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The Mathematics of OZ: Mental Gymnastics from Beyond the Edge

Clifford A. Pickover
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Charles Ashbacher
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If you open this book, inject your brain with some caffeine, grab a pencil and paper and prepare yourself for some serious thinking, for Cliff Pickover has created a book full of 108 challenging puzzles in logic and mathematics. The premise is that aliens, the leader of which is a Dr. Oz, arrive on Earth, abduct Dorothy and her dog Toto and then subject her to a series of challenges. The puzzles are of various complexity, and Pickover has marked them with one through four stars, where four stars is the most difficult.

The problems are all fairly easy to understand, in most cases even mathematical novices will be able to understand the statement of the problem after reading the short leader. Solving it is of course another matter, but fortunately, detailed solutions to all of the problems are given at the end. In general, the problems are recreational in nature. There are several that involve moving through a maze in a certain way, many that involve pattern matching and some which require basic number theory.

A few are recycled oldies, for example number 78 is the problem where you have two containers with equal volumes of different liquids. A teaspoon of liquid is taken from one and placed in the other, that mixture is thoroughly stirred and then a teaspoon of the mixture is taken back to the original. The question is to determine which container is more contaminated. Several of the pattern matching problems involve some of the basic ideas for the differences in the patterns that I have seen in other problems over the years.

Nevertheless, the pace of the book allows you to easily overlook any of the problems you may have seen before. Each one begins with a quote and they are very good. In some cases, I enjoyed the quote more than I did the problem that followed it. The only negative point is that at the end of many problems, the villainous Dr. Oz gives an ultimatum concerning the solution. For example, in problem 89 he says, "If you get this problem wrong, I won't take you on a tour of the Tarantula Nebula." In my opinion, these phrases were a bit of a distraction. I kept looking for a clue to the solution in the ultimatums, but could never discern one. It would have been more interesting if there would have been a few more subtle clues in those threats, as the ultimatums did not seem to have anything to do with the problems.

Overall, this is a book of high quality puzzles, and even though many were familiar to me, I still read it with great interest. The solutions are very complete and understandable, although there are some solutions that I groaned about. While I never disputed their correctness, the pattern was one that was subject to more than one interpretation.

Charlie Ashbacher ( is the principal of Charles Ashbacher Technologies, a company that offers state of the art computer training. He is also an adjunct instructor at Mount Mercy College in Cedar Rapids, Iowa, and at the end of this academic year, he will be three courses short of having taught every class in the math and computer science majors. A co-editor of the Journal of Recreational Mathematics, he is the author of four books in mathematics and one in computer programming.

Travel guide; Preface; Introduction; Puzzles: 1. The yellow-brick road; 2. Animal array; 3. An experiment with Kansas; 4. An experiment with signs; 5. The logic of greenness; 6. Magical maze; 7. Kansas railway contraction; 8. The problem of the bones; 9. Square overdrive; 10. Squares and cubes; 11. Plex’s matrix; 12. Chaos at the clock factory; 13. The upsilon configuration; 14. Bone toss; 15. Animal farm courthouse; 16. Omega sphere; 17. Leg bone shatter produces triangle; 18. Z-bar ranch; 19. Mystery of phasers; 20. Salty number cycle; 21. Where are the composites?; 22. Brain trip; 23. The gaps of omicron; 24. Hutchinson problem; 25. Flint hill series; 26. Wacky tiles; 27. Toto clone puzzle; 28. Legion's number; 29. The problems of the tombs; 30. Plex’s tiles; 31. Phasers on targets; 32. The chamber of death and despair; 33. Zebra irrationals; 34. Creatures in resin; 35. Prime-poor equations; 36. Number satellite; 37. Flatworm math; 38. Regolith paradox; 39.; 40. Entroy; 41. Animal gap; 42. Arranging alien heads; 43. Ramanujan congruences and the quest for transcendence; 44. Getting noticed; 45. Juggler numbers; 46. Friends from Mars; 47. Phi in four 4s; 48; On planet zyph; 49. The jellyfish of europa; 50. Archeological dissection; 51. The gamma gambit; 52. Robot hand hive; 53. Ramanujan and the quattuordecillion; 54. Lunatic ferris wheel; 55. The ultimate spindle; 56. Prairie artifact; 57. Alien pellets; 58. The beauty of polygon slicing; 59. Cosmic call; 60. Knight moves; 61. Sphere; 62. Potawatomi target; 63. Sliders; 64. Swapping; 65. Triangle dissection; 66. A simple code; 67. Heterosquare; 68. Insertion; 69. Missing landscape; 70. The choice; 71. Animal selection; 72. The skeletal men of Uranus; 73. Hindbrain stimulation; 74. The arrays of absolution; 75. Trochophore abduction; 76. The dream pyramids of Missouri; 77. Mathematical flower petal; 78. Blood and water; 79. Cavern problems; 80. Three triplets; 81. Oos and oob gambit; 82. Napiform mathematics; 83. Toto, Mr. plex, elephant; 84. Witch overdrive; 85. What is art?; 86. Wendy magic square; 87. Heaven and hell; 88. The stars of heaven; 89. Vacation in the Tarantula nebula; 90. Hot lava; 91. Circular primes; 92. The truth about cats and dogs; 93. Disc mania; 94. N2+m2=s; 95. 2, 271, 2718281; 96. Android watch; 97. Knight moves; 98. Pool table gambit; 99. A connection between pi and e; 100. Venusian number bush; 101. Triangle cave; 102. Rat attack; 103. The scarecrow formula; 104. Circle math; 105. A, AB, ABA; 106. Ants and cheese; 107. The omega crystal; 108. Attack of undulating undecamorphs; Epilog; Further exploring; For further reading; About the author.