The Master Book of Mathematical Recreations.pdf

The Master Book of Mathematical Recreations.pdf


Praised for its "exceptionally good value" by the Journal of Recreational Math, this volume offers original puzzles as well as new approaches to classic conundrums. Puzzle fans will enjoy mastering these domino puzzles, noughts and crosses, and other challenges. They'll also appreciate the numerous worked examples, which will improve their puzzle-solving strategies and mathematical skills. 

Dutch mathematician Frederik Schuh (1875–1966) received his PhD in algebraic geometry from Amsterdam University and taught at the Delft University of Technology.

[Asterisks indicate sections that involve algebraic formulae.]
Chapter I: Hints for Solving Puzzles
I. Various Kinds of Puzzles
  1. Literary puzzles
  2. Pure puzzles
  3. Remarks on pure puzzles
  4. Puzzle games
  5. Correspondences and differences between puzzles and games
II. Solving by Trial
  6. Trial and error
  7. Systematic trial
  8. Division into cases
  9. Example of a puzzle tree
III. Classification System
  10. Choosing a classification system
  11. Usefulness of a classification system
  12. More about the classification system
IV. Solving a Puzzle by Simplification
  13. Simplifying a puzzle
  14. Example of how to simplify a puzzle
  15. Remarks on the seven coins puzzle
  16. Reversing a puzzle
  17. Example of reversing a puzzle
V. Solving a Puzzle by Breaking It Up
  18. Breaking a puzzle up into smaller puzzles
  19. Application to the crossing puzzle
  20. Number of solutions of the crossing puzzle
  21. Restrictive condition in the crossing puzzle
  22. Shunting puzzle
VI. Some Puzzles with Multiples
  *23. Trebles puzzle
  *24. Breaking up the trebles puzzle
  *25. Trebles puzzle with larger numbers
  *26. Doubles puzzle with 7-digit numbers
  *27. Remarks on the numbers of §26
  *28. Quintuples puzzle
Chapter II: Some Domino Puzzles
I. "Symmetric Domino Puzzle, with Extensions"
  29. Symmetric domino puzzle
  30. Extended symmetric domino puzzle
  *31. Another extension of the symmetric domino puzzle
II. Doubly Symmetric Domino Puzzle
  *32. First doubly symmetric domino puzzle
  *33. Doubly symmetric domino puzzle without restrictive condition
  *34. Connection with the puzzle of §32
  35. Second doubly symmetric domino puzzle
  36. Puzzle with dominos in a rectangle
III. Smallest and Largest Numbers of Corners
  37. Salient and re-entrant angles
  38. Puzzle with the smallest number of angles
  39. Puzzle with the largest number of angles
Chapter III: The Game of Noughts and Crosses
I. Description of the Game
  40. Rules of the game
  41. Supplement to the game
  42. Consequences of the rules
II. Considerations Affecting Values of the Squares
  43. Value of a square
  44. Remarks on the value of a square
III. Directions for Good Play
  45. Semi-row or threat
  46. Double threat
  47. Combined threat
  48. Replying to a double threat
  49. Further directions for good play
IV. Some Remarks on Good Play
  50. Remarks on the double threat
  51. Connection with the value of a move
V. General Remarks on the Analysis of the Game
  52. Preliminary remarks
  53. Diagrams
  54. Tree derived from the diagrams
VI. Partial Analysis of the Game
  55. "John starts with the central square 5, Peter replies with the corner square 1"
  56. "John starts with the corner square 1, Peter replies with the central square 5"
  57. "John starts with the border square 2, Peter replies with the central square 5"
  58. Equitable nature of the game
VII. Complete Analysis of the Game
  59. "John starts with the central square 5, Peter replies with the border square 2"
  60. "John starts with the corner square 1, Peter replies with the border square 2"
  61. "John starts with the corner square 1, Peter replies with the corner square 3"
  62. "John starts with the corner square 1, Peter replies with the border square 6"
  63. "John starts with the corner square 1, Peter replies with the corner square 9"
  64. Results of John's first move 1
  65. "John starts with the border square 2, Peter replies with the corner square 1"
  66. "John starts with the border square 2, Peter replies with the border square 4"
  67. "John starts with the border square 2, Peter replies with the corner square 7"
  68. "John starts with the border square 2, Peter replies with the border square 8"
  69. Results of John's first move 2
VIII. Modification of the Game of Noughts and Crosses
  70. First modification of the game
  71. Second modification of the game
  72. Conclusions from the trees of §71
IX. Puzzles Derived from the game
  *73. Possible double threats by John
  *74. Possible double threats by Peter
  *75. Some more special puzzles
