Hello, kingman!

I have a very primitive solution.

Place a pencil on , the eraser at , the point at

Rotate the pencil about so that the eraser is at .

Rotate the pencil about so that the point is at

Rotate the pencil about so that the eraser is at

Rotate the pencil about so that the point is at

Rotate the pencil about so that the eraser is at

Rotate the pencil about so that the point is at

Rotate the pencil about so that the eraser is at

We find that the pencil has undergone rotations.

Therefore: .