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Table of Contents |
ii |
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Preface |
xi |
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To the Instructor
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xv |
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1 |
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1.1 Squaring the rectangle |
2 |
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1.1.1 Continue experimenting |
3 |
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1.1.2 Focus on the smallest square |
3
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1.1.3 Where is the smallest square |
4 |
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1.1.4 What are the neighbors of the smallest square? |
5 |
1.1.5 Is there a five square tiling? |
7 |
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1.1.6 Is there a six, seven, or nine square tiling? |
9 |
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1.2 A solution? |
10 |
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1.2.1 Bouwkamp codes |
12 |
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13 |
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1.3 Tiling by cubes |
14 |
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1.4 Tilings by equilateral triangles |
15 |
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1.5 Supplementary material |
16 |
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1.5.1 Squaring the square |
16 |
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1.5.2 Additional problems |
19 |
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1.6 Answers to problems |
20 |
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2 Pick's Rule
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29 |
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2.1 Polygons |
30 |
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2.1.1 On the grid |
30 |
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2.1.2 Polygons |
30 |
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2.1.3 Inside and outside |
31 |
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2.1.4 Splitting a polygon |
32 |
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2.1.5 Area of a polygonal region |
33 |
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33 |
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2.2 Some methods of calculating areas |
36 |
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2.2.1 An ancient Greek method |
37 |
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2.2.2 Grid point credit—a new fast method? |
38 |
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2.3 Pick credit |
41 |
| 2.3.1 Experimentation and trial-and-error |
41 |
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2.3.2 Rectangles and triangles |
44 |
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2.3.3 Additivity |
45 |
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2.4 Pick’s formula |
46 |
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2.4.1 Triangles solved |
47 |
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2.4.2 Proving Pick’s formula in general |
48 |
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2.5 Summary |
49 |
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2.6 Supplementary material |
50 |
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2.6.1 A bit of historical background |
50 |
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2.6.2 Can’t be useful though |
51 |
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2.6.3 Primitive triangulations |
51 |
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2.6.4 Reformulating Pick’s theorem |
54 |
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2.6.5 Gaming the proof of Pick’s theorem |
54 |
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2.6.6 Polygons with holes |
56 |
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2.6.7 An improved Pick count |
58 |
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2.6.8 Random grids |
60 |
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2.6.9 Additional problems |
62 |
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2.7 Answers to problems |
63 |
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3 Nim |
95 |
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3.1 Care for a game of tic-tac-toe? |
96 |
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3.2 Combinatorial games |
97 |
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3.2.1 Two-marker games |
98 |
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3.2.2 Three-marker games |
99 |
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3.2.3 Strategies? |
100 |
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3.2.4 Formal strategy for the two-marker game |
101 |
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3.2.5 Formal strategy for the three-marker |
102 |
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3.2.6 Balanced and unbalanced positions |
102 |
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3.2.7 Balanced positions in subtraction games |
106 |
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3.3 Game of binary bits |
107
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3.3.1 A coin game |
107 |
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3.3.2 A better way of looking at the coin game |
108 |
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3.3.3 Binary bits game |
109 |
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3.4 Nim |
113 |
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3.4.1 The mathematical theory of Nim |
113 |
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3.4.2 2–pile Nim |
114 |
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3.4.3 3–pile Nim |
115 |
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3.4.4 More three-pile experiments |
116 |
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3.4.5 The near-doubling argument |
117 |
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3.5 Nim solved by near-doubling |
120 |
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3.5.1 Review of binary arithmetic |
121 |
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3.5.2 Simple solution for the game of Nim |
123 |
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3.5.3 Déjà vu? |
124 |
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3.6 Return to marker games |
126 |
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3.6.1 Mind the gap |
127 |
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3.6.2 Strategy for the 6–marker game |
129 |
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3.6.3 Strategy for the 5–marker game |
131 |
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3.6.4 Strategy for all marker games |
131 |
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3.7 Misère Nim |
132 |
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3.8 Reverse Nim |
133 |
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3.8.1 How to reverse Nim |
133 |
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3.8.2 How to play Reverse Misère Nim |
135 |
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3.9 Summary and Perspectives |
136 |
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3.10 Supplementary material |
136 |
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3.10.1 Another analysis of the game of Nim |
137 |
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3.10.2 Grundy number |
137 |
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3.10.3 Nim-sums computed |
139 |
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3.10.4 Proof of the Sprague-Grundy theorem |
140 |
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3.10.5 Why does binary arithmetic keep coming up? |
142 |
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3.10.6 Another solution to Nim |
143 |
3.10.7 Playing the Nim game with nim-sums
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143 |
| 3.10.8 Obituary notice of Charles L. Bouton |
145 |
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3.11 Answers to problems |
148 |
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181 |
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4.1 Linking circles |
182 |
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4.1.1 Simple, closed curves |
183 |
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4.1.2 Shoelace model |
183 |
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4.1.3 Linking three curves |
184 |
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4.1.4 3–1 and 3–2 configurations |
185 |
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4.1.5 A 4–3 configuration |
185 |
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4.1.6 Not so easy? |
185 |
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4.1.7 Finding the right notation |
186 |
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4.2 Algebraic systems |
188 |
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4.2.1 Some familiar algebraic systems |
188 |
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4.2.2 Linking and algebraic systems |
189 |
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4.2.3 When are two objects equal? |
189 |
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4.2.4 Inverse notation |
190 |
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4.2.5 The laws of combination |
191 |
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4.2.6 Applying our algebra to linking problems |
191 |
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4.3 Return to the 4–3 configuration |
192 |
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4.3.1 Solving the 4–3 configuration |
192 |
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4.4 Constructing a 5–4 configuration |
194 |
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4.4.1 The plan |
194 |
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4.4.2 Verification |
194 |
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4.4.3 How about a 6–5 configuration? |
195 |
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4.4.4 Improving our notation again |
196 |
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4.5 Commutators |
196 |
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4.6 Moving on |
197 |
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4.6.1 Where we are |
198 |
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4.6.2 Constructing a 4–2 configuration |
198 |
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4.6.3 Constructing 5–2 and 6–2 configurations |
199 |
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4.7 Some more constructions |
199 |
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4.8 The general construction |
199 |
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4.8.1 Introducing a subscript notation |
200 |
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4.8.2 Product notation |
201 |
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4.8.3 Subscripts on subscripts |
202 |
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4.9 Groups |
203 |
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4.9.1 Rigid Motions |
205 |
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4.9.2 The group of linking operations |
206 |
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4.10 Summary and perspectives |
207 |
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4.11 A Final Word |
209 |
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4.11.1 As mathematics develops |
209 |
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4.11.2 A gap? |
210 |
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4.11.3 Is our linking language meaningful? |
212 |
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4.11.4 Avoid knots and twists |
212 |
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4.11.5 Now what? |
214 |
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4.12 Answers to problems |
215 |
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229 |
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A.1 Quitting smoking by the inductive method |
230 |
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A.2 Proving a formula by induction |
230 |
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A.3 Setting up an induction proof |
232 |
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A.3.1 Starting the induction somewhere else |
232 |
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A.3.2 Setting up an induction proof (alternative method) |
232 |
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A.4 Answers to problems |
235 |
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B Nim, A Game with a Complete Mathematical Theory |
239 |
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Bibliography |
245 |
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Index |
247 |
