Proofs that Really Count: The Art of Combinatorial Proof (Dolciani Mathematical Expositions)

Proofs that Really Count: The Art of Combinatorial Proof (Dolciani Mathematical Expositions)

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Proofs that Really Count: The Art of Combinatorial Proof (Dolciani Mathematical Expositions)

Proofs that Really Count: The Art of Combinatorial Proof (Dolciani Mathematical Expositions)

Written by Arthur T. Benjamin
4.85 53,176 ratings
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About this ebook:
Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. The book emphasizes numbers that are often not thought of as numbers that count: Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels, from high school math students to professional mathematicians.

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Pages N/A
Published August 1, 2003
ISBN 765859
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Arthur T. Benjamin
About the Author
Arthur T. Benjamin

After spending years working in the corporate world, Arthur T. Benjamin finally decided to pursue a lifelong passion for writing. They currently reside in a quiet suburb where they spend most of their time outlining new stories, reading extensively, and occasionally attempting to bake.

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Dorothy Martin
Dorothy Martin July 14, 2024
I thoroughly enjoyed the elegance of it. The protagonist was compelling, though some subplots felt slightly overshadowed. I would highly recommend this to discerning readers.
Mark Garcia
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A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece.
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Absolutely brilliant. The prose is exquisite. It’s accessible yet undeniably sophisticated. An effortless 5 stars.
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Such a rich narrative. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece.
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Masterpiece from start to finish. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem.
Edward Sanchez
Edward Sanchez April 29, 2025
An acceptable diversion. The tempo was somewhat languid, and I found my mind wandering. Not poor, simply average. Unlikely to reread, but a reasonable expenditure of time.
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Jessica Robinson November 26, 2025
I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece.
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A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers.
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Absolutely brilliant. The author orchestrated a breathtaking finale. Truly a work of art. I will undoubtedly revisit this masterpiece.
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Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece.
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Masterpiece from start to finish. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I will undoubtedly revisit this masterpiece.
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Masterpiece from start to finish. I became deeply invested in the protagonist's journey. A masterclass in storytelling. Worth every moment spent reading.
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I find myself somewhat conflicted. It verged on repetitive in the second act. A more concise edit would have served it well. Perhaps more aligned with a different audience's tastes.
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Exceeded all my expectations. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem.
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I was completely blown away. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I am now a lifelong fan of this author.
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Masterpiece from start to finish. The author orchestrated a breathtaking finale. Truly a work of art. I am now a lifelong fan of this author.
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Honestly one of the most premium reading experiences lately. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem.
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A very strong entry. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A very distinguished novel.
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I thoroughly enjoyed the elegance of it. It held my rapt attention, even when certain twists were telegraphed. I would highly recommend this to discerning readers.
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Excellent, though slightly flawed. A brilliant concept executed with flair. Just a few pacing choices I would change. A wonderful companion for a quiet weekend.
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Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An absolute must-read for connoisseurs of the genre.
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A profoundly satisfying read. The author orchestrated a breathtaking finale. Truly a work of art. An effortless 5 stars.
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A highly sophisticated read. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A wonderful companion for a quiet weekend.
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I was completely blown away. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece.
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Such a rich narrative. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading.
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Honestly one of the most premium reading experiences lately. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An effortless 5 stars.
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I thoroughly enjoyed the elegance of it. A brilliant concept executed with flair. Just a few pacing choices I would change. I would highly recommend this to discerning readers.
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I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An absolute must-read for connoisseurs of the genre.
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Masterpiece from start to finish. The prose is exquisite. It’s accessible yet undeniably sophisticated. A true literary gem.
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I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece.
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A profoundly satisfying read. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An effortless 5 stars.
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Such a rich narrative. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I will undoubtedly revisit this masterpiece.
