Too much mathematical rigor teaches rigor mortis: the fear of making
an unjustified leap even when it lands on a correct result. Instead of
paralysis, have courage—shoot first and ask questions later. Although
unwise as public policy, it is a valuable problem-solving philosophy, and
it is the theme of this book: how to guess answers without a proof or an
Educated guessing and opportunistic problem solving require a toolbox.
A tool, to paraphrase George Polya, is a trick I use twice. This book
builds, sharpens, and demonstrates tools useful across diverse fields of
human knowledge. The diverse examples help separate the tool—the
general principle—from the particular applications so that you can grasp
and transfer the tool to problems of particular interest to you.
The examples used to teach the tools include guessing integrals without
integrating, refuting a common argument in the media, extracting
physical properties from nonlinear differential equations, estimating drag
forces without solving the Navier–Stokes equations, finding the shortest
path that bisects a triangle, guessing bond angles, and summing infinite
series whose every term is unknown and transcendental.