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Journals ACM SIGACT News Volume 36 Issue 1

Review 2 of Proofs that Really Count: The Art of Combinatorial Proof Author: Arthur T. Benjamin and Jennifer J. Quinn Publisher: MAA, 2003 $43.95, Hardcover Reviewer: William Gasarch Abbott has been teaching Costello combinatorics. Abbott: Costello, how many subsets are there of {1, . . . , n}? Costello: Oh. You can either choose 0 elements, or choose 1 element, or choose 2 elements, etc. So the answer is n n . i=0 i Abbott: Well . . . let me show you a di €erent way to do it. The number 1 is either in the set A or not, so thats 2 choices. Then the number 2 is either in the set A or not, so thats 2 choices, etc. So the nal answer is 2 2 = 2n . So, Costello, you did the problem your way, I did it my way, and we got di €erent answers. What can you conclude? Costello: That one of us is wrong? Abbott: No. We ™ve shown. n n = 2n . i=0 i Costello: Really! I don ™t believe that! Prove it!! Abbott: We did! Costello: When? Abbott: Just now. Costello: What!? Abbott: Whats on Second. Costello:

Review of Proofs that Really Count: The Art of Combinatorial Proof by Arthur T. Benjamin and Jennifer J. Quinn; MAA, 2003


Review 2 of Proofs that Really Count: The Art of Combinatorial Proof Author: Arthur T. Benjamin and Jennifer J. Quinn Publisher: MAA, 2003 $43.95, Hardcover Reviewer: William Gasarch Abbott has been teaching Costello combinatorics. Abbott: Costello, how many subsets are there of {1, . . . , n}? Costello: Oh. You can either choose 0 elements, or choose 1 element, or choose 2 elements, etc. So the answer is n n . i=0 i Abbott: Well . . . let me show you a di €erent way to do it. The number 1 is either in the set A or not, so thats 2 choices. Then the number 2 is either in the set A or not, so thats 2 choices, etc. So the nal answer is 2 2 = 2n . So, Costello, you did the problem your way, I did it my way, and we got di €erent answers. What can you conclude? Costello: That one of us is wrong? Abbott: No. We ™ve shown. n n = 2n . i=0 i Costello: Really! I don ™t believe that! Prove it!! Abbott: We did! Costello: When? Abbott: Just now. Costello: What!? Abbott: Whats on Second. Costello: Who? Abbott: Who ™s on rst. Costello: (Ignoring reference) Usually when I do a math problem two ways and get two answers I assume one of them is wrong and try to nd my error. Its better than what a friend of mine did in elementary algebra ” do a problem three times and then take the average. Abbott: In math you can sometimes prove that two things are the same by solving the same problem two di €erent ways. Costello: No way! Abbott: Way! Costello: I ™d like to read more about this. Do you have a book to recommend? Abbott: Yes. You should read Proofs that Really Count: The Art of Combinatorial Proof by Arthur T. Benjamin and Jennifer J. Quinn. Costello: Can you give me some examples from the book? Abbott: Okay. Here is one involving Fibonacci Numbers. Recall that the Fibonaci numbers are de ned by f0 = 0, f1 = 1, and ( €n ¥ 2)[fn = fn ’1 + fn ’2 ]. I...

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Review of Proofs that Really Count: The Art of Combinatorial Proof by Arthur T. Benjamin and Jennifer J. Quinn; MAA, 2003

Gasarch, William
ACM SIGACT News , Volume 36 (1)
Association for Computing Machinery – Mar 1, 2005

More Info
  • Publisher Association for Computing Machinery
  • Copyright The ACM Portal is published by the Association for Computing Machinery. Copyright © 2010 ACM, Inc.
  • Subject Combinatorial algorithms
  • ISSN 0163-5700
  • D.O.I. 10.1145/1052796.1052800
  • Publisher site Get PDF  

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