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 NOVEMBER/DECEMBER 2015

THE MATH COLUMN
Logical Thinking – An Important Goal, Especially in Mathematics
By Dr. Alfred Posamentier

One of the responsibilities of teachers – in particular math teachers – is to teach students to think logically.  Sometimes this can be done in a very entertaining (or perhaps a bit frustrating) fashion. Here are two examples that will bring students to think logically and perhaps help their mental training!

Try to find the mistake here – a paradox

A customer walks into a bookshop and buys a book for \$10.  The next day he returns to the bookshop and returns the book he bought the previous day.  He then selected a book costing \$20 and simply walks out with it.  His reasoning is that he paid for the \$10 book on the first day, and then returned the \$10 book, thereby leaving \$10.00 cash plus the \$10.00 book behind.  With this \$20.00 credit, he then took a \$20.00 book and considered it an even trade.  Is this correct? If not where is the error? There is obviously a subtle mistake made for the reader to discover.  (Hint: Try doing this by replacing “a \$10 book” with “two \$5 bills.” The mistake should then become clear.)

The Paradox of the Missing Dollar

Three men plan to spend one night in a hotel room.  They pay \$60.00 for the hotel room. Just as they were about to leave their room, the receptionist noticed that the cost for the room was \$55.00 per night.  The receptionist sends the bellhop to the room to return the \$5.00 of overpayment. However, the bellhop decides to give each the three guests \$1.00, and keeps the remaining \$2.00 for himself.  Therefore, each of the three guests has paid only \$19.00 for the room. The sum of these three payments is therefore 3 x \$19 = \$57.  This plus the \$2.00 that the bellhop kept only totals to \$59.00.  Where is the missing dollar?  Is there some mistake?

After a somewhat bewildered reaction to this transaction, we offer the following explanation of the mistake: It is totally meaningless to add the \$2.00 that the bellhop took to the \$57.00 paid by the three men.  The correct calculation is as follows: three men paid \$57.00 for the room, of which \$55.00 went to the receptionist and \$2.00 went to the bellhop.

Another way of looking at this (unmistakenly), is to note that the three guests got a refund of \$3.00 which when added to the \$55.00 they originally paid for the room and the \$2.00 the bellhop took, use a total of \$60.00. Such calculating mistakes are not uncommon, yet should not be accepted casually.

There are lots of such conundrums that – when presented properly – can truly help develop a student’s logical thinking.  A book that can provide more examples that show the pitfalls in mathematics work is “Magnificent Mistakes in Mathematics” by A. S. Posamentier and I. Lehmann (Prometheus Books, 2013). #