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Cofman J. Ч What to Solve? Problems and Suggestions for Young Mathematicians
Cofman J. Ч What to Solve? Problems and Suggestions for Young Mathematicians

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Ќазвание: What to Solve? Problems and Suggestions for Young Mathematicians

јвтор: Cofman J.

јннотаци€:

This book provides a wide variety of mathematical problems for teenagers and students to help stimulate interest in mathematical ideas outside of the classroom. Problems in the text vary in difficulty from the easy to the unsolved, but all will encourage independent investigation, demonstrate different approaches to problem-solving, and illustrate some of the famous dilemmas that well-known mathematicians have attempted to solve. Helpful hints and detailed discussions of solutions are included, making this book a valuable resource for schools, student teachers, and college mathematics courses, as well as for anyone fascinated by mathematical ideas.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1990

 оличество страниц: 263

ƒобавлена в каталог: 22.01.2014

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$\pi$, the number, Archimedes' algorithm for calculating      137
$\pi$, the number, Leibniz' series for      138 164
$\pi$, the number, problems      137Ч140
$\pi$, the number, solutions      160Ч166
Abel, Niels Henrik      234
Algebra, fundamental theorem of      140
Algebraic structures      144 214Ч218
Archimedes of Syracuse      137 146 234
Argand diagrams      220Ч221
Arithmetic progressions      12Ч13 41Ч43 133Ч134
Axioms      145
Axonometry      141Ч142 166Ч167
Binomial coefficients      38 230
Block designs      183 192Ч194 231 232
Block designs, balanced incomplete      231Ч232
Bolyai, Janos      145 234
Buffon's Needle Problem      139Ч140 165Ч166
Buffon, Comte de, Georges Louis Leclerc      139 234
Catalan's problem      149Ч150 172Ч173
Cayley, Arthur      145 235 240
Centres of gravity      131
Chameleons problem      97 115
Chessboards, generalized      22Ч23 74Ч77
Chessboards, king moves on      11 12 39Ч40
Chessboards, knight moves on      10Ч11 35Ч37
Chessboards, natural numbers on      91Ч92
Chessboards, number patterns from      11Ч12 37Ч40
Chessboards, rook moves on      11 22Ч23 37Ч38 74Ч77
Circles and lattice points      186 199
Circles, carrying at least three points      182Ч183 191Ч192
Circles, circumferences of      137
Circles, circumscribed about polygons      221
Circles, construction of      96 112
Circles, inscribed in polygons      221
Circles, inscribed in triangles      25Ч26 85Ч87
Circles, inversion with respect to      225Ч226
Circles, properties of points on      21 69Ч70
Circles, rolling      97 116Ч117
Coefficients of polynomials      213
Coefficients, binomial      38 230
Coefficients, multinomial      230
Coefficients, trinomial      40 76 230
Combinations      229
Combinatorics      149 183 229
complex numbers      208 220
Complex numbers, applications of problems      140Ч145
Complex numbers, applications of solutions      166Ч168
Composite numbers      208
Congruences      208
Contradiction, proof by      94
Contradiction, proof by, problems      99
Contradiction, proof by, solutions      127
Converses      4Ч5
Converses, problems      23Ч27
Converses, solutions      77Ч88
Coordinate systems      141 219Ч221
Coordinate systems, barycentric      107Ч110 220
Coordinate systems, Cartesian      70 98 103 120Ч121 186 219Ч220
Copernicus, Nicholas      97 235
Counting, the art of      149
Counting, the art of, problems      149Ч153
Counting, the art of, solutions      172Ч179
Cube numbers      41 210
Cube numbers, sum of      45
Cubes, division of      98 118Ч120
Cycles      105 231
de Bruijn, N.G.      183 234
de Fermat, Pierre      187 188 236
Desargues, Girard      235
Desargues, theorem of, in plane      97 113Ч115
Desargues, theorem of, in space      25 85
Descartes, Rene      235
Diophantine equations      203 235
Diophantus of Alexandria      235
Dirichlet's principle      184
Dirichlet, Peter Gustav Lejeune      134 235
Dodecahedra      112
Dominoes      95 99Ч101
Duality, principle of      171
