Cofman J. — Numbers and shapes revisited: More problems for young mathematicians :: Электронная библиотека попечительского совета мехмата МГУ

Главная Ex Libris Книги Журналы Статьи Серии Каталог Wanted Загрузка ХудЛит Справка Поиск по индексам Поиск Форум

Авторизация

Поиск по указателям

Cofman J. — Numbers and shapes revisited: More problems for young mathematicians

Обсудите книгу на научном форуме

Нашли опечатку?
Выделите ее мышкой и нажмите Ctrl+Enter

Название: Numbers and shapes revisited: More problems for young mathematicians

Автор: Cofman J.

Аннотация:

Numbers and Shapes Revisited is the ideal guide for high school and undergraduate students seeking to understand the connections between the wide range of mathematical methods and concepts that they may come across in their curriculum. Topics include elementary number theory, classical algebra, euclidean geometry, group theory, and combinatorics. Stimulating and enjoyable, the book will promote independent thinking and the ability to pose and answer questions.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 319

Добавлена в каталог: 16.02.2014

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

-bonacci sequence      4
-dimensional hypercubes      76—7
-dimensional hypercubes, problems      77—9
-dimensional hypercubes, solutions      237—42
-dimensional Pascal pyramid      82—3
-dimensional real euclidean spaces      71—6
-dimensional ‘map-colouring’ problems      115
-gons      see Polygons
24-point sphere [of tetrahedron]      65 224
3-bonacci cuboids      12—13 141—2
3-bonacci hexagon      9 133—5
3-bonacci numbers      9 25—6 125
36 officers problem      103
Affine planes      108 109
Alternating groups      92
Appel, Kenneth      113
Archimedes of Syracuse      66
Associative operations      88 256 257
Barr, Mark      7
Bernoulli, Jakob      33
Besicovitch, A. S.      115
Binary operation      88
Binary relations [on the set]      46
Binet’s formula      6 17 19
Binet’s formula, generalization      252
Binet’s formula, proof      130
Binomial coefficients      81 243
Bos $\acute{e}$ — Shrikhande — Parker theorem      104
Cauchy — Bunyakowski inequality      54
Cauchy, Augustin Louis      52
Cauchy’s inequality      51
Cauchy’s inequality, generalization      53—5
Cauchy’s inequality, problems      51—5
Cauchy’s inequality, proofs      51—2 193—7
Cauchy’s inequality, solutions      193—201
Cayley, Arthur      112
Centroid of tetrahedron      64 220
Centroid of triangle      64
Chinese remainder theorem      45 47—8
Chinese remainder theorem, applications      48—50
Chinese remainder theorem, problems      46—50
Chinese remainder theorem, solutions      187—92
Cissoid of Diodes      59 60 211
Coefficients, binomial      81 243
Coefficients, multinomial      81 243
Coefficients, trinomial      81 243 249
Colouring/painting problems      5—6 103 112—15
Colouring/painting problems, solutions      125—6 287—94
Combinatorics      103—20
Combinatorics, problems      105—12 114 118—21
Combinatorics, solutions      273—300
Congruences      46
Congruences, solution of      47 188 190
Connected graphs      115 116
