No math required. It is impossible to draw this figure without taking the pen off the paper, redrawing any lines, or other trickery. Explain why it is impossible. (Answer & Solution).
Integral calculus and trigonometry required. Two circles of radius one are adjancent at point (0,0). Let the center of the first circle be at point (-1,0). Label the point on the edge of the first circle at (0,0) x. The second circle is centered at (1,0). Rotate the first circle around the second, like a gear, keeping the second circle stationary. If you traced the path of point x as the first circle traveled around the second what would be the length of the path point x traced? The figure above shows the path in red. (Answer), (Solution).
Modular arithmetic suggested. In a spirograph toy you first select two circular gears. Keeping one gear stationary, you let the second gear travel around the first one until it arrives back at its initial position and orientation. By tracing the path with a pen through a hole in the second gear you can make designs like the one above. Assuming the notches in the gears are equally spaced apart, and that there are x notches in the stationary gear, and y notches in the moving gear, how many trips around the first gear will the second gear need to make before it arrives back at its initial position and orientation? (Answer).
Michael Shackleford, A.S.A.