  *76. Possible cases of a treble threat
  77. Remark on the treble threat
Chapter IV: Number Systems
I. Counting
  78. Verbal counting
  79. Numbers in written form
  80. Concept of a digital system
II. Arithmetic
  81. Computing in a digital system
  *82. Changing to another number system
III. Remarks on Number Systems
  83. The only conceivabe base of a number system is 10
  84. Comparison of the various digital systems
  85. Arithmetical prodigies
IV. More about Digital Systems
  86. Origin of our digital system
  97. Forerunners of a digital system
  88. Grouping objects according to a number system
Chapter V: Some Puzzles Related to Number Systems
I. Weight Puzzles
  89. Bachet's weights puzzle
  90. Weights puzzles with weights on both pans
  91. Relation to the ternary system
II. Example of a Binary Puzzle
  92. Disks puzzle
  93. Origin of the disks puzzles
III. Robuse and Related Binary Puzzle
  94. Robuse
  95. Transposition puzzles
  *96. Other transposition puzzles
CHAPTER VI: Games with Piles of Matches
I. General Observations
  97. General remarks
  98. Winning situations
II. Games with One Pile of Matches
  99. Simplest match game
  100. Extension of the simplest match game
  101. More difficult game with one pile of matches
III. Games with Several Piles of Matches
  102. Case of two piles
  103. Case of more than two piles and a maximum of 2
  104. Case of more than two piles and a maximum of 3
  *105. Case of more than two piles and a maximum of 4 or 5
  *106. "As before, but the last match loses"
IV. Some Other Match Games
  107. Game with two piles of matches
  108. Game with three piles of matches
  *109. Extension of four or five piles
  *110. Modification of the game with three piles of matches
  111. Match game with an arbitary number of piles
  *112. Case in which loss with the last match is a simpler game
V. Game of Nim
  113. General remarks
  114. Game of nim with two piles
  115. Some winning situations
VI. Game of Nim and the Binary System
  116. Relation to the binary system
  117. Proof of the rule for the winning situations
  118. Remarks on the correct way of playing
  119. Case in which the last match loses
  120. Simplest way to play
VII. Extension or Modification of the Game of Nim
  121. Extension of the game of nim to more than three piles
  *122. Further extension of the game of nim
  *123. Special case of the game of §122
  *124. Modification of the game of nim
Chapter VII: Enumeration of Possibilities and the Determination of Probabilities
I. Number of Possibilities
  125. Multiplication
  126. Number of complete permutations
  127. Number of restricted permutations
  128. Number of combinations
  129. "Number of permutations of objects, not all different "
  130. Number of divisions into piles
II. Determining Probabilities from Equally Likely Cases
  131. Notion of probability
  132. Origin of the theory of probability
  133. Misleading example of an incorrect judgment of equal likelihood
III. Rules of Calculating Probabilites
  134. Probability of either this or that; the addition rule
  135. Probability of both this and that; the product rule
  136. Examples of dependent events
  137. Maxima and minima of sequences of numbers
  138. Extension to several events
  139. Combination of the sum rule and product rule
  140. More about maxima and minima in a sequence of numbers
IV. Probabilities of Causes
  141. A posteriori probability: the quotient rule
  142. Application of the quotient rule
  143. Another application
Chapter VIII: Some Applications of the Theory of Probability
I. Various Questions on Probabilities
  144. Shrewd prisoner
  145. Game of kasje
  *146. Simplification of the game kasje
  147 Poker dice
  148. Probabilities in poker dice
II. Probabilities in Bridge
  149. Probability of a given distribution of the cards
  150. A posteriori probability of a certain distribution of the cards
  151. Probabilities in finessing
Chapter IX: Evaluation of Contingencies and Mean Values
I. Mathematical Expectation and Its Applications
  152. Mathematical Expectation
  153. Examples of mathematical expectation
  154. More complicated example
  155. Modification of the example §154
  156. Petersburg paradox
II. Further Application of Mathematical Expectation
  157. Appplication of mathematical expectation to the theory of probability
  158. Law of large numbers
  159. Probable error
  160. Remarks on the law of large numbers
  161. Further relevance of the law of large numbers
III. Average Values
  162. Averages
  163. Other examples of averages
  164. Incorrect conclusion from the law of large numbers
Chapter X: Some Games of Encirclement
I. Game of Wolf and Sheep
  165. Rules of the game of wolf and sheep
  166. Correct methods for playing wolf and sheep
