The Art and Craft of Problem Solving

362 INDEX

rules for, 94, 107
division algorithm
for integers, 83, 224, 226
for polynomials, 164
divisors
common, 60
number of, 30, 68, 195
of a product, 236
sum of, 235
domain of a function, 145
draw a picture strategy, 64, 75, 25 1, 340
drawing an auxiliary object, see strate-
gies, drawing an auxiliary object
drawing supplies, 256
dyadic rationals, 326

Eastern Europe, xi, 8
edge of graph, see graph theory
ego, 256
elegant solution, 147, 154
elementary symmetric functions, see
functions, symmetric
ellipse, 9, 73
empty set, 143
encoding, 196
Endurance, 23
equilateral triangle, 66, 84, 132
escribed circle, 28 1
essay-proof exam, 7
Euclid, 51
Euclidean algorithm, 228
Euler line, 290
Euler's formula
for ei6, 123, 131
for polyhedra, 25, 37, 93, 108
Euler's inequality, 279
Euler, Leonhard, 140, 247,254,347-349
Eulerian mathematics, 346-349
Eulerian path, 113-1 15
algorithmic construction, 114
exercise, x, 22
defined, I
versus problem, x, 1, 2,4, 15
extension of a side, 267
exterior angle, 259
extreme principle, 21, 42, 62, 73-83,1 12,
225, 228,229, 244

factor theorem, 152, 166, 349

factoring, see algebraic methods

Fermat's Last Theorem, 230

Fermat's little theorem, 232-233, 239

combinatorial proof, 249

induction proof, 234

Fibonacci numbers, 20, 52

definition, 10

divisibility properties, 229

formula, 141

in Pascal's triangle, 10, 24

recurrence relation for, 216

Fisk, S., 55

fixed point, 64, 324, 332, 34 1

floor function, 146

FOIL, 168, 194

forest, III, 113

functions

bijection, 145

continuous, 323

floor and ceiling, 146

generating, 132- 142

graph of, 35

growth rates, 174, 328

indicator, 146, 212

monotonic, 337

multiplicative, 235

number theoretic, 235-240

one-to-one, 145

onto, 145

symmetric, 71

uniformly continuous, 324

fundamental theorem, 336

of algebra, 90, 166

of arithmetic, 223, 354

of calculus, 315 , 345

Gallery problem, 38, 55, 109

Galois theory, 93

Gardner, Martin, 7

Gauss plane, 120

Gauss's lemma, 170, 229

Gauss, Carl, 26, 44, 67

Gaussian pairing tool, 67-69, 250

applications of, 68, 157

GCD, see greatest common divisor

generating functions, 132-142

and partitions, 136-141

and recurrence relations, 134- 135, 218

generatingfunctionology, 138

Geogebra, 256

Geometer, 256

Geometer's Sketchpad, 256

geometric interpretation, see reformulat­

ing a problem; draw a picture strat­

egy

of AM-GM inequality, 178

of Cauchy-Schwarz inequality, 187

of complex numbers, 120- 126, 128

of differentiation, 328

geometric mean, 177, 178

geometric series, see series, geometric

geometric series tool, 133, 346

geometry problem

characterization, 257

glide reflection, see transformations,

rigid motions, glide reflection

Gnepp, Andrei, 98

Go (board game), 23

golf, 23

graph (graph theory)

as opposed to multigraph, 109

bipartite, 119

connected, III

directed, 115

existence of cycles, 110, III, 113

forest, 111

tree, III

greatest common divisor, 28, 77, 83, 1 19,

225

Halmos, Paul, 12

Hamiltonian paths and cycles, 115

handshake lemma, III, 118

handshake problem, 2, 75, 109

harmonic series, 161, 162, 173, 328, 348

harmony

and symmetry, 63

Herrigel, Eugen, 23

heuristics, 3

hexagon, 66, 91

hockey stick property, 20 1, 205

Holmes, Sherlock, 23, 41

homothety, see

tions,homothety

Hong Kong, 80

Hunter, Denise, 15

hypothesis, 4, 26, 40

inductive, 45

need to strengthen, 49

ideas

new, 18, 21

stealing, 18

identity principle, 172

imagination, 4

transforma-

IMO, see International Mathematical

Olympaid

incenter, 266

incircle, 266

construction, 267

existence and uniqueness, 267

indicator function, 146, 212

indistinguishable objects, 189

induction, 45-50

standard, 45-47

strong, 47-50