Continuous functions      44
Coordinate hyperplanes      73
Crum, M.      115
Cubes, doubling of      58—9
Cubes, net of      239
Cubes, problems      62
Cubes, solutions      213—17
De Moivre’s Theorem      228
de Morgan, Augustus      112
Decimal expansions      34
Decimal fractions      32
Dehn, Max      66
Dehn’s condition      66 67 227
Delian problem      58—9
DePalluel      107
Derangements      105 274
Dihedral angles [of polyhedra]      67
Diodes      59
Dodecahedra      see Rhombic dodecahedra
Dots, patterns of      20—30
Dots, problems      23—30
Dots, solutions      151—64
Doubling of cube      58—9
Doubling of square      60—1
Equidecomposable shapes      66—9
Equilateral triangle, area      201—2
Equivalent relations      46
Euclid of Alexandria      56 71
Euclidean geometries      71
Euler, Leonhard      23 103 114
Eulerian circuits [of graphs]      116 118 295 296 297 298
Euler’s      36
Euler’s [figurate number] formula      23
Euler’s [figurate number] formula, proof      30 164
Euler’s [map-colouring] theorem      114 118
Euler’s [map-colouring] theorem, proof      119 290—1
Even permutation      92
Exclusion and inclusion, principle of      273
Facets [in hypercubes]      77
Farey series      35 172
Farey series, properties      173
Ferrer graphs      22 27—9 161—2
Ferrer graphs, advantage for studying partitions      27
fibonacci      3
Fibonacci gasket      17
Fibonacci numbers      3 25
Fibonacci numbers, generalized      4 6—8 81 84
Fibonacci numbers, Lucas’ result      85 251
Fibonacci rectangles      6—7 8 131—2
Fibonacci sequence      3—4
Fibonacci sequence, problems      5—19
Fibonacci sequence, solutions      125—50
Fibonacci square      9
Fibonacci triangle      13—14 250
Fibonacci triangle, modulo-2      16 146
Fifteen puzzle      97—8
Fifteen puzzle, problem      98
Fifteen puzzle, solution      266—8
Figurate numbers      20 (see also Pentagonal numbers; square numbers; triangular numbers)
Figurate numbers, Euler’s formula      23 30 164
Finite affine plane      110
Finite group, operation table      107 277—8
Five-colour problem      114
Four-colour problem      112—13
Four-dimensional hypercubes, visualization      79—80
Fractals      4 17
Fractions      31
Full symmetric group      90
Functional equations, groups used in solution      93—5
Gambler’s ruin problem      86—7
Gambler’s ruin problem, solution      252—4
Gardner, Martin      7
Generalized Fibonacci numbers      4 6—8 81 84
Generators of groups      92
Geodesics      231
Geometries, euclidean geometries      71
Golden ratio      7
Golden rectangle      7 10
Golden rectangle, generalization      7—8
Golden rectangle, problem covering      10—11 137—9
Golden spiral      11
graphs      115—17
Graphs, definitions      115—16
Graphs, problems      118—20
Graphs, solutions      295—300
Great circles [of spheres]      70
Gregory, David      69
Group axioms      88
groups      88—102
Groups, alternating groups      92
Groups, and lattice points      96—7
Groups, basic properties      89—92
Groups, generators      92
Groups, meaning of term      88
Groups, permutation groups      90
Groups, problems      89—92 95—7 98—102

