- 1 Lattice polygons
- 1.1 Pick’s Theorem: area of lattice polygon ....
- 1.2 Counting primitive triangles ...........
- 1.3 The Farey sequence ..................
- 2 Lattice points
- 2.1 Counting interior points of a lattice triangle ....
- 2.2 Lattice points on a circle ...........
- 3 Equilateral triangle in a rectangle
- 3.1 Equilateral triangle inscribed in a rectangle ....
- angle ......................... 3.2 Construction of equilateral triangle inscribed in a rect-
- 3.1 Equilateral triangle inscribed in a rectangle ....
- 4 Basic geometric constructions
- 4.1 Geometric mean ..................
- 4.2 Harmonic mean ..................
- 4.3 Equal subdivisions of a segment ...........
- 4.4 The Ford circles ..................
- 5 Greatest common divisor
- 5.1 gcd(a, b)as an integer combination ofaandb .....
- 5.2 Nonnegative integer combinations ofaandb ......
- 5.3 Cassini formula for Fibonacci numbers ....
- 5.4 gcd of generalized Fibonacci and Lucas numbers....
- 6 Pythagorean triples
- 6.1 Primitive Pythagorean triples ...........
- 6.2 Primitive Pythagorean triangles with square perimeters
- triangles ........................ 6.3 Lewis Carroll’s conjecture on triples of equiareal Pythagorean
- Pythagorean triangle ................. 6.4 Points at integer distances from the sides of a primitive
- 6.5 Dissecting a rectangle into Pythagorean triangles
- 7 The tangrams
- 7.1 The Chinese tangram .................
- 7.2 A British tangram .................
- 7.3 Another British tangram.................
- 8 The classical triangle centers
- 8.1 The centroid .................
- 8.2 The circumcircle and the circumcircle .......
- 8.3 The incenter and the incircle ..........
- 8.4 The orthocenter and the Euler line ...........
- 8.5 The excenters and the excircles ..........
- 9 The area of a triangle
- 9.1 Heron’s formula for the area of a triangle .......
- 9.2 Heron triangles .................
- 9.3 Heron triangles with consecutive sides ......
- 10 The golden section
- 10.1 The golden sectionφ ..................
- 10.2 Dissection of a square .................
- 10.3 Dissection of a rectangle.................
- 10.4 The golden right triangle ..........
- 10.5 What is the most non-isosceles triangle? ......
- 11 Constructions with the golden section
- 11.1 Construction of golden rectangle ...........
- section ........................ 11.2 Hofstetter’s compass-only construction of the golden
- section ........................ 11.3 Hofstetter’s 5-step division of a segment in the golden
- 11.4 Construction of regular pentagon ...........
- tagon ........................ 11.5 Ahlburg’s parsimonious construction of the regular pen-
- 11.6 Construction of a regular 17-gon ...........
- 11.1 Construction of golden rectangle ...........
- 12 Cheney’s card trick CONTENTS v
- 12.1 Principles ..................
- 12.2 Examples.........................
- 13 Digit problems
- rect answer? .................. 13.1 When can you cancel illegitimately and yet get the cor-
- 13.2 A Multiplication problem ...........
- 13.3 A division problem ..................
- 13.4 The most popular Monthly problem...........
- 13.5 The problem of 4n’s ..................
- 14 Numbers with many repeating digits
- 14.1 A quick multiplication..................
- 14.2 The repunits ..................
- 14.3 Squares of repdigits ..................
- 14.4 Sorted numbers with sorted squares ...........
- 15 Digital sum and digital root
- 15.1 Digital sum sequences ..................
- 15.2 Digital root ..................
- 15.3 The digital roots of the powers of 2 ...........
- 15.4 Digital root sequences..................
- 16 3-4-5 triangles in the square
- 17 Combinatorial games
- 17.1 Subtraction games ..................
- 17.1.1 The Sprague-Grundy sequence ....
- 17.1.2 Subtraction of square numbers ....
- 17.1.3 Subtraction of square numbers ....
- 17.2 The nim sum of natural numbers ...........
- 17.3 The gameNim ......................
- 17.4 Northcott’s variation of Nim ...........
- 17.5 Wythoff’s game ..................
- 17.1 Subtraction games ..................
- 18 Repunits
- 18.1 k-right-transposable integers ...........
