Acknowledgments |
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xi | |
Preliminaries |
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1 | (6) |
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What is a Mathematical Circle? |
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1 | (1) |
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2 | (1) |
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3 | (1) |
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4 | (1) |
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5 | (2) |
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7 | (202) |
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9 | (1) |
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A Few Words about Problem Sets |
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10 | (1) |
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Session 1 How to Solve a Problem |
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11 | (12) |
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11 | (1) |
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12 | (1) |
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Introductions and Math Warm-up |
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12 | (1) |
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What is a Mathematical Circle? |
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13 | (1) |
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Discussion of the Day: "What Does it Mean to Solve a Problem?" |
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13 | (6) |
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A Few Words about Problem Sets and Homework |
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19 | (4) |
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19 | (4) |
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Session 2 Knights and Liars |
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23 | (8) |
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23 | (1) |
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Problem Set From the Previous Class |
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24 | (7) |
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Discussion of the Day, Part 1 Cause and Effect Problems |
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24 | (1) |
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Discussion of the Day, Part 2 Knights-and-Liars Problems |
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25 | (3) |
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28 | (3) |
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Session 3 How to Turn Lies into Truth |
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31 | (6) |
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32 | (1) |
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Discussion of the Day: Negation of Logical Statements |
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33 | (4) |
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34 | (3) |
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Session 4 Mathematical Auction |
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37 | (6) |
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37 | (1) |
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The Game of Mathematical Auction |
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37 | (3) |
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Mathematical Auction Problems |
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40 | (3) |
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41 | (2) |
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Session 5 Word Problems and Common Sense |
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43 | (8) |
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43 | (1) |
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Discussion of the Day: "Draw a Picture---Solve the Problem!" |
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43 | (8) |
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Discussion of the Day, Part 2 Working Together |
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46 | (1) |
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47 | (4) |
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Session 6 More Word Problems |
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51 | (6) |
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51 | (1) |
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Discussion of the Day: Of Pigs and Hens |
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52 | (5) |
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53 | (4) |
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Session 7 Odd and Even Numbers I. Magic Paper Cups |
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57 | (6) |
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57 | (1) |
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Discussion of the Day: Odd and Even Numbers. Parity |
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58 | (5) |
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60 | (3) |
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Session 8 Odd and Even Numbers II. Definitions and Properties |
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63 | (8) |
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63 | (1) |
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Discussion of the Day: How to Define Odd and Even Numbers |
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63 | (2) |
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Parity of Sums and Differences |
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65 | (4) |
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69 | (2) |
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69 | (2) |
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Session 9 Halloween Math Hockey I |
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71 | (6) |
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71 | (1) |
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71 | (1) |
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72 | (5) |
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74 | (3) |
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Session 10 Odd and Even Numbers III. Alternations |
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77 | (8) |
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77 | (1) |
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Discussion of the Day: Alternations |
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77 | (8) |
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81 | (4) |
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Session 11 Weighings and Counterfeit Coins |
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85 | (6) |
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85 | (1) |
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Discussion of the Day: Balance Scales and Counterfeit Coins |
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86 | (5) |
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89 | (2) |
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Session 12 Mathematical Olympiad I Math Warm-up |
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91 | (6) |
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Event of the Day: Math Olympiad |
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92 | (1) |
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Math Olympiad I. First Page of Problems |
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93 | (1) |
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Math Olympiad I. Second Page of Problems |
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93 | (1) |
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Math Olympiad I. Additional Problems |
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94 | (3) |
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Session 13 Meet the Cube. First Lesson in 3D Geometry |
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97 | (8) |
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98 | (1) |
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Discussion of the Day: Cubes |
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98 | (2) |
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100 | (1) |
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"Meet the Cube" Exercises |
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101 | (4) |
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102 | (3) |
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Session 14 Cross Sections. Second Lesson in 3D Geometry |
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105 | (8) |
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105 | (1) |
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Discussion of the Day: Cross Sections |
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106 | (7) |
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110 | (3) |
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Session 15 Mathematical Auction |
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113 | (4) |
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113 | (1) |
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Event of the Day: The Game of Mathematical Auction |
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113 | (1) |
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Mathematical Auction Problems |
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114 | (3) |
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115 | (2) |
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Session 16 Combinatorics I |
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117 | (10) |
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117 | (1) |
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Discussion of the Day: Introduction to Combinatorics |
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117 | (10) |
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124 | (3) |
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Session 17 Combinatorics II |
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127 | (10) |
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127 | (1) |
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Discussion of the Day: The Slot Rule and Permutations without Repetitions |
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127 | (10) |
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134 | (3) |
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Session 18 Mathematical Hockey II |
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137 | (4) |
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137 | (1) |
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Event of the Day: Math Hockey Game |
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137 | (1) |
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138 | (3) |
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139 | (2) |
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Session 19 Numerical Puzzles I. Runaway Digits |
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141 | (6) |
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141 | (1) |
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Discussion of the Day: Runaway Digits |
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142 | (5) |
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144 | (3) |
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Session 20 Numerical Puzzles II. Encrypted Problems |
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147 | (6) |
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147 | (1) |
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Discussion of the Day: Encrypted Problems |
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147 | (6) |
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150 | (3) |
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Session 21 Mathematical Olympiad II |
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153 | (4) |
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Event of the Day: Math Olympiad |
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153 | (1) |
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Math Olympiad II. First Page of Problems |
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154 | (1) |
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Math Olympiad II. Second Page of Problems |
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155 | (1) |
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Math Olympiad II. Additional Problems |
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156 | (1) |
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Session 22 Divisibility I. Definition and Properties |
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157 | (8) |
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157 | (1) |
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Discussion of the Day: Divisibility |
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158 | (1) |
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159 | (1) |
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Divisibility: More Definitions |
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159 | (1) |
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Notation for Divisibility |
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160 | (1) |
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161 | (4) |
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163 | (2) |
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Session 23 Divisibility II. Prime Numbers and Prime Factorization |
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165 | (8) |
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165 | (1) |
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Discussion of the Day: Prime Numbers and Divisibility |
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165 | (1) |
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Prime Factor Decomposition |
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166 | (2) |
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168 | (1) |
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Prime Factorization and Divisibility |
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168 | (2) |
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The Prime Factorization of a Product |
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170 | (3) |
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171 | (2) |
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Session 24 Mathematical Auction |
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173 | (4) |
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173 | (1) |
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The Game of Mathematical Auction |
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173 | (1) |
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Mathematical Auction Problems |
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174 | (3) |
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174 | (3) |
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Session 25 Divisibility III. Divisibility Rules |
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177 | (8) |
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177 | (1) |
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Discussion of the Day: Divisibility Rules |
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177 | (4) |
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Divisibility and Cryptography |
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181 | (4) |
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182 | (3) |
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Session 26 Divisibility IV. Relatively Prime Numbers |
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185 | (6) |
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185 | (1) |
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Discussion of the Day: Relatively Prime Numbers |
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185 | (6) |
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188 | (3) |
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Session 27 Mathematical Games of Strategy I |
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191 | (6) |
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191 | (1) |
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Discussion of the Day: Games and Winning Strategies |
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191 | (6) |
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195 | (2) |
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Session 28 Mathematical Games of Strategy II |
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197 | (6) |
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197 | (1) |
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Discussion of the Day: Complements |
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197 | (2) |
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199 | (4) |
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200 | (3) |
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Session 29 Mathematical Olympiad III |
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203 | (6) |
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203 | (1) |
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Event of the Day: Math Olympiad |
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204 | (1) |
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Math Olympiad III. First Page of Problems |
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204 | (1) |
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Math Olympiad II. Second Page of Problems |
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205 | (1) |
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Math Olympiad II. Additional Problems |
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206 | (3) |
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Part 2 Mathematical Contests and Competitions |
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209 | (24) |
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211 | (2) |
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213 | (1) |
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Rules of Mathematical Auction |
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213 | (2) |
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A Sample Round of Mathematical Auction Game |
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215 | (1) |
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216 | (1) |
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216 | (1) |
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Example Math Auction Problems |
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217 | (2) |
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219 | (1) |
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Mathematical Hockey Rules |
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219 | (1) |
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220 | (3) |
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223 | (1) |
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223 | (1) |
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224 | (1) |
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Planning for an Oral Olympiad |
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225 | (1) |
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225 | (1) |
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226 | (1) |
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226 | (3) |
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Short Entertaining Math Games |
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229 | (1) |
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229 | (1) |
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229 | (1) |
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230 | (3) |
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Part 3 More Teaching Advice |
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233 | (16) |
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How to be a Great Math Circle Teacher |
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235 | (4) |
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239 | (1) |
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A Typical Session in Our Circle |
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239 | (2) |
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A Math Entertainment Session in Our Circle |
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241 | (1) |
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How to Organize a Session in Your Circle |
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242 | (1) |
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A Few More Ideas on the Layout of a Session |
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243 | (1) |
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How to Keep a Circle Journal |
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243 | (1) |
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How to Organize Problem-Solving Time |
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244 | (3) |
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247 | (2) |
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249 | (86) |
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Session 1 How to Solve a Problem |
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251 | (3) |
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Session 2 How to Solve a Problem and Explain a Solution |
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254 | (3) |
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Session 3 How to Turn Lies into Truth |
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257 | (2) |
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Session 4 Mathematical Auction |
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259 | (1) |
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Session 5 Word Problems and Common Sense |
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260 | (2) |
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Session 6 More Word Problems |
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262 | (4) |
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Session 7 Odds and Evens - Magic Paper Cups |
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266 | (3) |
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Session 8 Odds and Evens - Definitions and Properties |
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269 | (2) |
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271 | (1) |
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Session 9 Problem Set Solutions |
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272 | (2) |
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Session 10 Odds and Evens - Alternation |
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274 | (2) |
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Session 11 Weighing and Fake Coins |
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276 | (3) |
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Session 12 Math Olympiad I |
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279 | (4) |
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283 | (2) |
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Session 14 Cross Sections |
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285 | (3) |
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Session 15 Problem Set Solutions |
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288 | (3) |
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Session 16 Combinatorics I |
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291 | (4) |
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Session 17 Combinatorics II |
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295 | (3) |
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Session 18 Math Hockey II |
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298 | (1) |
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Session 18 Problem Set Solutions |
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299 | (2) |
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Session 19 Runaway Digits |
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301 | (2) |
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Session 20 Encrypted Problems |
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303 | (2) |
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Session 21 Math Olympiad II |
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305 | (5) |
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Session 22 Divisibility I |
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310 | (3) |
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Session 23 Divisibility II |
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313 | (3) |
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Session 24 Problem Set Solutions |
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316 | (2) |
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Session 25 Divisibility III |
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318 | (3) |
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Session 26 Divisibility IV |
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321 | (2) |
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Session 27 Mathematical Games of Strategy I |
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323 | (3) |
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Session 28 Mathematical Games of Strategy II |
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326 | (2) |
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328 | (7) |
Bibliography |
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335 | |