**Truncated icosahedron:** pentagon, hexagon, hexagon. Where have you seen this?

**Rhombicuboctahedron:** triangle, square, square, square.

**Rhombicosidodecahedron:** triangle, square, pentagon, square.

Now, make up some original polyhedra of your own. What will you name them?

Read all the posts from the December 2000/January 2001 issue of my *Mathematical Adventures of Alexandria Jones* newsletter.

“50/52 Weeks of Teddy – Merry Christmas” photo by Austin Kirk via Flickr (CC BY 2.0).

Want to help your kids learn math? Claim your free problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.

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**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.

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?

Alex wrote a holiday greeting on Face 1. Then she drew Christmas pictures on the other three faces of her card.

Read all the posts from the December 2000/January 2001 issue of my *Mathematical Adventures of Alexandria Jones* newsletter.

“Happy Holidays” photo by Mike Brand via Flickr (CC BY 2.0). Video by Shaireen Selamat of DynamicEducator.com.

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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 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.”

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.

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!”

*Mathematical Adventures of Alexandria Jones* newsletter.

“Christmas Window” photo by slgckgc via Flickr (CC BY 2.0). Video by Shaireen Selamat of DynamicEducator.com.

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