Problem Solving and Recreational Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2016 April 4, 2016 Contents 1 Arithmetic Problems 101 1.1 Reconstruction of division problems . ........101 1.1.1 AMM E1 . ....................101 1.1.2 AMM E10 ....................103 1.1.3 AMM E1111 . ................104 1.1.4 AMM E971 ....................105 1.1.5 AMM E198 ....................106 2 Digit problems 107 2.1 When can you cancel illegitimately and yet get the cor- rect answer? . ....................107 2.2 Repdigits . ....................109 2.3 Sums of squares of digits ................112 3 Representation of a number in different bases 115 3.1 Base b-representation of a number ............115 3.1.1 A number from its base b-representation . ....116 3.2 The Josephus problem . ................119 4 Representation of a number in different bases 121 4.1 Balanced division by an odd number . ........121 4.2 Balanced base b representation of a number . ....121 4.3 Arithmetic in balanced base 3 representations . ....123 4.3.1 A matrix card trick ................124 5 Cheney’s card trick 127 5.1 Three basic principles . ................127 5.1.1 The pigeonhole principle . ............127 5.1.2 Arithmetic modulo 13 ..............127 iv CONTENTS 5.1.3 Permutations of three objects . ..........128 5.2 Examples . ......................129 5.3 A variation: Cheney card trick with spectator choosing secret card . ......................131 6 The nim game 133 6.1 Thenimsum.......................133 6.2 Thenimgame......................134 7 Fibonacci and Lucas numbers 137 7.1 The Fibonacci sequence . ..............137 7.2 Some relations of Fibonacci numbers ..........140 7.3 Zeckendorff representations . ..............141 7.4 The Lucas numbers . ..................142 7.5 How can we know if a given integer is a Fibonacci number? . ......................143 7.6 Periodicity of Fibonacci sequence mod m .......144 8 Counting with Fibonacci numbers 145 8.1 Squares and dominos ..................145 8.2 Fat subsets of [n] .....................146 8.3 An arrangement of pennies . ..............147 8.4 Counting circular permutations . ..........149 9 Floor and Ceiling 201 9.1 Clock problems . ..................201 9.2 Some equations involving x ..............202 9.3 Number of trialing zeros in n! ..............203 9.4 Characterizations of the ceiling and floor functions . 204 10 Combinatorial games 207 10.1 Subtraction games . ..................207 10.2 The Sprague-Grundy sequence . ..........209 11 Permutations 213 11.1 The universal sequence of permutations . ......213 11.2 The position of a permutation in the universal sequence 214 11.3 The disjoint cycle decomposition of a permutation . 216 11.4 Lexicographical ordering of permutations . ......217 11.5 Dudeney’s puzzle . ..................221 CONTENTS v 11.6 Problems on permutations ................224 12 Perfect card shuffles 227 12.1 In-shuffles . ....................229 12.2 Out-shuffles . ....................231 12.3 Perfect shuffling a card to a specified position . ....232 13 Interpolation problems 233 13.1 What is f(n +1)if f(k)=2k for k =0, 1, 2 ...,n? . 233 1 13.2 What is f(n +1)if f(k)= k+1 for k =0, 1, 2 ...,n? . 235 13.3 Why is ex not a rational function? ............237 14 Transferrable numbers 239 14.1 Right-transferrable numbers . ............239 14.2 Left-transferrable integers ................241 15 The equilateral lattice L (n) 245 15.1 Counting triangles ....................245 15.2 Counting parallelograms . ................248 15.3 Counting regular hexagons . ............249 16 Counting triangles 253 16.1 Integer triangles of sidelengths ≤ n ...........253 16.2 Integer scalene triangles with sidelengths ≤ n .....254 16.3 Number of integer triangles with perimeter n ......255 16.3.1 The partition number p3(n) ............255 17 Pythagorean triangles 301 17.1 Primitive Pythagorean triples . ............301 17.1.1 Rational angles . ................302 17.1.2 Some basic properties of primitive Pythagorean triples . ....................302 17.2 A Pythagorean triangle with an inscribed square ....304 17.3 When are x2 − px ± q both factorable? . ........305 17.4 Dissection of a square into Pythagorean triangles ....305 18 The area of a triangle 307 18.1 Heron’s formula for the area of a triangle ........307 18.2 Heron triangles . ....................309 18.2.1 The perimeter of a Heron triangle is even ....309 vi CONTENTS 18.2.2 The area of a Heron triangle is divisible by 6 . 