Problem Solving and Recreational Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2012 Chapters 1–44 August 1 Monday 6/25 7/2 7/9 7/16 7/23 7/30 Wednesday 6/27 *** 7/11 7/18 7/25 8/1 Friday 6/29 7/6 7/13 7/20 7/27 8/3 ii Contents 1 Digit problems 101 1.1 When can you cancel illegitimately and yet get the cor- rectanswer? .......................101 1.2 Repdigits.........................103 1.3 Sortednumberswithsortedsquares . 105 1.4 Sumsofsquaresofdigits . 108 2 Transferrable numbers 111 2.1 Right-transferrablenumbers . 111 2.2 Left-transferrableintegers . 113 3 Arithmetic problems 117 3.1 AnumbergameofLewisCarroll . 117 3.2 Reconstruction of multiplicationsand divisions . 120 3.2.1 Amultiplicationproblem. 120 3.2.2 Adivisionproblem . 121 4 Fibonacci numbers 201 4.1 TheFibonaccisequence . 201 4.2 SomerelationsofFibonaccinumbers . 204 4.3 Fibonaccinumbersandbinomialcoefficients . 205 5 Counting with Fibonacci numbers 207 5.1 Squaresanddominos . 207 5.2 Fatsubsetsof [n] .....................208 5.3 Anarrangementofpennies . 209 6 Fibonacci numbers 3 211 6.1 FactorizationofFibonaccinumbers . 211 iv CONTENTS 6.2 TheLucasnumbers . 214 6.3 Countingcircularpermutations . 215 7 Subtraction games 301 7.1 TheBachetgame ....................301 7.2 TheSprague-Grundysequence . 302 7.3 Subtraction of powers of 2 ................303 7.4 Subtractionofsquarenumbers . 304 7.5 Moredifficultgames. 305 8 The games of Euclid and Wythoff 307 8.1 ThegameofEuclid . 307 8.2 Wythoff’sgame .....................309 8.3 Beatty’sTheorem . 311 9 Extrapolation problems 313 9.1 Whatis f(n + 1) if f(k)=2k for k =0, 1, 2 ...,n? . 313 1 9.2 Whatis f(n + 1) if f(k)= k+1 for k =0, 1, 2 ...,n? . 315 9.3 Whyis ex notarationalfunction? . 317 10 The Josephus problem and its generalization 401 10.1 TheJosephusproblem . 401 10.2 Chamberlain’ssolution . 403 10.3 The generalized Josephus problem J(n, k) .......404 11 The nim game 407 11.1 Thenimsum.......................407 11.2 Thenimgame ......................408 12 Prime and perfect numbers 411 12.1 Infinitudeofprimenumbers . 411 12.1.1 Euclid’sproof. 411 12.1.2 Fermatnumbers . 411 12.2 ThesieveofEratosthenes . 412 12.2.1 A visualization of the sieveof Eratosthenes . 412 12.3 Theprimenumbersbelow20000 . 414 12.4 Perfectnumbers . 415 12.5 Mersenneprimes. 416 12.6 CharlesTwiggonthefirst10perfectnumbers . 417 12.7 Primesinarithmeticprogression . 421 CONTENTS v 12.8 Theprimenumberspirals . 421 12.8.1 The prime number spiral beginningwith 17 . 422 12.8.2 The prime number spiral beginningwith 41 . 423 13 Cheney’s card trick 501 13.1 Threebasicprinciples . 501 13.1.1 Thepigeonholeprinciple . 501 13.1.2 Arithmetic modulo 13 ..............501 13.1.3 Permutationsofthreeobjects. 502 13.2 Examples.........................503 14 Variations of Cheney’s card trick 505 14.1 Cheney card trick with spectator choosing secret card . 505 14.2 A 3-cardtrick ......................507 15 The Catalan numbers 511 15.1 Numberofnonassociativeproducts . 511 16 The golden ratio 601 16.1 Divisionofasegmentinthegoldenratio . 601 16.2 Theregularpentagon . 603 16.3 Construction of 36◦, 54◦, and 72◦ angles . 604 16.4 Themostnon-isoscelestriangle . 