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



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

јвтор: Cofman J.

јннотаци€:

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: en

–убрика: ћатематика/

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

ed2k: ed2k stats

√од издани€: 1995

 оличество страниц: 319

ƒобавлена в каталог: 16.02.2014

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$k$-bonacci sequence      4
$n$-dimensional hypercubes      76Ч7
$n$-dimensional hypercubes, problems      77Ч9
$n$-dimensional hypercubes, solutions      237Ч42
$n$-dimensional Pascal pyramid      82Ч3
$n$-dimensional real euclidean spaces      71Ч6
$n$-dimensional Сmap-colouringТ problems      115
$n$-gons      see Polygons
24-point sphere [of tetrahedron]      65 224
3-bonacci cuboids      12Ч13 141Ч2
3-bonacci hexagon      9 133Ч5
3-bonacci numbers      9 25Ч6 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
BinetТs formula      6 17 19
BinetТs formula, generalization      252
BinetТs formula, proof      130
Binomial coefficients      81 243
Bos$\acute{e}$ Ч Shrikhande Ч Parker theorem      104
Cauchy Ч Bunyakowski inequality      54
Cauchy, Augustin Louis      52
CauchyТs inequality      51
CauchyТs inequality, generalization      53Ч5
CauchyТs inequality, problems      51Ч5
CauchyТs inequality, proofs      51Ч2 193Ч7
CauchyТs inequality, solutions      193Ч201
Cayley, Arthur      112
Centroid of tetrahedron      64 220
Centroid of triangle      64
Chinese remainder theorem      45 47Ч8
Chinese remainder theorem, applications      48Ч50
Chinese remainder theorem, problems      46Ч50
Chinese remainder theorem, solutions      187Ч92
Cissoid of Diodes      59 60 211
Coefficients, binomial      81 243
Coefficients, multinomial      81 243
Coefficients, trinomial      81 243 249
Colouring/painting problems      5Ч6 103 112Ч15
Colouring/painting problems, solutions      125Ч6 287Ч94
Combinatorics      103Ч20
Combinatorics, problems      105Ч12 114 118Ч21
Combinatorics, solutions      273Ч300
Congruences      46
Congruences, solution of      47 188 190
Connected graphs      115 116
Continuous functions      44
Coordinate hyperplanes      73
Crum, M.      115
Cubes, doubling of      58Ч9
Cubes, net of      239
Cubes, problems      62
Cubes, solutions      213Ч17
De MoivreТs Theorem      228
de Morgan, Augustus      112
Decimal expansions      34
Decimal fractions      32
Dehn, Max      66
DehnТs condition      66 67 227
Delian problem      58Ч9
DePalluel      107
Derangements      105 274
Dihedral angles [of polyhedra]      67
Diodes      59
Dodecahedra      see Rhombic dodecahedra
Dots, patterns of      20Ч30
Dots, problems      23Ч30
Dots, solutions      151Ч64
Doubling of cube      58Ч9
Doubling of square      60Ч1
Equidecomposable shapes      66Ч9
Equilateral triangle, area      201Ч2
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
EulerТs      36
EulerТs [figurate number] formula      23
EulerТs [figurate number] formula, proof      30 164
EulerТs [map-colouring] theorem      114 118
EulerТs [map-colouring] theorem, proof      119 290Ч1
Even permutation      92
Exclusion and inclusion, principle of      273
Facets [in hypercubes]      77
Farey series      35 172
Farey series, properties      173
Ferrer graphs      22 27Ч9 161Ч2
Ferrer graphs, advantage for studying partitions      27
fibonacci      3
Fibonacci gasket      17
Fibonacci numbers      3 25
Fibonacci numbers, generalized      4 6Ч8 81 84
Fibonacci numbers, LucasТ result      85 251
Fibonacci rectangles      6Ч7 8 131Ч2
Fibonacci sequence      3Ч4
Fibonacci sequence, problems      5Ч19
Fibonacci sequence, solutions      125Ч50
Fibonacci square      9
Fibonacci triangle      13Ч14 250
Fibonacci triangle, modulo-2      16 146
Fifteen puzzle      97Ч8
Fifteen puzzle, problem      98
Fifteen puzzle, solution      266Ч8
Figurate numbers      20 (see also Pentagonal numbers; square numbers; triangular numbers)
Figurate numbers, EulerТs formula      23 30 164
Finite affine plane      110
Finite group, operation table      107 277Ч8
Five-colour problem      114
Four-colour problem      112Ч13
Four-dimensional hypercubes, visualization      79Ч80
Fractals      4 17
Fractions      31
Full symmetric group      90
Functional equations, groups used in solution      93Ч5
GamblerТs ruin problem      86Ч7
GamblerТs ruin problem, solution      252Ч4
Gardner, Martin      7
Generalized Fibonacci numbers      4 6Ч8 81 84
Generators of groups      92
Geodesics      231
Geometries, euclidean geometries      71
Golden ratio      7
Golden rectangle      7 10
Golden rectangle, generalization      7Ч8
Golden rectangle, problem covering      10Ч11 137Ч9
Golden spiral      11
graphs      115Ч17
Graphs, definitions      115Ч16
Graphs, problems      118Ч20
Graphs, solutions      295Ч300
Great circles [of spheres]      70
Gregory, David      69
Group axioms      88