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Honestly one of the most premium reading experiences lately. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. Worth every moment spent reading.
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I was completely blown away. I adored how the story unfolded. It kept me captivated without sacrificing elegance. Worth every moment spent reading.
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Exceeded all my expectations. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece.
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Excellent, though slightly flawed. The protagonist was compelling, though some subplots felt slightly overshadowed. A wonderful companion for a quiet weekend.
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I was completely blown away. I adored how the story unfolded. It kept me captivated without sacrificing elegance. A true literary gem.
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Took a moment to build, but the payoff was spectacular. It held my rapt attention, even when certain twists were telegraphed. A wonderful companion for a quiet weekend.
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Such a rich narrative. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading.
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Took a moment to build, but the payoff was spectacular. The middle had a deliberate pace, but the climax was phenomenal. A very distinguished novel.
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Took a moment to build, but the payoff was spectacular. The middle had a deliberate pace, but the climax was phenomenal. A wonderful companion for a quiet weekend.
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Jacob Adams April 02, 2025
Such a rich narrative. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading.
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It was adequate. The tempo was somewhat languid, and I found my mind wandering. Not poor, simply average. Unlikely to reread, but a reasonable expenditure of time.
Amanda Moore
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Absolutely brilliant. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author.
Steven Williams
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Such a rich narrative. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece.
Paul Moore
Paul Moore August 16, 2025
I thoroughly enjoyed the elegance of it. A brilliant concept executed with flair. Just a few pacing choices I would change. A very distinguished novel.
Mary Nelson
Mary Nelson October 03, 2025
Excellent, though slightly flawed. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A solid 4 stars from me.
Donna King
Donna King November 11, 2024
It was adequate. It verged on repetitive in the second act. A more concise edit would have served it well. Perfectly average.
Margaret Lopez
Margaret Lopez January 16, 2025
Masterpiece from start to finish. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I will undoubtedly revisit this masterpiece.
Margaret Allen
Margaret Allen December 24, 2025
I was completely blown away. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece.
Richard Robinson
Richard Robinson April 14, 2024
I thoroughly enjoyed the elegance of it. It held my rapt attention, even when certain twists were telegraphed. Quite impressive overall.
Thomas Rivera
Thomas Rivera July 18, 2025
Took a moment to build, but the payoff was spectacular. The protagonist was compelling, though some subplots felt slightly overshadowed. Quite impressive overall.
Elizabeth Martin
Elizabeth Martin December 07, 2024
I was completely blown away. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I am now a lifelong fan of this author.
Elizabeth Garcia
Elizabeth Garcia April 10, 2024
A highly sophisticated read. The protagonist was compelling, though some subplots felt slightly overshadowed. I would highly recommend this to discerning readers.
James Mitchell
James Mitchell December 09, 2024
A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading.
Nancy Flores
Nancy Flores September 12, 2024
Exceeded all my expectations. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece.
Deborah Wright
Deborah Wright November 24, 2024
I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading.
Deborah Ramirez
Deborah Ramirez October 17, 2024
I thoroughly enjoyed the elegance of it. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers.
Joshua Garcia
Joshua Garcia November 03, 2024
Such a rich narrative. I became deeply invested in the protagonist's journey. A masterclass in storytelling. Worth every moment spent reading.
Stephanie Martinez
Stephanie Martinez October 12, 2024
Excellent, though slightly flawed. The middle had a deliberate pace, but the climax was phenomenal. A solid 4 stars from me.
Edward Martinez
Edward Martinez June 14, 2024
Honestly one of the most premium reading experiences lately. The prose is exquisite. It’s accessible yet undeniably sophisticated. An absolute must-read for connoisseurs of the genre.
Ryan Edwards
Ryan Edwards October 04, 2025
Excellent, though slightly flawed. It held my rapt attention, even when certain twists were telegraphed. A solid 4 stars from me.
Kimberly Collins
Kimberly Collins November 15, 2024
I was completely blown away. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece.
Gary Cruz
Gary Cruz February 27, 2025
Masterpiece from start to finish. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An absolute must-read for connoisseurs of the genre.
Sharon Wilson
Sharon Wilson December 18, 2024
A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. A solid 4 stars from me.
Nancy Wright
Nancy Wright August 26, 2024
A highly sophisticated read. The protagonist was compelling, though some subplots felt slightly overshadowed. A very distinguished novel.
Amanda Clark
Amanda Clark January 11, 2024
Exceeded all my expectations. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author.
Richard Robinson
Richard Robinson September 08, 2024
Absolutely brilliant. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. Worth every moment spent reading.
Matthew Moore
Matthew Moore April 23, 2025
Excellent, though slightly flawed. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers.
Joshua Taylor
Joshua Taylor February 13, 2025
A decent, if standard, experience. The prose was competent, yet the characters lacked the depth I usually seek. A fair 3 stars.
Carol Garcia
Carol Garcia March 08, 2024
Masterpiece from start to finish. I adored how the story unfolded. It kept me captivated without sacrificing elegance. Worth every moment spent reading.
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