Here is my explanation (note, I’m pretty sure I have a mistake in toppling 20)

Here is a video that Numberphile put together describing them as well:

Reflection questions:

Is it possible for a pile to topple differently? Does it matter if different stages appear different or is it necessary for the toppling to be consistent?

What would an isometric (triangular) grid toppling looks like? How about a hexagonal one?

How might you meld this with computer programming?

Is there a less or more illuminative coloring for the sand piles? What kinds of coloring may make the sand pile structures more obvious or less?

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Draw a large shape. Then place a large circle inside that shape, touching the edges of the original shape. Then draw the next largest circle you can, and repeat drawing the next largest circle you can. (See video below for examples.)

If you’re interested in some of the math behind this, this is a great article, featuring the work of mathematician Hee Oh.

Vi Hart also has a great video along these lines, demonstrating some serious doodle skills.

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

5.) Use appropriate tools strategically. What tools make this easier or more difficult? Is it necessary or better to use a compass and be completely precise, or is it better to “eye-ball” it?

7.) Look for and make use of structure. How does the radius of the largest inner circle affect the radii of the other circles?

8.) Look for and express regularity in repeated reasoning Compare multiple drawings. How does the choice of largest inner circle affect the final outcome? The amount of positive to negative area?