Groups, solutions      255—72
Groups, use for solution of functional equations      93—5
Guthrie, Francis      112
H $\ddot{o}$ lder’s inequality      54
Haken, Wolfgang      113
Hamilton, William Rowan      120
Hamiltonian circuits [of graphs]      120 298—9
Heawood, P. J.      114
Higher-dimensional euclidean geometries      71
Higher-dimensional spaces      71—87
Higher-dimensional spaces, problems      73—80 82—7
Higher-dimensional spaces, solutions      235—54
Hilbert, David      66
Hilbert’s third problem      66
Honeybee cells      61
Hoppe, R.      69
House-painting problems      5—6
House-painting problems, solutions      125—6
Hypercubes      76
Hypercubic lattice paths      80—7
Hyperplanes      73
Inclusion and exclusion, principle of      273
Inequalities      51
Inequalities, problems      51—7
Instant-insanity puzzle      120—1
Instant-insanity puzzle, problem      121
Instant-insanity puzzle, solution      299—300
Inversion, of permutation groups      91
Irrational numbers, Kronecker’s theorem      40—2
Isoperimetric theorem      51
Isoperimetric theorem, problems      56—7
Isoperimetric theorem, proofs      51
Isoperimetric theorem, solutions      201—7
Isoperimetric theorem, special cases      56—7
Isosceles triangle, area      202
K $\ddot{o}$ nig      62
Kaplansky      106 276
Kepler, Johannes      61 62
Kronecker, Leopold      37
Kronecker’s theorem      40—2
Kronecker’s theorem, further applications      42—4
Kronecker’s theorem, problems      41—4
Kronecker’s theorem, proof      40—1
Kronecker’s theorem, solutions      182—6
Latin rectangles      115
Latin rectangles, normalized Latin rectangles      105
Latin squares      103—7
Latin squares, mutually orthogonal Latin squares      108 279
Latin squares, orthogonal Latin squares      104
Latin squares, problem      105—12
Latin squares, solutions      273—87
Lattice parallelograms      36 96
Lattice paths      80—7
Lattice paths, problems      82—5 86 87
Lattice paths, solutions      242—52
Lattice points      31 34—6
Lattice points, invisible      49—50
Lattice points, visible      35 48 172
Leonardo of Pisa      3 (see also Fibonacci)
Light rays linear congruences      47
Light rays, reflection      37 (see also Non-periodic light ray paths; reflected)
Logarithmic spiral      11
Lucas, $\acute{E}$ douard      85 106 251
M $\acute{e}$ nage numbers      106
M $\acute{e}$ nage numbers, formula      106
M $\acute{e}$ nage numbers, formula, proof      276
Maclaurin, Colin      212
Mandelbrot, Benoit      17
Map, definitions      113
Map-colouring problems      103 112—15
Map-colouring problems, solutions      287—94
Married-couples problem      106
Married-couples problem, solution      274
Memory wheels      116—17
Memory wheels, problems      118—19
Memory wheels, solutions      294—8
Midspheres      230—1
Monge, point of      65
Multinomial coefficients      81 243
Mutually orthogonal Latin squares      108 279
Natural numbers, partitioning      22 26 28 29—30
Newton, Isaac      60 69
Nine-point circles      64 223
Non-periodic light ray paths      40—2
Normalized Latin rectangles      105
Odd permutation      92
Officers problem      103 104
Operation table of finite group      107 277—8
Operation table of set      255 256
Oriented circuits      116
Oriented graphs      116 295
Orthocentre of tetrahedron      64 65 220—2
Orthocentre of triangle      64
Orthogonal Latin squares      104
Orthogonality condition [for euclidean real planes]      75
Painting/colouring problems      5—6 103 112—15
Painting/colouring problems, solutions      125—6 287—94
Partitions of natural numbers      22 26 28 29—30
Pascal pyramid      82 244
Pascal’s triangle      14 16
Pascal’s triangle, modulo-2      16 145
Pascal’s triangle, problems      14—17 42
Pascal’s triangle, solutions      142—5
Pebble-heap problems      23—6
Pebble-heap problems, solutions      151—60
Pentagonal numbers      20
Pentagonal numbers, generalized      30
Periodic decimal numbers      31—3
Periodic fractions      32
Permutation groups      90
Permutation groups, inversion of      91
Pick’s Theorem      36
Platonic solids      120 298
Platonic solids, problem      120
Platonic solids, solution      298—9 (see also Cubes; dodecahedra; tetrahedral)
Polygons, equidecomposability      66
Polygons, partition of      26
Polyhedra, dihedral angles      67
Polyhedra, equidecomposability      66—9
Principle of inclusion and exclusion      273
Projective planes      108 111
Pyramids, equidecomposability      66 68
Pyramids, volume calculation      66
Pythagorean numbers      20
R $\acute{e}$ aumur, Ren $\acute{e}$       62
Rado, R.      115
Rational numbers      31
Reading list      301—3
Recurrence relations for Fibonacci numbers      3 4 125 127 148
Recurrence relations, how to solve      17—19
References listed      301—3
Reflected light rays      37
Reflected light rays, problems      37—40
Reflected light rays, solutions      175—81
Repunits      31—3 167
Residue class      46
Rhombic dodecahedra      61—2 120 298
Rubik’s cube      98—101
Rubik’s cube, problems      101—2
Rubik’s cube, solutions      268—72
Seating-arrangement problem      106
Seating-arrangement problem, solutions      274 277
Set, binary operation on      88
Set, operation table for      255 256
Sierpi $\acute{n}$ ski gasket      16 17
Sierpi $\acute{n}$ ski, Waclaw      17
Silver hexagon, problem covering      11—12 139—41
Silver polygons      8
Sphere-packing problems      69
Spheres, problems      69—70

1 2

О проекте