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The Master Book of Mathematical Recreations

by

The Master Book of Mathematical Recreations Cover

 

Table of Contents

[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

Product Details

ISBN:
9780486221342
Translator:
Gobel, F.
Editor:
O'Beirne, T. H.
Author:
Schuh, Fred
Publisher:
Dover Publications
Location:
New York
Subject:
General science
Subject:
Mathematics
Subject:
Recreations & Games
Subject:
Mathematical recreations
Subject:
Puzzles
Subject:
mathematical puzzles
Subject:
Logic puzzles
Subject:
Mathematics-Games and Puzzles
Copyright:
Edition Description:
Trade Paper
Series:
Dover Recreational Math
Publication Date:
20111130
Binding:
Paperback
Language:
English
Illustrations:
Yes
Pages:
464
Dimensions:
8.5 x 5.38 in 1.08 lb

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