Written for aspiring Vex IQ teams, coaches and middle-and high schoolers, this is a handbook for designing compact, rigid and efficient gearboxes for robotics applications. The book presents a mathematically-rigorous analysis of multi-stage gear systems. Parts of the book are suitable for beginner teams and younger students - most chapters and concepts are summarized and simplified to a second grade level by Professor Gearhardt, who Loves To Simplify. This book is based upon an original STEM research project completed by the authors as part of the requirements of the 2014 VEX IQ Robotics Competition Challenge, Add It Up. This research project helped the team win the overall championship (the Excellence Award) at the 2014 VEX IQ State Championship in Virginia and propelled them to the VEX IQ World Championship. Team members have collectively won 9 robotics awards at regional, state and national levels over the past three years. This research is being published to help other teams use and build upon the authors' original research on designing gearboxes, and to inspire other top teams to publish and share their research. The book starts with a brief history of gears. Gears have been in use for almost 5,000 years You can learn about the various types of gears and the rules of gearing. Gears also occur in nature. Did you know that the insect Issus Coleoptratus uses skeletal gears to jump? Starting with a simple explanation of simple and compound gears, the book proceeds to explain the mechanical structure of multi-stage gear systems. The fundamental relation between torque and speed is next explored. When a gear system is used to increase torque by a certain factor, the angular velocity (or speed) is reduced by inverse of the same factor. Gear reductions and gear ratios are analyzed next. This is followed by consideration of the basic set of spur gears in the Vex IQ set. Did you know that just three types of gears can be used to construct 819 different gear permutations in 1-3 stages. Even more surprisingly, only a few of these are unique. Two new concepts are next introduced - Minimum Spanning Beam and Stack Height. The Minimum Spanning Beam is the smallest beam or plate that can span all the axles of a gearing permutation. It is a measure of the rigidity of the gearbox. The Stack Height is the distance from the bottom of the lowest gear to the top of the highest gear. It is a measure of the overall size of a gear permutation. Formulae for calculating the Minimum Spanning Beam and the Stack Height for various stages of gearing are derived. Finally, nonlinear stacks are considered. A link to the source data spreadsheet is provided for further study. An author biography is appended.