Recreational Mathematics
Paul Yiu
Department of Mathematics Florida Atlantic University
Summer 2003
Chapters 1–44
Version 031209 ii Contents
1 Lattice polygons 101 1.1 Pick’s Theorem: area of lattice polygon ...... 102 1.2 Counting primitive triangles ...... 103 1.3 The Farey sequence ...... 104
2 Lattice points 109 2.1 Counting interior points of a lattice triangle ...... 110 2.2 Lattice points on a circle ...... 111
3 Equilateral triangle in a rectangle 117 3.1 Equilateral triangle inscribed in a rectangle ...... 118 3.2 Construction of equilateral triangle inscribed in a rect- angle ...... 119
4 Basic geometric constructions 123 4.1 Geometric mean ...... 124 4.2 Harmonic mean ...... 125 4.3 Equal subdivisions of a segment ...... 126 4.4 The Ford circles ...... 127
5 Greatest common divisor 201 5.1 gcd(a, b) as an integer combination of a and b .....202 5.2 Nonnegative integer combinations of a and b ...... 203 5.3 Cassini formula for Fibonacci numbers ...... 204 5.4 gcd of generalized Fibonacci and Lucas numbers ....205
6 Pythagorean triples 209 6.1 Primitive Pythagorean triples ...... 210 6.2 Primitive Pythagorean triangles with square perimeters 211 iv CONTENTS
6.3 Lewis Carroll’s conjecture on triples of equiareal Pythagorean triangles ...... 212 6.4 Points at integer distances from the sides of a primitive Pythagorean triangle ...... 213 6.5 Dissecting a rectangle into Pythagorean triangles . . . . 214
7 The tangrams 225 7.1 The Chinese tangram ...... 226 7.2 A British tangram ...... 227 7.3 Another British tangram ...... 228
8 The classical triangle centers 231 8.1 The centroid ...... 232 8.2 The circumcircle and the circumcircle ...... 233 8.3 The incenter and the incircle ...... 234 8.4 The orthocenter and the Euler line ...... 235 8.5 The excenters and the excircles ...... 236
9 The area of a triangle 301 9.1 Heron’s formula for the area of a triangle ...... 302 9.2 Heron triangles ...... 303 9.3 Heron triangles with consecutive sides ...... 304
10 The golden section 309 10.1 The golden section ϕ ...... 310 10.2 Dissection of a square ...... 311 10.3 Dissection of a rectangle ...... 313 10.4 The golden right triangle ...... 314 10.5 What is the most non-isosceles triangle? ...... 316
11 Constructions with the golden section 321 11.1 Construction of golden rectangle ...... 322 11.2 Hofstetter’s compass-only construction of the golden section ...... 323 11.3 Hofstetter’s 5-step division of a segment in the golden section ...... 325 11.4 Construction of regular pentagon ...... 327 11.5 Ahlburg’s parsimonious construction of the regular pen- tagon ...... 328 11.6 Construction of a regular 17-gon ...... 329 CONTENTS v
12 Cheney’s card trick 335 12.1 Principles ...... 336 12.2 Examples ...... 337
13 Digit problems 401 13.1 When can you cancel illegitimately and yet get the cor- rect answer? ...... 402 13.2 A Multiplication problem ...... 403 13.3 A division problem ...... 404 13.4 The most popular Monthly problem ...... 407 13.5 The problem of 4 n’s...... 408
14 Numbers with many repeating digits 415 14.1 A quick multiplication ...... 416 14.2 The repunits ...... 417 14.3 Squares of repdigits ...... 418 14.4 Sorted numbers with sorted squares ...... 419
15 Digital sum and digital root 423 15.1 Digital sum sequences ...... 424 15.2 Digital root ...... 425 15.3 The digital roots of the powers of 2 ...... 426 15.4 Digital root sequences...... 427
16 3-4-5 triangles in the square 431
17 Combinatorial games 501 17.1 Subtraction games ...... 502 17.1.1 The Sprague-Grundy sequence ...... 503 17.1.2 Subtraction of square numbers ...... 504 17.1.3 Subtraction of square numbers ...... 505 17.2 The nim sum of natural numbers ...... 509 17.3 The game Nim ...... 510 17.4 Northcott’s variation of Nim ...... 511 17.5 Wythoff’s game ...... 512
18 Repunits 517 18.1 k-right-transposable integers ...... 518 18.2 k-left-transposable integers ...... 519 18.3 Sam Yates’ repunit riddles ...... 520 vi CONTENTS
