Quotations X: The Royal Road

More quotations especially for teachers:

There is no Royal Road to Geometry.


Teaching is the royal road to learning.

Jessamyn West

The title which I most covet is that of teacher. The writing of a research paper and the teaching of freshman calculus, and everything in between, falls under this rubric. Happy is the person who comes to understand something and then gets to explain it.

Marshall Cohen

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Hints and Solutions: Patty Paper Trisection

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 and Solutions: Patty Paper Trisection

Puzzle: Patty Paper Trisection

[Feature photo above by Michael Cory via Flickr (CC BY 2.0).]


One of the great unsolved problems of antiquity was to trisect any angle using 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 or something.

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

Historical Tidbits: Alexandria Jones

[Read the story of the pharaoh’s treasure: Part 1, Part 2, and Part 3.]

Alexandria JonesHere are a few more tidbits from math history, along with links to relevant Internet sites or books, and three more math puzzles for you to try. I hope you find them interesting.

Next time, a new adventure (sort of)…

Continue reading Historical Tidbits: Alexandria Jones

The Secret of the Pharaoh’s Treasure, Part 3

[In the last 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.]

Back at the camp, Fibonacci Jones stacked multi-layer sandwiches while Alexandria poured milk and set the table for supper.

“Geometry,” Fibonacci said.


Geo means earth, and metry means to measure. So geometry means to measure the earth. That is what the Egyptian rope stretches did.”

Alex thought for a moment. “So in the beginning, math was just surveying?”

“And taxes…”

Continue reading The Secret of the Pharaoh’s Treasure, Part 3

The Secret of the Pharaoh’s Treasure, Part 2

[In the last episode, Alexandria Jones, daughter of the world-famous archaeologist, caught her father’s arch-enemy trying to uncover the Pharaoh’s Treasure.]

…”I can’t believe it!” Simon Skulk threw down the last stone in disgust and walked away. At the mouth of the cave, he turned back and shook his fist. “You haven’t seen the last of me, Alexandria Jones.”

Her muscles aching, Alex sank to the ground and hugged her dog. The she gave him a little push toward the front of the cave. “Rammy, go get Dad.”

Ramus barked once and took off running.

Alex turned back to look at the Pharaoh’s Treasure. Where the last stone had stood was a hole. In the hole lay three wooden sticks, like tent pegs, and a long loop of rope with 12 evenly-spaced knots.

What could it be?

Continue reading The Secret of the Pharaoh’s Treasure, Part 2