Interactive problem-solving atlas
In my experience a hard problem is hard because the decisive structure goes unnoticed in the first thirty seconds, not because the algebra is inherently deep.
The book, rebuilt for retrievalI have made all 54 chapters searchable by visible cue and domain, each with the first move that unlocks its family of problems.
Order of attack
I start by naming the object, its domain, its symmetries, its singularities, and the form a plausible answer must have. Simplify the representation before applying a named method.
read → classify → normalise → choose → compute → verifyI insist on this order because it keeps a familiar technique from becoming a reflex applied to the wrong shape.
Cue
A cue is a visible feature: a Riemann-sum grid, a derivative quotient, reflection symmetry, a missing variable, a repeating move, or a hidden modulus.
Method
Use a method as a mechanism, not a spell. That is the distinction I care most about here: preserve its assumptions and state what it transforms the problem into.
Trap
Division can delete solutions, substitutions can change domains, interchange can lack justification, and a numerical answer can hide the wrong scale.
Verification
I count verification as part of the method, not as an afterthought. Use the cheapest independent check that attacks your likeliest error: sign, units, boundary values, direct differentiation, asymptotic scale, or a second representation.
The universal five