20 Rows of Pegs, 1,048,576 Paths in One Triangle

About this piece

One ball, one peg. It can go left or right, so I let it go both ways.

Then another row of pegs, and another. By row 20 there are 1,048,576 different ways down.

I counted how many paths go through each peg and the numbers made a triangle. I colored every path by how often it turned right, cut a square and a couple of cubes into blocks, leaned the board, and dropped 256,000 real balls to see if they'd agree. Then the triangle turned into a town with a house and a school, and I started closing roads.

A million ways down, and most of them end up near the middle. Nobody told them to.