18.4 Recurring decimals ...... 522 18.5 The period length of a prime ...... 523
19 More digital trivia 531
20 The shoemaker’s knife 535 20.1 Archimedes’ twin circles ...... 536 20.2 Incircle of the shoemaker’s knife ...... 537 20.2.1 Archimedes’ construction ...... 537 20.2.2 Bankoff’s constructions ...... 538 20.2.3 Woo’s three constructions ...... 539 20.3 More Archimedean circles ...... 540
21 Infinitude of prime numbers 601 21.1 Proofs by construction of sequence of relatively prime numbers ...... 602 21.2 Somos sequences ...... 603 21.3 F¨urstenberg’s topological proof made easy ...... 604
22 Strings of prime numbers 611 22.1 The prime number spiral ...... 612 22.2 The prime number spiral beginning with 17 ...... 613 22.3 The prime number spiral beginning with 41 ...... 614
23 Strings of composites 621 23.1 Strings of consecutive composite numbers ...... 622 23.2 Strings of consecutive composite values of n2 +1 . . . 623 23.3 Consecutive composite values of x2 + x +41 .....624
24 Perfect numbers 627 24.1 Perfect numbers ...... 628 24.2 Charles Twigg on the first 10 perfect numbers . . . . . 629 24.3 Abundant and deficient numbers ...... 630 24.3.1 Appendix: Two enumerations of the rational num- bers in (0,1) ...... 633
25 Routh and Ceva theorems 701 25.1 Routh theorem: an example ...... 702 25.2 Routh theorem ...... 703 25.3 Ceva Theorem ...... 704 CONTENTS vii
26 The excircles 711 26.1 Feuerbach theorem ...... 712 26.2 A relation among the radii ...... 713 26.3 The circumcircle of the excentral triangle ...... 714 26.4 The radical circle of the excircles ...... 715 26.5 Apollonius circle: the circular hull of the excircles . . . 716
27 Figurate numbers 719 27.1 Triangular numbers ...... 719 27.2 Special triangular numbers ...... 719 27.3 Pentagonal numbers ...... 721 27.4 The polygonal numbers Pn,k ...... 722 27.4.1 Appendix: Solution of Pell’s equation ...... 723
28 Polygonal triples 725 28.1 Double ruling of S ...... 726 28.2 Primitive Pythagorean triple associated with a k-gonal triple ...... 727 28.3 Triples of triangular numbers ...... 727 28.4 k-gonal triples determined by a Pythagorean triple . . . 728 28.5 Correspondence between (2h +1)-gonal and 4h-gonal triples ...... 730
29 Sums of consecutive squares 801 29.1 Sum of squares of natural numbers ...... 802 29.2 Sums of consecutive squares: odd number case . ....803 29.3 Sums of consecutive squares: even number case ....805
30 Sums of powers of natural numbers 807
31 A high school mathematics contest 809
32 Mathematical entertainments 817 32.1 2 ...... 818 32.2 2 ...... 820
33 Maxima and minima without calculus 903
34 A British test of teachers’ mathematical background 911 viii CONTENTS
35 A mathematical contest 915
36 Some geometry problems from recent journals 919
37 The Josephus problem and its generalization 1001 37.1 The Josephus problem ...... 1001 37.2 Generalized Josephus problem J(n, k) ...... 1003
38 Permutations 1005 38.1 The universal sequence of permutations ...... 1005 38.2 The position of a permutation in the universal sequence 1007
39 Cycle decompositions 1009 39.1 The disjoint cycle decomposition of a permutation . . . 1009 39.2 2 ...... 1010 39.3 Dudeney’s Canterbury puzzle 3 ...... 1013 39.4 The game of ropes and rungs ...... 1015
40 Graph labelling 1017 40.1 Edge-magic triangle ...... 1017 40.2 Face-magic tetrahedron ...... 1018 40.3 Magic cube ...... 1019 40.4 Edge-magic heptagon ...... 1020 40.5 Edge-magic pentagram ...... 1021 40.6 A perfect magic circle ...... 1022
41 Card tricks from permutations 1101
42 Tetrahedra 1107 42.1 The isosceles tetrahedron ...... 1107 42.2 The volume of an isosceles tetrahedron ...... 1108 42.3 Volume of a tetrahedron ...... 1109 42.4 2 ...... 1110
43 Lewis Carroll’s unused geometry pillow problem 1113
44 Japanese Temple Geometry 1115 Chapter 1
Lattice polygons
1 Pick’s Theorem: area of lattice polygon 2 Counting primitive triangles 3 The Farey sequences Appendix: Regular solids Exercise Project: A cross number puzzle