By method · Volume 8, Problem-Solving Methods and Strategies
23 method lines
Volumes 1–7 are the territory; Volume 8 is the lines that run across it. Each idea turns up once in algebra, geometry, number theory and combinatorics. Pick a line and ride it to the end.
Part 1 · Algebra
M1Completing the Square§1.1 · 5 stopsFill the missing corner, then take it away — a perfect square minus a number.M2Substitution§1.2 · 5 stopsCall the repeated block t, solve the easier problem in t, then switch back.M3Discriminant and Vieta§1.3 · 5 stopsNo solving: Δ says whether roots exist; Vieta gives their sum and product.M4The Principal Variable§1.4 · 4 stopsSeveral letters? Choose the main one: cubic in x may be linear in a.M5Symmetry§1.5 · 4 stopsIf swapping letters changes nothing, prove it for one and get the rest free.M6Bounding§1.6 · 4 stopsCannot compute it? Squeeze it between bounds until only a few integers fit.
Part 2 · Geometry
M7The Area Method§2.1 · 5 stopsFind one area two ways and set them equal; trade length ratios for area ratios.M8Translation, Rotation and Reflection§2.2 · 6 stopsMove a piece of the figure without changing it, so scattered parts meet.M9Building Similar Triangles§2.3 · 4 stopsFind similar triangles that hold the four segments, or build a pair.M10Using Circle Properties§2.4 · 5 stopsA circle carries angles around. No circle in sight? Find four concyclic points.M11Algebra for Geometry§2.5 · 4 stopsCall the unknown length x and let the figure write the equation for you.
Part 3 · Number theory
M12Parity§3.1 · 4 stopsAsk only “odd or even?” If no move changes it, some goals are out of reach.M13Prime Factor Analysis§3.2 · 4 stopsWrite integers as products of primes and compare only the exponents.M14Choosing a Good Modulus§3.3 · 4 stopsEqual numbers leave equal remainders. Find a modulus where they cannot.M15Estimation§3.4 · 4 stopsIntegers come one step at a time. Trap the unknown, then check what is left.
Part 4 · Combinatorics
M16The Pigeonhole Principle§4.1 · 3 stopsMore objects than boxes means a box holds two. The art is building boxes.M17Coloring and Assigning Values§4.2 · 3 stopsColor the cells or give each a number; what a move can change shows up at once.M18Cases and Enumeration§4.3 · 4 stopsSplit by one rule, no overlaps and no gaps, so each case is an easy problem.M19Recursion and Induction§4.4 · 3 stopsShrink the problem to a smaller copy of itself, then nail it by induction.M20The Extremal Principle§4.5 · 3 stopsWatch the largest, smallest or most extreme one: contradictions start there.M21From Local to Global§4.6 · 3 stopsCombine all the local pieces; count one whole two ways and the answers agree.M22Generalize and Specialize§4.7 · 4 stopsStuck on the general case? Try a special one. Messy numbers? Use letters.M23Guess, Then Prove§4.8 · 5 stopsTry small cases, guess the pattern boldly, then prove it properly.