By method · Volume 8, Problem-Solving Methods and Strategies
20 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 · General strategies
M1ReductionCh. 1 · 5 stopsTurn a problem you cannot solve into one you already can, one step at a time.M2Proof by ContradictionCh. 2 · 4 stopsAssume the conclusion is false and follow it until it crashes into something.M3Mathematical InductionCh. 3 · 8 stopsTopple the first, let each topple the next. Pick the base, step and statement.M9Looking at the WholeCh. 9 · 4 stopsLook at the whole before the parts. Local trouble often cancels out overall.M10Choosing Good NotationCh. 10 · 6 stopsNew notation can turn a hard problem easy: it puts the structure on the page.M11Combining Numbers and FiguresCh. 11 · 6 stopsRead formulas as pictures: radicals are distances, ratios are slopes.
Part 2 · Combinatorial principles
M4The Pigeonhole PrincipleCh. 4 · 4 stopsMore objects than boxes means a box holds two. The art is building boxes.M5Inclusion–ExclusionCh. 5 · 3 stopsAdd the parts, subtract what you overcounted, add back what you oversubtracted.M6The Extremal PrincipleCh. 6 · 4 stopsLook at the largest, smallest or farthest one. It is the most constrained.M7ParityCh. 7 · 3 stopsKeep only odd or even. Moves that change the value often keep its parity.
Part 3 · Correspondence and counting
M8The Area MethodCh. 8 · 4 stopsCompute one area in two ways to get an equation; read length ratios as areas.M12Correspondence and PairingCh. 12 · 5 stopsCount a hard set through an easy twin; pair terms so each pair sums alike.M13RecursionCh. 13 · 6 stopsLet aₙ be the answer for size n; split on the last element to get a recurrence.M16Counting in Two WaysCh. 16 · 4 stopsCount one quantity in two ways and equate: rows of a table, then columns.
Part 4 · Construction and change
M14ColoringCh. 14 · 3 stopsColor cells, points or numbers so each move's effect is fixed, then count.M15Assigning ValuesCh. 15 · 4 stopsGive objects clever values (±1, 0/1, roots of unity) and let arithmetic decide.M17Step-by-Step AdjustmentCh. 17 · 4 stopsFreeze all but two variables and move those two to improve, until you cannot.M18ConstructionCh. 18 · 5 stopsBuild what you are looking for: an example, a function, a figure.M19Invariants and MonovariantsCh. 19 · 4 stopsMoves vary endlessly; watch a quantity that never changes or only goes one way.M20Graph-Theoretic MethodsCh. 20 · 3 stopsObjects become vertices, relations edges; degrees and triangles are visible.