Automated Generation of Anomalous Cancellations

.pdf   .ps   .tex

First Written: Sept. 5, 2017

Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

When my beloved master, Doron Zeilberger (who kindly wrote the foreword to this article), visited Israel in July 2017, he bought a fascinating `popular' book on game theory (in Hebrew) by the great expositor Haim Shapira. In his book, Shapira commented how gullible his students are. Once he had to `simplify' the fraction 64/16 by `cancelling out' the 6,

6 4 /1 6 = 4/1   ,

getting that it equals 4, that is the correct answer. None of his students said anything for a whole minute, until one brave student objected that this `simplification', while it does give you the right answer, is not exactly kosher. Shapira was wondering if there were more examples. BTW, this is a good way to shut students up when they complain about getting zero credit for a problem where they got the right answer (by sheer luck), using an erroneous method.

When Doron came back home, he asked Neil Sloane whether he has heard about it, and sure enough, Neil told him about OEIS sequence A159975, that lead him to a wonderful article by R.P. Boas, who in addition to being a great analyst was also a master expositor (BTW, his son Harold Boas, is also a great expositor (and mathematician)). R.P. Boas already used a computer to generate some data, but back in 1979 could only go that far. Here we go much further, and readers are welcome to go even further using the Maple package below.

Added Oct. 5, 2017: On Oct. 2, 2017, DZ gave a lecture for the Rutgers Undergraduate Math Association, and at the end he gave this quiz. Quite a few people came close, but only seven people got perfect score. Here there are (in alphabetical order)

• Steve Hsu

• Eric Giovannini

• Daniel Solano

• Lauren Squillace

• Sohum Sanghvi

• Tiancheng Xue

Here are scans of their nice Solutions.

It was a tough choice who the best entry is, but DZ decided on Eric Giovannini, who won the book "Mathematical Plums", edited by Ross Honsberger.

# Sample Input and Output for AnomalousCancellation.txt

• If you want to see a computer-generated book including all the anomalous cancellations (in base 10) with the denominator with four digits

the input file generates the output file.

[Note: this is .txt version of the above TeX-typset article ],

• If you want to see all examples of anomalous cancellation with the denominator with two digits, for all bases up to 150 (extending R.P. Boas' tables in reference [B]

the input file generates the output file.

• If you want to see all anomalous cancellations where the denominator has up to three digits for all bases from 2 to 20

the input file generates the output file.

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger