# Research Paper: Math Problems and Concepts in Teaching

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Mathematics of Mathematic Puzzles

Using mathematics puzzles is a frequently-used deployed pedagogical device to teach critical concepts to math students of all ages. "Understanding in mathematics is born not only from formulas, definitions and theorems but, and even more so, from those networks of related problems… Mathematicians seek knowledge. In search of knowledge, they enjoy themselves tremendously inventing and solving new problems" (Bogomolny 2015). The mathematics of puzzles is also designed to challenge the student's instinctive conceptions of how the world works. For example consider the question posed 1702 of how long would a rope encircling the earth have to be "for it to be one foot off the ground all the way around the equator -- this can be used to illustrate the geometric concept of finding the circumference (Pickover 2010). The student's intuitive instinct would be that the rope must be extremely long but in fact the answer is only 2pi. "If r is the radius of the Earth, and 1 + r is the radius in feet of the enlarged circle, we can compare the rope circumference before (2pir) and after (2pi (1 + r))" (Pickover 2010).

Other examples of these teaching devices are the multiplying 'wheat on a chessboard' dilemma. In this problem, the grains of wheat, beginning with one, are doubled every square leading to such a proliferation of grains that it would be impossible to provide that much wheat in reality, thus illustrating the concept of geometric growth (Pickover 2010). There is also the barber's paradox "which involves a town with one male barber who, every day, shaves every man who doesn't shave himself, and no one else" which is used to illustrate the concept of set theory (Pickover 2010).

The applicability of apparently pointless mathematical queries to vital foundational concepts in math has also come to light in the evolving discipline of game theory, frequently used by economists to illustrate how people make choices. For example, in the classical case of the Prisoner's Dilemma, two separately-imprisoned individuals are placed in a cell. If neither confess, both men go free, if both confess, both get a reduced sentence but still do jail time, while if only…
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