608 17 Medians and angle Bisectors 609 17.1 Apollonius’Theorem . 609 17.2 Anglebisectortheorem . 611 17.3 Theanglebisectors . 612 17.4 Steiner-LehmusTheorem . 613 18 Dissections 615 18.1 Dissection of the 6 6 square. .. .. .. .615 × 18.2 Dissectionof a 7 7 squareintorectangles. 617 18.3 Dissectarectangletoformasquare× . 619 18.4 Dissectionofasquareintothreesimilarparts . 620 19 Pythagorean triangles 701 19.1 PrimitivePythagoreantriples . 701 19.1.1 Rationalangles . 702 vi CONTENTS 19.1.2 Some basic properties of primitive Pythagorean triples.......................702 19.2 A Pythagorean trianglewithan inscribedsquare . 704 19.3 When are x2 px q bothfactorable? . 705 19.4 Dissection of− a square± into Pythagorean triangles . 705 20 Integer triangles with a 60◦ or 120◦ angle 707 20.1 Integer triangles with a 60◦ angle ............707 20.2 Integer triangles with a 120◦ angle ...........710 21 Triangles with centroid on incircle 713 21.1 Construction . 714 21.2 Integertriangleswithcentroidontheincircle . 715 22 The area of a triangle 801 22.1 Heron’sformulafortheareaofatriangle . 801 22.2 Herontriangles. 803 22.2.1 TheperimeterofaHerontriangleiseven . 803 22.2.2 The area of a Heron triangle is divisible by 6 . 803 22.2.3 Heron triangles with sides < 100 .........804 22.3 Heron triangles with sides in arithmetic progression . 805 22.4 IndecomposableHerontriangles . 807 22.5 Herontriangleaslatticetriangle. 809 23 Heron triangles 811 23.1 Herontriangleswithareaequaltoperimeter . 811 23.2 Herontriangleswithintegerinradii . 812 23.3 Division of a triangle into two subtriangles with equal incircles .........................813 23.4 Inradiiinarithmeticprogression. 817 23.5 Herontriangleswithintegermedians . 818 23.6 Herontriangleswithsquareareas . 819 24 TriangleswithsidesandonealtitudeinA.P. 821 24.1 Newton’ssolution . 821 24.2 Thegeneralcase . 822 25 The Pell Equation 901 25.1 The equation x2 dy2 =1 ...............901 25.2 The equation x2 − dy2 = 1 ..............903 − − CONTENTS vii 25.3 The equation x2 dy2 = c ...............903 − 26 Figurate numbers 907 26.1 Whichtriangularnumbersaresquares?. 907 26.2 Pentagonalnumbers . 909 26.3 Almostsquaretriangularnumbers. 911 26.3.1 Excessivesquaretriangularnumbers . 911 26.3.2 Deficientsquaretriangularnumbers . 912 27 Special integer triangles 915 27.1 AlmostisoscelesPythagoreantriangles . 915 27.1.1 The generators of the almost isosceles Pythagorean triangles......................916 27.2 Integer triangles (a, a +1, b) with a 120◦ angle . 917 28 Heron triangles 1001 28.1 Herontriangleswithconsecutivesides . 1001 28.2 Heron triangles with two consecutivesquare sides . 1002 29 Squares as sums of consecutive squares 1005 29.1 Sumofsquaresofnaturalnumbers . 1005 29.2 Sumsofconsecutivesquares: oddnumbercase . 1008 29.3 Sumsofconsecutivesquares: evennumbercase . 1010 29.4 Sumsofpowersofconsecutiveintegers . 1012 30 Lucas’ problem 1013 30.1 Solution of n(n + 1)(2n +1)=6m2 for even n . 1013 30.2 The Pell equation x2 3y2 =1 revisited . 1014 − 30.3 Solution of n(n + 1)(2n +1)=6m2 for odd n . .1015 31 Some geometry problems 1101 32 Basic geometric constructions 1109 32.1 Somebasicconstructionprinciples . 