Edges      230Ч231
Erdoes, Paul      181 182 183 185 235
Euclid of Alexandria      133 145 153 236
Euclid's parallel postulate      145
Euler's problem on polygon division      150Ч151 174
Euler, Leonhard      150 188 236
Eulerian paths      100
Exceptions and special cases      3Ч4
Exceptions and special cases, problems      17Ч19
Exceptions and special cases, solutions      55Ч66
Extending the field of investigation      90Ч91
Extending the field of investigation, problems      96Ч97
Extending the field of investigation, solutions      110Ч115
Extremal elements, use of      91Ч92
Extremal elements, use of, problems      97Ч98
Extremal elements, use of, solutions      117Ч118
Factors      208
Faltings, Gerd      188 236
Fermat's last theorem      187Ч188 201Ч204
Fermat's little theorem      24 81 155 157
Fibonacci numbers      15 38 48 210 236
Fibonacci numbers, properties of      8 32
Fibonacci, son of Bonaccio      236
Figurate numbers      208Ч210
Fox, Captain      140
Fractions      205
Fractions in simplest terms      16 53
Fractions, decimal      205Ч206
Fractions, successors of      16Ч17 53Ч54
Gabriel-Marie, Fr      127
Gallai, problem of Sylvester Ч Gallai      180Ч183 188Ч194
Gallai, Tibor      181 236
Gauss' fundamental theorem of axonometry      140Ч142 166Ч167
Gauss, Carl Friedrich      140 236
Generalizing given problems      4
Generalizing given problems, problems      19Ч23
Generalizing given problems, solutions      66Ч77
Geometry, Euclidean problems      145Ч146 180Ч183
Geometry, Euclidean solutions      168Ч170 188Ч192
Geometry, non-Euclidean      145
Geometry, non-Euclidean, problems      146Ч149
Geometry, non-Euclidean, solutions      170Ч172
graphs      230Ч231
Graphs, connected      104 231
Graphs, oriented      99Ч101 231
Gregory's arc tangent series      138Ч139 161Ч164
Gregory, James      138 237
groups      214Ч216
Hamilton, William Rowen      143 144 237
Hanani, Haim      183 194 232 237
Heron of Alexandria      126 237
Hexagons, regular      121
Hilbert, David      237Ч238 240 244
Hurwitz, Adolf      134 238
Hyperbolic geometry      145
imaginary numbers      140 207
incidence      148 170Ч172
Incident pairs      123 124
Induction, mathematical      94
Induction, mathematical, problems      98Ч99
Induction, mathematical, solutions      121Ч124
Infinite descent, method of      92Ч94
Infinite descent, method of, problems      98
Infinite descent, method of, solutions      118Ч121
Integers, positive      205
Integers, positive, as solutions to equations      96 97 102Ч104 117
Invariants of transformations, use of      91
Invariants of transformations, use of, problems      97
Invariants of transformations, use of, solutions      115Ч117
inversion      191 225Ч227
iterating      1Ч2
Iterating, problems      5Ч10
Iterating, solutions      27Ч37
Ladder, sliding      91
Lagrange's identity      142Ч144
Lagrange, Joseph Louis      134 142 238
Language, different, expressing problem in      89Ч90
Language, different, expressing problem, problems      95Ч96
Language, different, expressing problem, solutions      99Ч110
Lattice points      98 120Ч121 186Ч187 199Ч200
Leibniz' series for $\pi$      138 164
Leibniz' theorem      135 157Ч158
Leibniz, Gottfried Wilhelm      8 135 138 238
Light rays      126
Limits      232Ч233
Line segment, division of      96 107Ч110 122Ч123
Line, ideal      147 148
Lobachevsky, Nicolai Ivanovitch      145 238
Loops      231
Markoff numbers      85
Markoff, A.A.      238
Matijasevic, Jurii Vladimirovic      136 238
Matrices      21Ч22 73Ч74 212Ч213
Minkowski, Hermann      180 186 238
Moebius      107
Motzkin, T.S.      181 239
Mouseholes problem      18 58Ч59
n-gons      see "Polygons"
Natural numbers      205
Natural numbers as solutions of equations      19 63Ч64 187Ч188 201Ч204
Natural numbers as sums of square numbers      142Ч145 167Ч168
Natural numbers, number expressed as sum of      19 64Ч65
Natural numbers, odd divisors of      17 54Ч55
Natural numbers, prime divisors of      154Ч155
Natural numbers, square numbers between      110Ч112
Natural numbers, sum of powers of      98 121Ч122
Necessary conditions      4Ч5
News transmission problem      96 104Ч106
