Many of these puzzles are entirely original; others have been treated in as original a way as possible, including domino puzzles, noughts and crosses, puzzles in mechanics, more. Puzzle fans will not only enjoy mastering the puzzles, they'll find their ability to solve mathematical problems increasing. "Exceptionally good value." — Journal of Recreational Math.
A wealth of original challenges and new approaches to classic conundrums includes numerous worked examples and a focus on the mathematics behind the puzzles. "Exceptionally good value." — Journal of Recreational Math.
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