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). #