groups      88Ч102
Groups, alternating groups      92
Groups, and lattice points      96Ч7
Groups, basic properties      89Ч92
Groups, generators      92
Groups, meaning of term      88
Groups, permutation groups      90
Groups, problems      89Ч92 95Ч7 98Ч102
Groups, solutions      255Ч72
Groups, use for solution of functional equations      93Ч5
Guthrie, Francis      112
H$\ddot{o}$lderТs inequality      54
Haken, Wolfgang      113
Hamilton, William Rowan      120
Hamiltonian circuits [of graphs]      120 298Ч9
Heawood, P. J.      114
Higher-dimensional euclidean geometries      71
Higher-dimensional spaces      71Ч87
Higher-dimensional spaces, problems      73Ч80 82Ч7
Higher-dimensional spaces, solutions      235Ч54
Hilbert, David      66
HilbertТs third problem      66
Honeybee cells      61
Hoppe, R.      69
House-painting problems      5Ч6
House-painting problems, solutions      125Ч6
Hypercubes      76
Hypercubic lattice paths      80Ч7
Hyperplanes      73
Inclusion and exclusion, principle of      273
Inequalities      51
Inequalities, problems      51Ч7
Instant-insanity puzzle      120Ч1
Instant-insanity puzzle, problem      121
Instant-insanity puzzle, solution      299Ч300
Inversion, of permutation groups      91
Irrational numbers, KroneckerТs theorem      40Ч2
Isoperimetric theorem      51
Isoperimetric theorem, problems      56Ч7
Isoperimetric theorem, proofs      51
Isoperimetric theorem, solutions      201Ч7
Isoperimetric theorem, special cases      56Ч7
Isosceles triangle, area      202
K$\ddot{o}$nig      62
Kaplansky      106 276
Kepler, Johannes      61 62
Kronecker, Leopold      37
KroneckerТs theorem      40Ч2
KroneckerТs theorem, further applications      42Ч4
KroneckerТs theorem, problems      41Ч4
KroneckerТs theorem, proof      40Ч1
KroneckerТs theorem, solutions      182Ч6
Latin rectangles      115
Latin rectangles, normalized Latin rectangles      105
Latin squares      103Ч7
Latin squares, mutually orthogonal Latin squares      108 279
Latin squares, orthogonal Latin squares      104
Latin squares, problem      105Ч12
Latin squares, solutions      273Ч87
Lattice parallelograms      36 96
Lattice paths      80Ч7
Lattice paths, problems      82Ч5 86 87
Lattice paths, solutions      242Ч52
Lattice points      31 34Ч6
Lattice points, invisible      49Ч50
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 112Ч15
Map-colouring problems, solutions      287Ч94
Married-couples problem      106
Married-couples problem, solution      274
Memory wheels      116Ч17
Memory wheels, problems      118Ч19
Memory wheels, solutions      294Ч8
Midspheres      230Ч1
Monge, point of      65
Multinomial coefficients      81 243
Mutually orthogonal Latin squares      108 279
Natural numbers, partitioning      22 26 28 29Ч30
Newton, Isaac      60 69
Nine-point circles      64 223
Non-periodic light ray paths      40Ч2
Normalized Latin rectangles      105
Odd permutation      92
Officers problem      103 104
Operation table of finite group      107 277Ч8
Operation table of set      255 256
Oriented circuits      116
Oriented graphs      116 295
Orthocentre of tetrahedron      64 65 220Ч2
Orthocentre of triangle      64
Orthogonal Latin squares      104
Orthogonality condition [for euclidean real planes]      75
Painting/colouring problems      5Ч6 103 112Ч15
Painting/colouring problems, solutions      125Ч6 287Ч94
Partitions of natural numbers      22 26 28 29Ч30
Pascal pyramid      82 244
PascalТs triangle      14 16
PascalТs triangle, modulo-2      16 145
PascalТs triangle, problems      14Ч17 42
PascalТs triangle, solutions      142Ч5
Pebble-heap problems      23Ч6
Pebble-heap problems, solutions      151Ч60
Pentagonal numbers      20
Pentagonal numbers, generalized      30
Periodic decimal numbers      31Ч3
Periodic fractions      32
Permutation groups      90
Permutation groups, inversion of      91
PickТs Theorem      36
Platonic solids      120 298
Platonic solids, problem      120
Platonic solids, solution      298Ч9 (see also Cubes; dodecahedra; tetrahedral)
Polygons, equidecomposability      66
Polygons, partition of      26
Polyhedra, dihedral angles      67
Polyhedra, equidecomposability      66Ч9
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      301Ч3
Recurrence relations for Fibonacci numbers      3 4 125 127 148
Recurrence relations, how to solve      17Ч19
References listed      301Ч3
Reflected light rays      37
Reflected light rays, problems      37Ч40
Reflected light rays, solutions      175Ч81
Repunits      31Ч3 167
Residue class      46
Rhombic dodecahedra      61Ч2 120 298
RubikТs cube      98Ч101
RubikТs cube, problems      101Ч2
RubikТs cube, solutions      268Ч72
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      11Ч12 139Ч41
Silver polygons      8
Sphere-packing problems      69
Spheres, problems      69Ч70
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