1109 32.2 Geometricmean . .1110 32.3 Harmonicmean . .1111 32.4 A.M G.M. H.M. ..................1112 ≥ ≥ viii CONTENTS 33 Construction of a triangle from three given points 1115 33.1 Someexamples . .1115 33.2 Wernick’sconstructionproblems . 1117 34 The classical triangle centers 1201 34.1 Thecentroid . .. .. .. .. .. .. .1201 34.2 Thecircumcircleandthecircumcenter . 1202 34.3 Theincenterandtheincircle . 1203 34.4 TheorthocenterandtheEulerline. 1204 34.5 Theexcentersandtheexcircles . 1205 35 The nine-point circle 1207 35.1 Thenine-pointcircle. 1207 35.2 Feuerbach’stheorem. 1208 35.3 Lewis Carroll’s unused geometry pillowproblem . 1209 35.4 Johnson’stheorem . 1211 35.5 Triangles with nine-point center on the circumcircle . 1212 36 The excircles 1213 36.1 Arelationamongtheradii . 1213 36.2 Thecircumcircleoftheexcentraltriangle . 1214 36.3 Theradicalcircleoftheexcircles . 1215 36.4 Apollonius circle: the circular hull of the excircles . 1216 36.5 Three mutually orthogonal circles with given centers . 1217 37 The Arbelos 1301 37.1 Archimedes’twincircletheorem . 1301 37.2 Incircleofthearbelos . 1302 37.2.1 Constructionofincircleofarbelos . 1304 37.3 Archimedeancirclesinthearbelos . 1304 37.4 Constructionsoftheincircle. 1307 38 Menelaus and Ceva theorems 1309 38.1 Menelaus’theorem . 1309 38.2 Ceva’stheorem. .1311 39 Routh and Ceva theorems 1317 39.1 Barycentriccoordinates . 1317 39.2 Cevianandtraces . .1318 39.3 Areaandbarycentriccoordinates . 1320 CONTENTS ix 40 Elliptic curves 1401 40.1 AproblemfromDiophantus. 1401 40.2 Dudeney’spuzzleofthedoctorofphysic . 1403 40.3 Grouplawon y2 = x3 + ax2 + bx + c .........1404 41 Applications of elliptic curves to geometry problems 1407 41.1 Pairs of isoscelestriangleand rectanglewithequal perime- tersandequalareas . .1407 41.2 Triangles with a median, an altitude, and an angle bi- sectorconcurrent. 1409 42 Integer triangles with an altitude equal to a bisector 1411 42.1 Aquarticequation . .1411 42.2 Transformation of a quartic equation into an elliptic curve1413 43 The equilateral lattice L (n) 1501 43.1 Countingtriangles . 1501 43.2 Countingparallelograms. 1504 43.3 Countingregularhexagons . 1505 44 Counting triangles 1509 44.1 Integer triangles of sidelengths n ...........1509 44.2 Integer scalene triangles with sidelengths≤ n . .1510 44.3 Number of integer triangles with perimeter≤n . .1511 44.3.1 The partition number p3(n) ............1511 Chapter 1 Digit problems 1.1 When can you cancel illegitimately and yet get the correct answer? Let ab and bc be 2-digit numbers. When do such illegitimate cancella- tions as ab ab a bc = bc6 = c , 6 a allowing perhaps further simplifications of c ? 16 1 19 1 26 2 49 4 Answer. 64 = 4 , 95 = 5 , 65 = 5 , 98 = 8 . Solution. We may assume a, b, c not all equal. 10a+b a Suppose a, b, c are positive integers 9 such that 10b+c = c . (10a + b)c = a(10b + c), or (9a + b≤)c = 10ab. If any two of a, b, c are equal, then all three are equal. We shall therefore assume a, b, c all distinct. 9ac = b(10a c). If b is not divisible− by 3, then 9 divides 10a c = 9a +(a c). It follows that a = c, a case we need not consider. − − It remains to consider b =3, 6, 9. Rewriting (*) as (9a + b)c = 10ab. If c