Newton, Isaac      239
Norms      144
Number sequences      232Ч233
Number series      7Ч8
Number theory, analytic      134
Operations, algebraic      214
Pappus Of Alexandria      19 146 239
parallelograms      18 19Ч20 48 57Ч58 66Ч67
Pascal's triangle      37 38 211 239
Pascal, Blaise      239
Paths      104Ч106 231
Patterns, search for      2Ч3
Patterns, search for, problems      11Ч17
Patterns, search for, solutions      37Ч55
Peano, Giuseppe      239
Pentagonal numbers      209
Pentagons      24 79Ч80
Pentagons, regular      8 30 31 98 120Ч121
Pentagrams      8 30 31
permutations      229 230
Permutations, zigzag      151Ч153 175Ч177
Physics, employment of      95
Physics, employment of, problems      99
Physics, employment of, solutions      125Ч132
Pigeon-hole principle      184 194Ч197
Plane, division of      21 71Ч72
Planes in three-dimensional space      181Ч183 189Ч191
Planes, projective      146Ч149
Plutarch      24
Point sets, convex      221Ч223
Points, ideal      147 148
Polygons (n-gons)      24 80Ч81 99 109Ч110 125 221
Polygons (n-gons), convex      18 56Ч57 150Ч151 174 221
Polygons (n-gons), regular      15 47Ч48 98 120Ч121 137 160Ч161
Polyhedra      19 65Ч66
Polynomial equations      140
Polynomials      98 121Ч122 134Ч135 155Ч157 213Ч214
Polynomials with prime number values      135Ч136 157Ч158
Positive numbers, products of      20Ч21 68Ч69
Prime numbers      24 208
Prime numbers, problems on      10 13 24 96 133Ч136
Prime numbers, problems on, solutions to      34 43 110 153Ч159
Prime numbers, relatively      15 53 208
Prime numbers, relatively, properties of      25 82 89Ч90 188 203Ч204
Probability theory      139
Projection, stereographic      112 227Ч228
Pyramids      20 67Ч68
Pythagoras of Samos      239
Pythagoras of Samos, theorem of      239
Quadrilaterals, areas of      19 61Ч62
Quadrilaterals, cyclic      62 129 223
Quaternions      143Ч144 167Ч168 208 225
Ramsey numbers      184Ч186 197Ч199
Ramsey, Frank Plumpton      185 239Ч240
Rational numbers      205 207
Real numbers      207
Rectangles      18 24 58 79
Reflection      228Ч229
Regions, division of plane into      21 71Ч72
Regions, division of space into      21 72Ч73
Rhombuses      15 47Ч48
Rotation      143 144 223Ч225
Scalars      216
Schoenberg, Isaac      186Ч187 240
Schwartz      129
sec x      151 153 178Ч179
Sequences, number      110Ч112 232Ч233
Series, number      7Ч8
Series, number, harmonic      212
Shoemaker's knife      146 168Ч170
Sierpiriski, Waclaw      71 154 186 240 244 246
Space, Division of      21 72Ч73
Sphere, cut by planes      26Ч27 87Ч88
Sphere, inversion with respect to      226Ч227
Sphere, properties of points on      21 70
Square numbers      19 24 42 43 62Ч63 209
Square numbers between terms of sequences      96 110Ч112
Square numbers, natural numbers as sums of      142Ч145 167Ч168
Square numbers, prime numbers as sums/differences of      136 158Ч159
Square, Latin      21
squares      15 18 23 48Ч50 58 78Ч79
Steinhaus, Hugo      186 240 242
Structures, algebraic      144 214Ч218
Sufficient conditions      4Ч5
Sundaram's sieve      13 43
Sylvester, James Joseph      181 240
Sylvester, problem of Sylvester Ч Gallai      180Ч183 188Ч194
Tan x      151 153 177Ч179
Tetrahedra      98Ч99 109 110 124 132
Tetrahedral numbers      209
transformations, geometric      223Ч229
Translation      228
tree diagrams      17 25 53 82Ч83 85
Trees      105Ч106
Triangle, harmonic      8 211Ч212
Triangle, Pascal's      37 38 211 239
Triangles, areas of      97Ч98 117Ч118
Triangles, circles inscribed in      25Ч26 85Ч87
Triangles, congruent      77 78
Triangles, construction of, problems      6Ч7 96 97Ч98 99
Triangles, construction of, solutions      29 107Ч108 118 125Ч130
Triangles, division of sides of      99 131Ч132
Triangles, equilateral      121
Triangles, isosceles      14 18 23 45Ч47 77Ч78
Triangles, right-angled      17Ч18 55Ч56 103Ч104
Triangular numbers      24 41 47 209
Tribonacci numbers      40
Trinomial coefficients      40 76 230
Triples, Pythagorean      21 210Ч211
tripods      141 142
Vector spaces, real      216Ч218
Vectors      35 69Ч70 216Ч218
Vertices      230
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