The Pharaoh’s Treasure: Historical Puzzles

Posted on by Denise Gaskins
Eratosthenes teaching in Alexandria

When two non-mathematicians meet on the street and one says, “I’ve got problems,” the other answers, “I’m so sorry for you.”

    When two mathematicians meet and one says, “I’ve got problems,” the other says, “Oh, goody!”

    —Herta Taussig Freitag

    Whenever you get a group of mathematicians together, at least one of them will suggest a puzzle to play with. This is how mathematics grows, puzzle upon puzzle, generation after generation.

    And since human nature hasn’t changed, we know that the same thing happened in the ancient Ptolemaic Egyptian of Alexandria.

    So let’s explore a few of the math stumpers that enchanted those old Greek scholars…

    Continue reading The Pharaoh’s Treasure: Historical Puzzles

    Thinking Thursday: Linear Crossings

    Posted on by Denise Gaskins
    Thinking Thursday math journal prompt

    Writing to Learn Math: People learn math by playing with ideas. A math journal can be like a science lab book.

    Do you want your children to develop the ability to reason creatively and figure out things on their own?

    Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

    Continue reading Thinking Thursday: Linear Crossings

    Thinking Thursday: Math Quilt

    Posted on by Denise Gaskins
    Thinking Thursday math journal prompt

    Writing to Learn Math: When students create their own math, they forge a personal connection to mathematical concepts and relationships. And it’s fun!

    Do you want your children to develop the ability to reason creatively and figure out things on their own?

    Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

    Continue reading Thinking Thursday: Math Quilt

    Ostomachion: Divide the Square

    Posted on by Denise Gaskins
    archimedes, studying

    “Puzzles are made of the things that the mathematician, no less than the child, plays with, and dreams and wonders about, for they are made of things and circumstances of the world he [or she] lives in.”

    —Edward Kasner

    Long ages ago, one of the greatest mathematicians of all time played with a children’s puzzle.

    The puzzle was called “The Ivory Challenge”—or something like that, translations vary. Ostomachion in Greek. Fourteen ivory polygons (flat shapes with straight sides) were supposed to fit into a square box.

    Archimedes solved that puzzle. But like any good mathematician, he didn’t stop there. The solution to any puzzle is only the beginning, a doorway to further adventures.

    So Archimedes asked himself one of the big questions of mathematics: “How many ways?”

    The Ostomachion is a rather complex puzzle, and our students may struggle to figure out where to start. One of the classic tools in any mathematician’s toolbox is to make it simpler. So let’s begin our own investigation with a simpler challenge:

    • How many ways can you bisect a square?

    Continue reading Ostomachion: Divide the Square

    Reasoning about Ratios and Proportions

    Posted on by Denise Gaskins
    photo of sailboat on the Nile river

    “The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver.”

    —I. N. Herstein

    Whenever I give a problem or puzzle in an Alexandria Jones story, I’ll try to post the answer soon afterward.

    But don’t peek!

    In the old days of my snail mail newsletters, the answers always came in the next issue, so students had two months to let the puzzle ruminate without temptation.

    After all, if you only read the answer I give, you miss out on the fun of solving the puzzle. Follow the Math Adventurer’s Rule:

    • Figure it out for yourself!

    Give the problem a good workout on your own. See what you can notice about the puzzle. Wonder about how it connects to the larger world of mathematics.

    Play with the ideas and find out where they may lead.

    Then check the answer just to prove you got it right, and maybe to see whether we did it the same way. There’s always more than one way to approach any math problem, and each method offers its own insights.

    Continue reading Reasoning about Ratios and Proportions

    The Pharaoh’s Treasure, Part 2

    Posted on by Denise Gaskins
    photo of Egyptian sphinx

    “One factor that has remained constant through all the twists and turns of the history of physical science is the decisive importance of the mathematical imagination.”

    —Freeman Dyson

    In the previous episode, Alexandria Jones discovered a mysterious treasure: three wooden sticks, like tent pegs, and a long loop of rope with 12 evenly spaced knots. Her father explained that it was an ancient Egyptian surveyor’s tool, used to mark right angles.

    loop of rope and 3 pegs

    Continue reading The Pharaoh’s Treasure, Part 2

    Math Game Monday: Symmetry Challenge

    Posted on by Denise Gaskins
    Learn a new game with Math Game Monday

    This game builds visual thinking skills.

    Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

    So what are you waiting for? Let’s play some math!

    Symmetry Challenge

    Math Concepts: symmetry, geometry.

    Players: only two.

    Equipment: graph paper or blank hundred chart, colored pencils or markers.

    Continue reading Math Game Monday: Symmetry Challenge

    Math Game Monday: Area Battle

    Posted on by Denise Gaskins
    Learn a new math game every week, for free

    This game pushes students to consider area and perimeter beyond the basic shapes.

    Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

    So what are you waiting for? Let’s play some math!

    Area Battle

    Math Concepts: area, square units, perimeter, regular and irregular polygons.

    Players: two to four.

    Equipment: homemade game cards, pencils or markers.

    Continue reading Math Game Monday: Area Battle

    Hints for the Patty Paper Trisection

    Posted on by Denise Gaskins
    drafting tools

    No peeking! This post is for those of you who have given the trisection proof a good workout on your own.

    If you have a question about the proof or a solution you would like to share, please post a comment here.

    But if you haven’t yet worked at the puzzle, go back and give it a try.

    When someone just tells you the answer, you miss out on the fun. Figure it out for yourself — and then check the answer just to prove that you got it right.

    Continue reading Hints for the Patty Paper Trisection

    Puzzle: Patty Paper Trisection

    Posted on by Denise Gaskins
    student using drafting tools

    One of the great unsolved problems of antiquity was to trisect any angle, to cut it into thirds with only the basic tools of Euclidean geometry: an unmarked straight-edge and a compass.

    Like the alchemist’s dream of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight-edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator.

    One “cheat” that works is to fold your paper.

    I will show you how it works, and your job is to show why.

    Continue reading Puzzle: Patty Paper Trisection