A technique discovered in 1939 can be used to build a device that is driven by standard circular motion (as in a drill press) and drills exact square holes. This device is quite different from the classic design by Watts, which uses a Reuleaux triangle and drills a hole that is almost, but not exactly, square. We describe the device in detail, deriving the equations needed to make a computer simulation. Then we investigate the extension to other polygons and show how the basic idea can be combined with some knowledge of envelopes to produce a device that traces out an exact hexagon. The simulations are available at http://demonstrations.wolfram.com/GenevaDriveMalteseCross, http://demonstrations.wolfram.com/DrillingASquareHole, http://demonstrations.wolfram.com/SquareHoleDrillInThreeDimensions, and http://demonstrations.wolfram.com/DrillingAHexagonalHole.