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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
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  • Songs by Artist

    Songs by Artist

    Songs by Artist Title Title (Hed) Planet Earth 2 Live Crew Bartender We Want Some Pussy Blackout 2 Pistols Other Side She Got It +44 You Know Me When Your Heart Stops Beating 20 Fingers 10 Years Short Dick Man Beautiful 21 Demands Through The Iris Give Me A Minute Wasteland 3 Doors Down 10,000 Maniacs Away From The Sun Because The Night Be Like That Candy Everybody Wants Behind Those Eyes More Than This Better Life, The These Are The Days Citizen Soldier Trouble Me Duck & Run 100 Proof Aged In Soul Every Time You Go Somebody's Been Sleeping Here By Me 10CC Here Without You I'm Not In Love It's Not My Time Things We Do For Love, The Kryptonite 112 Landing In London Come See Me Let Me Be Myself Cupid Let Me Go Dance With Me Live For Today Hot & Wet Loser It's Over Now Road I'm On, The Na Na Na So I Need You Peaches & Cream Train Right Here For You When I'm Gone U Already Know When You're Young 12 Gauge 3 Of Hearts Dunkie Butt Arizona Rain 12 Stones Love Is Enough Far Away 30 Seconds To Mars Way I Fell, The Closer To The Edge We Are One Kill, The 1910 Fruitgum Co. Kings And Queens 1, 2, 3 Red Light This Is War Simon Says Up In The Air (Explicit) 2 Chainz Yesterday Birthday Song (Explicit) 311 I'm Different (Explicit) All Mixed Up Spend It Amber 2 Live Crew Beyond The Grey Sky Doo Wah Diddy Creatures (For A While) Me So Horny Don't Tread On Me Song List Generator® Printed 5/12/2021 Page 1 of 334 Licensed to Chris Avis Songs by Artist Title Title 311 4Him First Straw Sacred Hideaway Hey You Where There Is Faith I'll Be Here Awhile Who You Are Love Song 5 Stairsteps, The You Wouldn't Believe O-O-H Child 38 Special 50 Cent Back Where You Belong 21 Questions Caught Up In You Baby By Me Hold On Loosely Best Friend If I'd Been The One Candy Shop Rockin' Into The Night Disco Inferno Second Chance Hustler's Ambition Teacher, Teacher If I Can't Wild-Eyed Southern Boys In Da Club 3LW Just A Lil' Bit I Do (Wanna Get Close To You) Outlaw No More (Baby I'ma Do Right) Outta Control Playas Gon' Play Outta Control (Remix Version) 3OH!3 P.I.M.P.