309 18.2.3 Heron triangles with sides < 100 .........310 18.3 Heron triangle as lattice triangle . ..........311 18.4 Indecomposable Heron triangles . ..........312 18.5 Heron triangles with area equal to perimeter ......314 18.6 Heron triangles with integer inradii . ..........315 18.7 Heron triangles with square areas . ..........316 19 The golden ratio 317 19.1 Division of a segment in the golden ratio . ......317 19.2 The regular pentagon ..................319 19.3 Construction of 36◦, 54◦, and 72◦ angles . ......320 19.4 The most non-isosceles triangle . ..........324 20 The Arbelos 325 20.1 Archimedes’ twin circle theorem . ..........325 20.2 Incircle of the arbelos ..................326 20.2.1 Construction of incircle of arbelos . ......327 20.3 Archimedean circles in the arbelos . ..........328 21 Menelaus and Ceva theorems 331 21.1 Menelaus’ theorem . ..................331 21.2 Ceva’s theorem ......................333 22 Routh and Ceva theorems 339 22.1 Barycentric coordinates . ..............339 22.2 Cevian and traces . ..................340 22.3 Area and barycentric coordinates . ..........342 23 Perimeter - area bisectors of a triangle 347 23.1 Construction of perimeter-area bisectors in angle A . 348 23.2 Isosceles triangles . ..................349 23.3 Scalene triangles . ..................350 23.4 Triangles with exactly two perimeter-area bisectors . 352 23.5 Integer triangles . ..................353 23.5.1 Triangles with Da < 0 ..............353 23.5.2 Triangles with Da =0 ..............354 23.5.3 Triangles with Da > 0 ..............354 24 Very Small Integer Triangles 357 CONTENTS vii 25 The games of Euclid and Wythoff 401 25.1 The game of Euclid . ................401 25.2 Wythoff’s game . ....................403 25.3 Beatty’s Theorem ....................405 26 Number sequences 407 26.1 The Farey sequence . ................407 26.2 The number spiral ....................410 26.3 The prime number spirals ................412 26.4 The prime number spiral beginning with 17 . ....413 26.5 The prime number spiral beginning with 41 . ....414 26.6 Two enumerations of the rational numbers in (0,1) . 416 27 Inequalities 417 27.1 The inequality AGH2 ..................417 27.2 Proofs of the A-G inequality . ............420 27.3 The AGH mean inequality . ............422 27.4 Cauchy-Schwarz inequality . ............424 27.5 Convexity . ....................427 27.6 The power mean inequality . ............429 27.7 Bernoulli inequality . ................430 27.8 Holland’s Inequality . ................430 27.9 Miscellaneous inequalities ................432 27.10 A useful variant of AG2 .................433 28 Calculus problems 435 28.1 Philo’s line . ....................435 28.1.1 Triangle with minimum perimeter ........436 28.2 Maxima and minima without calculus . ........440 29 449 29.1 Quartic polynomials and the golden ratio ........449 30 Infinite Series 453 30.1 Power series . ....................453 30.2 The Bernoulli numbers and the tangent function ....455 30.3 The Euler numbers and the secant function . ....457 30.4 Summation of finite series ................458 viii CONTENTS 31 The art of integration 461 31.1 Indefinite integrals . ..................461 31.2 An iterated integral and a generalization . ......463 31.2.1 Crux Math. 3913 . ..............463 31.2.2 A generalization . ..............464 32 Rearrangements of the alternating harmonic series 467 32.1 Riemann’s theorem on conditionally convergent series . 467 32.2 Rearrangements of the alternating harmonic series . 468 Chapter 1 Arithmetic Problems 1.1 Reconstruction of division problems 1.1.1 AMM E1 This is Problem E1 of the AMERICAN MATHEMATICAL MONTHLY: x 7 xxx xxx) xxxxxxxx xxxx xxx xxx xxxx xxx xxxx xxxx Clearly, the last second digit of the quotient is 0. Let the divisor be the 3-digit number d. Consider the 3-digit number in the seventh line, which is a multiple of d. Its difference from the 4-digit number in the sixth line is a 2-digit number. This must be 9xx. This cannot be the same as the 3-digit number in the fifth line, since the difference between the 3-digit numbers in the fourth and fifth lines is a 3-digit number. Therefore, in the quotient, the digit after 7 is a larger one, which must be smaller than the first and the last digits, since these give 4-digit multiples of d. It follows that the quotient is 97809. 102 Arithmetic Problems Since 8d is a 3-digit number 9xx, the 4-digit number in the third and bottom lines is 9d =10xx or 11xx. From this 8d must be 99x, and therefore 992 = 8 × 124. 97809 124)12128316 1116 968 868 1003 992 1116 1116 1.1 Reconstruction of division problems 103 1.1.2 AMM E10 This is Problem E10 of the MONTHLY, by Fitch Cheney. In this case, not even one single digit is given. xxxxxx xxx) xxxxxxxx xxx xxxx xxx xxxx xxx xxxx xxxx 104 Arithmetic Problems 1.1.3 AMM E1111 This is said to be the most popular MONTHLY problem. It appeared in the April issue of 1954. Our good friend and eminent numerologist, Professor Euclide Paracelso Bombasto Umbugio, has been busily engaged test- ing on his desk calculator the 81 · 109 possible solutions to the problem of reconstructing the following exact long division in which the digits indiscriminately were each replaced by x save in the quotient where they were almost entirely omitted. xx8xx xxx)xxxxxxxx xxx xxxx xxx xxxx xxxx Deflate the Professor! That is, reduce the possibilities to (18 · 109)0. Martin Gardner’s remark: Because any number raised to the power of zero is one, the reader’s task is to discover the unique reconstruction of the problem.