- 18.2 k-left-transposable integers ...........
- 18.3 Sam Yates’ repunit riddles ...........
- 18.4 Recurring decimals ................. vi CONTENTS
- 18.5 The period length of a prime ...........
- 19 More digital trivia
- 20 The shoemaker’s knife
- 20.1 Archimedes’ twin circles ...........
- 20.2 Incircle of the shoemaker’s knife ..........
- 20.2.1 Archimedes’ construction ..........
- 20.2.2 Bankoff’s constructions ...........
- 20.2.3 Woo’s three constructions ..........
- 20.3 More Archimedean circles ..........
- 21 Infinitude of prime numbers
- numbers ........................ 21.1 Proofs by construction of sequence of relatively prime
- 21.2 Somos sequences .................
- 21.3 F ̈urstenberg’s topological proof made easy .......
- 22 Strings of prime numbers
- 22.1 The prime number spiral.................
- 22.2 The prime number spiral beginning with 17 ......
- 22.3 The prime number spiral beginning with 41 ......
- 23 Strings of composites
- 23.1 Strings of consecutive composite numbers ......
- 23.2 Strings of consecutive composite values ofn^2 +1
- 23.3 Consecutive composite values ofx^2 +x+41 .....
- 24 Perfect numbers
- 24.1 Perfect numbers .................
- 24.2 Charles Twigg on the first 10 perfect numbers
- 24.3 Abundant and deficient numbers ...........
- bers in (0,1) ................. 24.3.1 Appendix: Two enumerations of the rational num-
- 25 Routh and Ceva theorems
- 25.1 Routh theorem: an example ...........
- 25.2 Routh theorem .................
- 25.3 Ceva Theorem .................
- 26 The excircles CONTENTS vii
- 26.1 Feuerbach theorem ..................
- 26.2 A relation among the radii ...........
- 26.3 The circumcircle of the excentral triangle ....
- 26.4 The radical circle of the excircles ...........
- 26.5 Apollonius circle: the circular hull of the excircles
- 27 Figurate numbers
- 27.1 Triangular numbers ..................
- 27.2 Special triangular numbers ...........
- 27.3 Pentagonal numbers ..................
- 27.4 The polygonal numbersPn,k...............
- 27.4.1 Appendix: Solution of Pell’s equation ....
- 28 Polygonal triples
- 28.1 Double ruling ofS ....................
- triple ......................... 28.2 Primitive Pythagorean triple associated with ak-gonal
- 28.3 Triples of triangular numbers ...........
- 28.4 k-gonal triples determined by a Pythagorean triple
- triples ......................... 28.5 Correspondence between(2h+1)-gonal and 4 h-gonal
- 28.1 Double ruling ofS ....................
- 29 Sums of consecutive squares
- 29.1 Sum of squares of natural numbers ...........
- 29.2 Sums of consecutive squares: odd number case ....
- 29.3 Sums of consecutive squares: even number case ....
- 30 Sums of powers of natural numbers
- 31 A high school mathematics contest
- 32 Mathematical entertainments
- 32.1 2 .........................
- 32.2 2 .........................
- 33 Maxima and minima without calculus
- 34 A British test of teachers’ mathematical background
- 35 A mathematical contest viii CONTENTS
- 36 Some geometry problems from recent journals
- 37 The Josephus problem and its generalization
- 37.1 The Josephus problem
- 37.2 Generalized Josephus problemJ(n, k)
- 38 Permutations
- 38.1 The universal sequence of permutations
- 38.2 The position of a permutation in the universal sequence
- 39 Cycle decompositions
- 39.1 The disjoint cycle decomposition of a permutation
- 39.2
- 39.3 Dudeney’s Canterbury puzzle
- 39.4 The game of ropes and rungs
- 40 Graph labelling
- 40.1 Edge-magic triangle
- 40.2 Face-magic tetrahedron
- 40.3 Magic cube........................
- 40.4 Edge-magic heptagon
- 40.5 Edge-magic pentagram
- 40.6 A perfect magic circle
- 41 Card tricks from permutations
- 42 Tetrahedra
- 42.1 The isosceles tetrahedron
- 42.2 The volume of an isosceles tetrahedron
- 42.3 Volume of a tetrahedron
- 42.4
- 43 Lewis Carroll’s unused geometry pillow problem
- 44 Japanese Temple Geometry
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