Hi All,
I'd like to solve some math puzzles, especially in the context of probability theory, but I'm open to other areas too. The kind of problems that does not require much knowledge of mathematics, except, perhaps, for a basic background.
I found this one helpful
http://www.amazon.com/Fifty-Challenging-Problems-Probability-Solutions/dp/0486653552
but I was wondering if you folks could recommend others.
Thank you
Problems and Snapshots from the World of Probability by Gunnar Blom, Lars Holst, and Dennis Sandell is a book that I like very much. Their problems are pretty challenging and assume a working knowledge of basic probability.
From the Russian masters of recreational mathematics: Challenging Mathematical Problems with Elementary Solutions, Vol. 1: Combinatorial Analysis and Probability Theory by Yaglom & Yaglom.
"Mathematical circles" by Fomin,Genkin,Itenberg is a nice collection of puzzles|problems.
Raymond Smullian's many books of logic puzzles are quite nice (if you like that sort of thing).
David Singmaster, Problems for Metagrobologists.
"This book is a collection of over 200 problems that David Singmaster has composed since 1987. Some of the math problems have appeared in his various puzzle columns for BBC Radio and TV, Canadian Broadcasting, Focus (the UK popular science magazine), Games and Puzzles, the Los Angeles Times, Micromath, the Puzzle a Day memo pad and the Weekend Telegraph. While some of these are already classics, many of the puzzles have not been published elsewhere previously.
"Puzzle enthusiasts of all ages will find here arithmetic problems, properties of digits; monetary problems; alpha-metics; Diophantine problems; magic figures; sequence problems; logical problems; geometric problems; physics problems; combinatorial problems; geographic problems; calendar problems; clock problems; dissection problems and verbal problems."
Contents:
General Arithmetic Puzzles
Properties of Digits
Magic Figures
Monetary Problems
Diophantine Recreations
Alphametics
Sequence Puzzles
Logic Puzzles
Geometrical Puzzles
Geographic Problems
Calendrical Problems
Clock Problems
Physical Problems
Combinatorial Problems
Some Verbal Puzzles
Paul R Halmos, Problems for Mathematicians, Young and Old.
"A collection of math problems for people of varying skills from high school through professional level, organized into fourteen categories, such as matrices, space, probability, and puzzles, and including hints and solutions."
Donald J Newman, A Problem Seminar.
"There was once a bumper sticker that read, "Remember the good old days when air was clean and sex was dirty?" Indeed, some of us are old enough to remember not only those good old days, but even the days when Math was fun(!), not the ponderous THEOREM, PROOF, THEOREM, PROOF, . . . , but the whimsical, "I've got a good problem. " Why did the mood change? What misguided educational philosophy transformed graduate mathematics from a passionate activity to a form of passive scholarship? In less sentimental terms, why have the graduate schools dropped the Problem Seminar? We therefore offer "A Problem Seminar" to those students who haven't enjoyed the fun and games of problem solving.
CONTENTS
Preface v
Format I
Problems 3
Estimation Theory 11
Generating Functions 17
Limits of Integrals 19
Expectations 21
Prime Factors 23
Category Arguments 25
Convexity 27
Hints 29
Solutions 41
FORMAT This book has three parts: first, the list of problems, briefly punctuated by some descriptive pages; second, a list of hints, which are merely meant as words to the (very) wise; and third, the (almost) complete solutions. Thus, the problems can be viewed on any of three levels: as somewhat difficult challenges (without the hints), as more routine problems (with the hints), or as a textbook on "how to solve it" (when the solutions are read). Of course it is our hope that the book can be enjoyed on any of these three levels."
Stephen Barr, Mathematical Brain-Benders; also, The Man in the Milk Carton.
"Challenge yourself with over 100 fresh paradoxes, puzzles, riddles, conundrums, word and number games for the jaded, skeptical puzzlist. Over 100 pages of comprehensive answers. Approximately 300 illustrations."
Hungarian Problem Books, I, II, III, IV.
"The Eötvös Mathematics Competition is the oldest high school mathematics competition in the world, dating back to 1894. This book is a continuation of Hungarian Problem Book III and takes the contest through 1963. Forty-eight problems in all are presented in this volume. Problems are classified under combinatorics, graph theory, number theory, divisibility, sums and differences, algebra, geometry, tangent lines and circles, geometric inequalities, combinatorial geometry, trigonometry and solid geometry. Multiple solutions to the problems are presented along with background material. There is a substantial chapter entitled "Looking Back," which provides additional insights into the problems. Hungarian Problem Book IV is intended for beginners, although the experienced student will find much here. Beginners are encouraged to work the problems in each section, and then to compare their results against the solutions presented in the book. They will find ample material in each section to help them improve their problem-solving techniques."
Loren C Larson, Problem-Solving Through Problems.
Vaderlind, Guy, and Larson, The Inquisitive Problem Solver.
Hugo Steinhaus, One Hundred Problems in Elementary Mathematics.
Heinrich Dörrie, 100 Great Problems of Elementary Mathematics.
Martin Gardner, ed., Mathematical Puzzles of Sam Loyd.
Peter Winkler, Mathematical Puzzles; also, Mathematical Mind-Benders.
Henry E Dudeney, Amusements in Mathematics.
Konhauser, Velleman, and Wagon, Which Way Did the Bicycle Go?
Oliver Roeder, The Riddler.
Ditmarsch and Kooi, One Hundred Prisoners and a Light Bulb.
Time permitting, I'll edit in some links to more information about these books.
On the site of the International Mathematics Olympiad all past problems can be found. They tend to be quite difficult but no advanced math is required.
There are some good math puzzles from interviews over here:
David Williams has a book called Probability with Martingales which includes some excellent puzzles as exercises. These are often easy to state but require real trickery to prove.
In the spirit of doetoe's answer, I also recommend the Putnam. They don't have a ton of probability based puzzles but they do have some. That website also has old AIME problems and solutions. In both cases the problems are tricky and fun.