Secret of row 10: a new visual key to ancient Pascalian puzzles

From Mathologer.

Today’s video is about a recent chance discovery (2002) that provides a new beautiful visual key to some hidden self-similar patterns in Pascal’s triangle and some naturally occurring patterns on snail shells. Featuring, Sierpinski’s triangle, Pascal’s triangle, some modular arithmetic and my giant pet snail shell.

Thank you very much to Marty for all his help with finetuning the script for this video and to Steve Humble and Erhard Behrends for making some photos available to me.



P.S.: The article I mentioned in this video is: Steve Humble, Erhard Behrends, ”Triangle Mysteries“, The Mathematical Intelligencer 35 (2), 2013, 10-15. There is also a followup article:
”Pyramid Mysteries“, The Mathematical Intelligencer 36 (3), 2014, 14 – 19.
And there is a book by Erhard Behrends that has a couple of chapters dedicated to this topic: The Math Behind the Magic: Fascinating Card and Number Tricks and How They Work: 🙂

A Wolfram demonstration project that implements the 3-color game:

Philip Smolen contributed this animation

Someone pointed out these links to some code wars problems:

Juan Mir Pieras pointed out these earlier references: (Problem of the month June 2001)( The webpage says the problem is from "Crux Mathematicorum 27:3 (April 2001) pages 204-205 – it is problem 3 from the Ninth Annual Konhauser Problemfest (Carleton College, prepared by David Savitt and Russell Mann."

Today’s t-shirt: