Mathematical Explorations for Young Minds: Adventures on the Island of Oxisz (Problem Books in Mathematics)

This book presents a carefully curated selection of mathematical problems suitable for young minds aged 12 and beyond. Most of these problems have been adapted from the Dürer Competition, a venerable mathematics competition in Hungary renowned for fostering teamwork and encouraging creative thinking in the realm of mathematical problem-solving. Within the pages of this book, readers are invited to accompany Albrecht, Tarkal, and Zsordi on 14 captivating journeys to the enigmatic Island of Oxisz, where they will encounter a total of 84 thought-provoking problems. Some of these challenges are intricately interconnected, elevating the level of difficulty. The concluding section provides comprehensive solutions and helpful hints, illuminating the paths taken by the three friends to tackle the problems. No specialized prior knowledge is required beyond a solid foundation in mathematics from the first seven years of school. Nevertheless, many problems in the book demand original ideas or clever combinations of elementary principles. This book offers students, mathematics educators, and trainers alike a captivating glimpse into a distinctive competition, making it a valuable resource for cultivating problem-solving skills at an early age. Preface Acknowledgments About the Authors Problems 1. Discover the Island of Oxisz 1.1 Embassies on the Island of Oxisz 1.2 Zoo Runway 1.3 Incomplete Divisor Sequences 1.4 Office 1.5 Bishops and Rooks 1.6 Rock–Paper–Scissors with a Twist 2. Painting with a Boat and a Barrel 2.1 Lifeboat 2.2 Barrels of Wine 2.3 Train Cars 2.4 Contemporary Painter 2.5 The Logo of the ``Super(stitious)man'' 2.6 The Game of Covering Numbers 3. Skiing Between Fréz and Gard 3.1 Skiing Competition 3.2 Shopping Carts 3.3 Connecting a Grid into a Polygon 3.4 Buses Connecting Villages 3.5 Who Is My Bigger Neighbor? 3.6 Placing Tokens in Four Squares 4. Football in Brassó 4.1 Stagecoaches 4.2 Pyramid of Numbers 4.3 Photocopied Polygon 4.4 Polygon with Many Concave Angles 4.5 Mixed up Kits 4.6 Placing Tokens on a Board with Five Squares 5. Can I Lock the Cat in the Mill? 5.1 Magic? 5.2 Tortoise, Cat, Table 5.3 Central Lock 5.4 Nine Men's Morris Without a Mill 5.5 Length of Line Segments in a Circle 5.6 Jump over the Other Piece! 6. The Trick to Making Moles Disappear 6.1 Blocks of Light Bulbs 6.2 Rectangle with a Zigzag 6.3 Group Stage Qualification 6.4 Mole and Bulldozer 6.5 The Magic Trick of Rodolfo 6.6 The Game of Turning Tokens Upside Down 7. The Rejuvenating Grandma Is Baking a Pie 7.1 Grandma Is Getting Younger 7.2 Too Many Knaves 7.3 Finding the Missing Sums 7.4 Dividing the Pie 7.5 Crouching and Standing Students 7.6 The Game of Two Heaps 8. Reconcilable Differences 8.1 Erring Organizers? 8.2 Find the Connection 8.3 Without Adjacent Obtuse Angles 8.4 Quadrilaterals That Are Small and Big at the Same Time 8.5 The Number of Odd Divisors 8.6 Policemen in Cubetown 9. How Many Centimeters Is a Dessert? 9.1 Boys, Girls, T-Shirts 9.2 Diagonal Halving an Octagon 9.3 Missing Marks on a Ruler 9.4 The Diet of Arthur Noodle 9.5 Dissection into Rectangle 9.6 Double Opening Move Tic-Tac-Toe 10. Self-intersecting Tic-Tac-Toe 10.1 Ice Creams on a Stick 10.2 Enthusiasm and Half-heartedness 10.3 Quadraples of Kritor and Lidek 10.4 Uniquely Self-intersecting Polygon 10.5 Prime Orders of Divisors 10.6 Anti-Tic-Tac-Toe 11. Flowers on the Tiles of the Housing Estates 11.1 Housing Estate 11.2 Bouquets of Flowers 11.3 Tile 11.4 The Colorful Flag 11.5 Short Heights, Large Area 11.6 Taking Away Tokens on the Same Line 12. Trolleybus on the Oktogon 12.1 Trolleybus, Bus, Tram 12.2 The Octagon Shaped Rug 12.3 Filling in Tables with 1s and 2s 12.4 Pairwise Products 12.5 Numbers with an Alternating Sequence of Divisors 12.6 Duel of Five Cards Against Five Cards 13. Bloody Serious Problems 13.1 Bidding on Each Other 13.2 Queen, Rook, Bishop, Knight 13.3 Polygon with Many Right Angles 13.4 Lucky Numbers in Oxisz 13.5 Blood Donors and Raspberry Juices 13.6 37 Tokens in Three Heaps 14. Icing on the Cake—Really Difficult Problems 14.1 Quadruples of Kritor, Lidek and Timol 14.2 Meandering Zigzag 14.3 Mole and Bulldozer 14.4 Dividing the Cake 14.5 Last Digits of Divisors 14.6 Heap Dividing Game, an Arbitrary Number of Tokens in 3 Heaps Solutions 1. Discover the Island of Oxisz 2. Painting with a Boat and a Barrel 3. Skiing Between Fréz and Gard 4. Football in Brassó 4.1 Stage-coaches 5. Can I Lock the Cat in the Mill? 6. The Trick to Making Moles Disappear 7. The Rejuvenating Grandma Is Baking a Pie 8. Reconcilable Differences 9. How Many Centimeters Is a Dessert? 10. Self-intersecting Tic-Tac-Toe 11. Flowers on the Tiles of the Housing Estates 12. Trolleybus on the Oktogon 13. Bloody Serious Problems 14. Icing on the Cake—Really Difficult Problems Sources