Consider all the powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, and so on...
The unit digits follow the progression 1, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6... Nothing too amazing, a nice cyclic relationship, and except for the priming 1, all evenly distributed. But consider the leading digit. In the limit, what are the distribution of the leading digits? I computed a table:
The puzzle is to verify and to explain this distribution. Neat stuff.
Some patents are just too much fun, and the pat2pdf script allows you to look them up and get a look at them. Today's fun patent is for the Slinky. The real invention was the machine that can take 80 feet of steel wire and coil it into a Slinky in 10 seconds. Now that's an invention.
In the modern era, these things are made of plastic, which just seems too wimpy for me. Give me the classic, brittle spring steel version every time. They are also really good at teaching kids about transverse waves or experimenting with odd sounds generated by suspending them like this one.