This is a book about mathematics appreciation via discovery, rather than about practical mathematics. It considers several problems that don't appear to be amenable to ordinary arithmetic, algebraic or geometric techniques. It then guides the reader through the process of discovering the solution to each problem, using creative methods and simple techniques that arise naturally. It also indicates how each solution leads to new questions, provides a bit of history of the problem, and discusses a few related problems of current interest that have not yet been solved.
This book is an outgrowth of classes given at the University of California, Santa Barbara, mainly for students who had little mathematical background. Many of the students indicated they never understood what mathematics was all about (beyond what they learned in algebra and geometry). Was there any more math- ematics to be discovered or created? How could one actually discover or create new mathematics? In order to give these students some sort of answers to such questions, we designed a course in which the students could actually participate in the discovery of mathematics. This version of our textbook has all illustrations in color. A less expensive grayscale version is also available.
Conditions of Use
This book is licensed under a Creative Commons License (CC BY-NC-SA). You can download the ebook Mathematical Discovery for free.
- Title
- Mathematical Discovery
- Publisher
- ClassicalRealAnalysis
- Author(s)
- Andrew M Bruckner, Brian S Thomson, Judith B Bruckner
- Published
- 2011-09-29
- Edition
- 1
- Format
- eBook (pdf, epub, mobi)
- Pages
- 268
- Language
- English
- ISBN-10
- 1463730578
- ISBN-13
- 9781463730574
- License
- CC BY-NC-SA
- Book Homepage
- Free eBook, Errata, Code, Solutions, etc.
Table of Contents Preface To the Instructor Tilings Squaring the rectangle Continue experimenting Focus on the smallest square Where is the smallest square What are the neighbors of the smallest square? Is there a five square tiling? Is there a six, seven, or nine square tiling? A solution? Bouwkamp codes Summary Tiling by cubes Tilings by equilateral triangles Supplementary material Squaring the square Additional problems Answers to problems Pick's Rule Polygons On the grid Polygons Inside and outside Splitting a polygon Area of a polygonal region Area of a triangle Some methods of calculating areas An ancient Greek method Grid point credit—a new fast method? Pick credit Experimentation and trial-and-error Rectangles and triangles Additivity Pick's formula Triangles solved Proving Pick's formula in general Summary Supplementary material A bit of historical background Can't be useful though Primitive triangulations Reformulating Pick's theorem Gaming the proof of Pick's theorem Polygons with holes An improved Pick count Random grids Additional problems Answers to problems Nim Care for a game of tic-tac-toe? Combinatorial games Two-marker games Three-marker games Strategies? Formal strategy for the two-marker game Formal strategy for the three-marker game Balanced and unbalanced positions Balanced positions in subtraction games Game of binary bits A coin game A better way of looking at the coin game Binary bits game Nim The mathematical theory of Nim 2–pile Nim 3–pile Nim More three-pile experiments The near-doubling argument Nim solved by near-doubling Review of binary arithmetic Simple solution for the game of Nim Déjà vu? Return to marker games Mind the gap Strategy for the 6–marker game Strategy for the 5–marker game Strategy for all marker games Misère Nim Reverse Nim How to reverse Nim How to play Reverse Misère Nim Summary and Perspectives Supplementary material Another analysis of the game of Nim Grundy number Nim-sums computed Proof of the Sprague-Grundy theorem Why does binary arithmetic keep coming up? Another solution to Nim Playing the Nim game with nim-sums Obituary notice of Charles L. Bouton Answers to problems Links Linking circles Simple, closed curves Shoelace model Linking three curves 3–1 and 3–2 configurations A 4–3 configuration Not so easy? Finding the right notation Algebraic systems Some familiar algebraic systems Linking and algebraic systems When are two objects equal? Inverse notation The laws of combination Applying our algebra to linking problems Return to the 4–3 configuration Solving the 4–3 configuration Constructing a 5–4 configuration The plan Verification How about a 6–5 configuration? Improving our notation again Commutators Moving on. Where we are. Constructing a 4–2 configuration. Constructing 5–2 and 6–2 configurations. Some more constructions. The general construction Introducing a subscript notation Product notation Subscripts on subscripts Groups Rigid Motions The group of linking operations Summary and perspectives A Final Word As mathematics develops A gap? Is our linking language meaningful? Avoid knots and twists Now what? Answers to problems Induction Quitting smoking by the inductive method Proving a formula by induction Setting up an induction proof Starting the induction somewhere else Setting up an induction proof (alternative method) Answers to problems Bibliography Index
