Category: Geometry

High School Students Finding New Proofs of Old Theorems (Part 3): Cayley’s Formula

This is a new favorite story to share with students: a high school student recently published an elementary proof of Cayley’s Formula from graph theory.

Professional article in the American Mathematical Monthly (requires a subscription): https://www.tandfonline.com/doi/full/10.1080/00029890.2025.2590423

By coincidence, this is one of the math theorems that was cited in the movie Good Will Hunting.

High School Students Finding New Proofs of Old Theorems (Part 2): Pythagorean theorem

This is a new favorite story to share with students: two high school students recently figured out multiple new proofs of the Pythagorean theorem.

Professional article in the American Mathematical Monthly (requires a subscription): https://maa.tandfonline.com/doi/full/10.1080/00029890.2024.2370240

Video describing one of their five ideas:

Interview in MAA Focus: http://digitaleditions.walsworthprintgroup.com/publication/?i=836749&p=14&view=issueViewer

Interview by 60 Minutes:

https://www.youtube.com/watch?v=VHeWndnHuQs

Praise from Michelle Obama: https://www.facebook.com/michelleobama/posts/i-just-love-this-story-about-two-high-school-students-calcea-johnson-and-nekiya-/750580956432311/

High School Students Finding New Proofs of Old Theorems (Part 1): Dividing a line segment with straightedge and compass

This is one of my all-time favorite stories to share with students: how a couple of ninth graders in 1995 played with Geometer’s Sketchpad and stumbled upon a brand-new way of using only a straightedge and compass to divide a line segment into any number of equal-sized parts. This article was published in 1997 and made quite a media sensation at the time.

1997_litchfieldd._euclidfibonaccisketchpadDownload

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, Part 2b, and Part 2c: The 1089 trick.

Part 3a, Part 3b, and Part 3c: A geometric magic trick.

Part 4a: Part 4b, Part 4c, and Part 4d: A trick using binary numbers.

Part 5a, Part 5b, Part 5c, and Part 5d: A trick using the rule for checking if a number is a multiple of 9.

Part 7: The Fitch-Cheney card trick, which is perhaps the slickest mathematical card trick ever devised.

Part 8a, Part 8b, and Part 8c: A trick using Pascal’s triangle.

Part 9: Mentally computing n given n^5 if 10 \le n \le 99.

Part 10: A mathematical optical illusion.

Part 11: The 27-card trick, which requires representing numbers in base 3.

Part 6: The Grand Finale.

And, for the sake of completeness, here’s a picture of me just before I performed an abbreviated version of this show for UNT’s Preview Day for high school students thinking about enrolling at my university.

magician

My Mathematical Magic Show: Part 10

This magic trick is an optical illusion instead of a pure magic trick, but it definitely is a crowd-pleaser. This illusion is called Sugihara’s Impossible Cylinder:

This is actually a mathematical magic trick. As detailed by David Richeson in Math Horizons, there is a fair amount of math that goes into creating this unique shape. He also provided this interacted Geogebra applet as well as a printable pdf file for creating this illusion.