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Mathematical Fallacies, Flaws and Flimflam (Spectrum)

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The creativity of students knows no bounds when it comes to excuses and the methods used to solve problems. While the wait for a collection of classic excuses continues, it is over for a collection of "creative" solutions. Generally, these are problems where a misapplication of mathematics somehow leads to a correct answer. In many cases, the mathematics needed to understand how the solution was arrived at using incorrect methods is more complex than that needed to solve it using "conventional" techniques.
There are times when it is necessary to read the solution several times very carefully before the flaw in the reasoning becomes apparent. Quite frankly, if some of these answers had appeared on my exams, I probably would have marked them as correct. Which only serves to enhance their charm.
The problems are taken from many different areas of mathematics, some of which have attracted a great deal of attention in the past. Classics such as trisecting an arbitrary angle and squaring the circle make their obligatory appearance. Once again proving that no matter how many times and ways something is verified.
there will always be people who believe only what they find convenient to believe. The others are taken from various undergraduate levels; basic algebra to advanced undergraduate mathematics. Of all the areas, those in probability seem to create the most controversy. The one that is rapidly becoming a classic is the "Monty Hall" or "car and goats" problem. The situation is one where a valuable prize is behind one of three doors with much less valuable objects behind the other two. With no knowledge of what is behind them.
you must choose one. Another entity, such as an announcer, knows what is behind each door. Therefore.
after your choice, you are shown what is behind one of the doors that you did not choose which also did not hide the prize. You are then given the opportunity to alter your choice to the other one that has not been revealed. The question then posed is. "Should you stick with your first choice or change to the other door?"
This problem is really an encapsulation of so many of the difficulties with probability problems, in that the false solutions appear so logical. Even experienced people often reach an inappropriate conclusion. The treatment here is well done, with several variants of the problem given.
Even though the problems presented here are all within the scope of undergraduate mathematics, I sometimes found myself thinking as hard about these problems as I have over problems that have appeared in other material containing more advanced topics. This is one more indication of how dynamic the field of mathematics is.

Published in Journal of Recreational Mathematics, reprinted with permission

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