What to Solve?: Problems and Suggestions for Young Mathematicians

Front Cover......Page 1
Title page......Page 3
Copyright......Page 4
Preface......Page 5
Acknowledgements......Page 9
Contents......Page 11
Introduction......Page 15
Section 1 : Iterating......Page 19
Section 2 : Search for patterns......Page 25
Section 3 : Exceptions & special cases......Page 31
Section 4 : Generalizing given problems......Page 33
Section 5 : Converse problems......Page 37
Section 1 : Iterating......Page 41
Section 2 : Search for patterns......Page 51
Section 3 : Exceptions & special cases......Page 69
Section 4 : Generalizing given problems......Page 80
Section 5 : Converse problems......Page 91
Introduction......Page 103
Section 1 : Expressing the problem in a different language......Page 109
Section 2 : Extending the field of investigation......Page 110
Section 4 : The use of extremal elements......Page 111
Section 6 : Mathematical induction......Page 112
Section 1 : Expressing the problem in a different language......Page 113
Section 2 : Extending the field of investigation......Page 124
Section 3 : The use of invariants of transformations......Page 129
Section 4 : The use of extremal elements......Page 131
Section 5 : The method of infinite descent......Page 132
Section 6 : Mathematical induction......Page 135
Section 8 : Employing physics......Page 139
1.1 Prime numbers in arithmetic progressions......Page 147
1.2 Wilson's theorem & results of Lagrange & Leibniz on prime numbers......Page 148
1.3 Polynomials with prime number values......Page 149
2.1 Archimedes' algorithm for calculating π......Page 151
2.2 God's delight in odd numbers : The Leibniz series for π, deduced from Gregory's arc tangent series......Page 152
2.3 π & probability : Buffon's needle problem......Page 153
3.1 Gauss' fundamental theorem of axonometry......Page 154
3.2 Lagrange's identity on products of sums of four squares treated by quaternions......Page 156
4.1 Euclidean geometry......Page 159
4.2 Projective planes......Page 160
5.1 In how many ways can a product of n factors be calculated by pairs?......Page 163
5.2 Euler's problem on polygon division......Page 164
5.3 The number of 'zigzag' permutations of the set {l,2,3, .. . ,n} leading to the secant & tangent series......Page 165
Section 1 : Problems on prime numbers......Page 167
Section 2 : The number π......Page 174
Section 3 : Applications of complex numbers & quaternions......Page 180
Section 4 : On Euclidean & non-Euclidean geometries......Page 182
Section 5 : The art of counting; results of Catalan, Euler & Gregory......Page 186
Section 1 : The problems of Sylvester-Gallai & related questions in Euclidean geometry & in combinatorics......Page 194
1.2 Two generalizations of Sylvester-Gallai's problem in Euclidean geometry......Page 195
1.3 The number of lines in L, & an intriguing discovery when generalization breaks down......Page 196
1.4 A generalization of Gallai's result in the theory of block designs......Page 197
2.2 Some Ramsey numbers......Page 198
3.2 Schoenberg's generalization of Steinhaus' problem......Page 200
Section 4 : Fermat's last theorem & related problems......Page 201
Section 1 : The problem of Sylvester & Gallai, & related questions......Page 202
Section 2 : The pigeon-hole principle & some Ramsey numbers......Page 208
Section 3 : Problems on lattice points......Page 213
Section 4 : Fermat's last theorem & related problems......Page 215
Appendix I : Definitions & basic results......Page 219
Appendix II : Notes on mathematicians mentioned in the text......Page 248
Appendix III : Recommended reading......Page 255
Index......Page 261