Geometric Magic Squares: A Challenging New Twist Using Colored Shapes Instead of Numbers (Dover Recreational Math)

Traditional magic squares are squares of numbers in which the rows, columns, and diagonals all add up to the same total. With a voluminous literature going back some 2,500 years, the universal assumption has ever been that magic squares are inherently arithmetical objects. In this innovative work by a British engineer, the author initiates a Copernican revolution in our understanding by replacing numbers with two-dimensional forms. The result is not merely a novel kind of geometrical magic square but a revelation that traditional magic squares are now better seen as the one-dimensional instance of this self-same geometrical activity. Title Page 2 Copyright Page 3 Contents 4 Dedication 5 Foreword 6 Part I: Geomagic Squares of 3×3 11 1. Introduction 11 2. Geometric Magic Squares 12 3. The Five Types of 3×3 Area Square 17 4. Construction by Formula 19 5. Construction by Computer 36 6. 3×3 Squares 38 7. 3×3 Nasiks and Semi-Nasiks 44 8. Special Examples of 3×3 Squares 53 Part II: Geomagic Squares of 4×4 62 9. Geo-Latin Squares 62 10. 4×4 Nasiks 67 11. Graeco-Latin Templates 73 12. Uniform Square Substrates 80 13. Dudeney’s 12 Graphic Types 81 14. The 12 Formulae 85 15. A Type I Geomagic Square 87 16. Self-Interlocking Geomagics 90 17. Form and Emptiness 99 18. Further Variations 103 Part III: Special Categories 110 19. 2×2 Squares 110 20. Picture-Preserving Geomagics 116 21. 3-Dimensional Geomagics 130 22. Alpha-Geomagic Squares 135 23. Normal Squares of Order-4 138 24. Eccentric Squares 155 25. Collinear Collations 166 26. Concluding Remarks 171 Appendix I. A Formal Definition of Geomagic Squares 178 Appendix II. Magic Formulae 183 Appendix III. New Advances with 4×4 Magic Squares 206 Appendix IV. The Dual of the Lo shu 219 Appendix V. The Lost Theorem 226 Glossary 236 References 243

Traditional magic squares employ a chessboard-like arrangement of numbers in which the total of all rows, columns, and diagonals add up to the same number. This innovative approach by a British engineer shines a new light on the centuries-old amusement, challenging puzzlists to think two dimensionally in solving dozens of creative puzzles with vivid geometric shapes instead of numbers.
Attempt The Twelve Pentominoes, a challenge consisting of the formation of a dozen different shapes from five-unit squares. Squeeze together the 16 pieces of a 4 x 4 self-interlocking square to form an empty 4 x 4 array. Take aim at 48 targets featuring astrological signs in a self-interlocking square. Suitable for puzzle lovers ages 12 and up, these and other colorful variations on conventional magic squares offer hours of captivating entertainment.

Traditional magic squares employ a chessboard-like arrangement of numbers in which the total of all rows, columns, and diagonals add up to the same number. This innovative approach by a Dutch engineer challenges puzzlists to think two dimensionally by replacing numbers with colorful geometric shapes. Dozens of creative puzzles, suitable for ages 12 and up