Here's one of my favorites. Simple, well executed, ingenious. The others are well worth watching.

(via @JadAbumrad)

It's based on a very clever idea: parallel tooth metal strips had numbers from 0 to 9 printed on them, as well as a "flag" (in my addiator's case a red arrow) to indicate that you need to carry the operation over to the next column. This mechanism was originally invented in the late 1800s by a Frenchman named Louis Troncet. I've included a picture from his 1889 patent that shows you the inner working of the device. You can view a number of his patents here.

The German company Addiator manufactured devices like mine from the 1920s to the early 1980s when digital calculators finally made them obsolete. Many other companies made them too, and not just to add / subtract either. 

This short video shows you how simple and effective Troncet's invention was.

Strogatz does a wonderful job sharing the wonders of maths for the lay person, starting from simple counting and finishing up with more complex topics like integration, probabilities, and some of David Hilbert's work.

Here's a beautiful example from an early column: Rock Groups. The question posed is why is the sum of consecutive odd numbers always a perfect square?

At first glance this is an interesting observation but hard to understand... Until you look at these equations graphically:

And then it makes perfect sense!

Another favorite focuses on limits and gives an elegant application to finding the area of a circle.

A series well worth reading and, if you have any, sharing with your children.