  167. Some wolf and sheep problems
  168. Even and odd positions
  169. Final remark on wolf and sheep
II. "Game of Dwarfs or "Catch the Giant!"
  170. Rules of the game
  171. Comparison with wolf and sheep
  172. Remarks on correct lines of play
  173. Correct way of playing
  174. Winning positions
  175. Positions where the dwarfs are to move
III. Further Considerations of the Game of Dwarfs
  176. "Remarks on diagrams D, E, and G"
  177. Critical positions
  178. More about the correct way of playing
  179. Trap moves by the giant
  180. Comparison of the game of dwarfs with chess
IV. Modified Game of Dwarfs
  181. Rules of the game
  182. Winning positons of the modified game
  183. Case in which the dwarfs have to move
  184. Dwarfs puzzle
  185. Remark on diagrams A-H
  186. Other opening moves of the giant
V. The Soldier's Game
  187. Rules of the game
  188. Winning positions
  189. Course of the game
  190. Other winning positions
  191. Modified soldier's game
Chapter XI: Sliding-Movement Puzzles
I. Game of Five
  192. Rules of the game
  193. Some general advice
  194. Moving a single cube
  195. Condition for solvability
II. Extensions of the Game of Five
  196. Some results summarized
  197. Proof of the assertions of §196
III. Fatal Fifteen
  198. Further extension of the game of five
  199. Proof of corresponding results
IV. Futher Considerations on Inversions
  200. Property of inversions
  *201 Cyclic permutation
  *202. Parity determination in terms of cyclic permutations
V. Least Number of Moves
  203. Determination of the least number of moves
  204. First example
  *205. Some more examples
VI. Puzzles in Decanting Liquids
  206. Simple decanting puzzle with three jugs
  207. Another decanting puzzle with three jugs
  208. Remarks on the puzzles of §§206 and 207
  209 Changes of the three jugs
  210. Further remarks on the three-jug puzzle
  211. Decanting puzzle with four jugs
  212. Another puzzle with four jugs
Chapter XII: Subtraction Games
I. Subtraction Game with a Simple Obstacle
  213. Subtraction games in general
  214. Subtraction games with obstacles
  215. Winning numbers when 0 wins
  216. Winning numbers when 0 loses
II. Subtraction Game with a More Complicated Obstacle
  217. Rules of the game
  218. Even-subtraction game
  219. Odd-subtraction game
III. "3-,5-,7- and 9-Subtraction Games"
  220. 3-subtraction game
  221. The other 3-subtraction games
  222. 5-subtraction game
  223. 7-subtraction game
  224. 9-subtraction game
IV. Subtraction Game where the Opener Loses
  225. Modified subtraction game
  226. "Modified 2-,3-, 4-, and 5-subtraction games"
  227. "Modified 6-, 7-, 9-, and 9-subtraction games"
  *228 Modified subtraction game with larger deductions
Chapter XIII: Puzzles with Some Mathematical Aspects
I. Simple Puzzles with Squares
  229. Puzzle with two square numbers of two or three digits
  230. Puzzle with three 3-digit squares
  231. Puzzle of §230 with initial zeros
II. Puzzle with 4-Digit Squares
  232. 4-digit squares
  233. Puzzle of the four-4digit squares
  234. Puzzle of §233 with zeros
III. A Curious Multiplication
  235. Multiplication puzzle with 20 digits
  236. Connection with remainders for divisions by 9
  237. Combination of the results of §§235 and 236
IV. Problem on Remainders and Quotients
  238. Arithmetical puzzle
  239. Variants of the puzzle of §238
  *240. Mathematical discussion of the puzzle
V. Commuter Puzzles
  241. Simple commuter puzzle
  242. More difficult commuter puzzle
  243. Solution of the puzzle of §242
VI. Prime Number Puzzles
  244. Prime number puzzle with 16 squares
  245. Solution of the puzzle of §244
  246. Examination of the five cases
  247. Puzzle of §244 with a restriction
  248. Prime number puzzle with 25 squares
  249. Puzzle with larger prime numbers
VII. Remarkable Divisibility
  250. Divisibility of numbers in a rectange
  251. Puzzle with multiples of 7
  252. Multiples of 7 puzzle with the largest sum
  253. Proof that the solutions found do in fact yield the largest sum
  254. Multiples of 7 with the maximum product
VIII. Multiplication and Division Puzzles
  255. "Multiplication puzzle "Est modus in rebus"
  256. Multiplication and division puzzle
  *257. Terminating division puzzle
  *258. Repeating division puzzle
IX. Dice Puzzles
  259 Symmetries of a cube
  *260. Group of symmetries
  *261. Symmetries of the regular octahedron
  262. Eight dice joined to make a cube
  *263. More difficult puzzle with eight dice
  *264 Which are the invisible spot numbers?
Chapter XIV: Puzzles of Assorted Types
I. Network Puzzle
  265. Networks
  266. Puzzle on open and closed paths
  267. Relation to the verti


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