{"id":24000,"date":"2013-09-06T18:28:14","date_gmt":"2013-09-06T22:28:14","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=24000"},"modified":"2014-06-24T13:36:53","modified_gmt":"2014-06-24T18:36:53","slug":"gardners-tribes","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/09\/06\/gardners-tribes\/","title":{"rendered":"Gardner’s Two Tribes"},"content":{"rendered":"

\"imgres\"<\/a>Mathematical conundrums are not always the product of academic rigor. \u00a0 Sometimes they come from a recreational playground built on the foundation of logic and unique understanding. \u00a0 Beyond numbers and sets, mathematical thinking is rooted in the ability to extract patterns and make non-intuitive connections.\u00a0 The late Scientific American Mathematical Games<\/i> columnist and writer, Martin Gardner was a master at communicating such patterns. \u00a0Subsequently, his expositions influenced generations of math hobbyists and professional mathematicians alike. In reviewing Gardner’s book, The Colossal Book Of Mathematics<\/em>, \u00a0Dr. Persi Diaconis of Stanford University stated , \u201cWarning: Martin Gardner has turned dozens of innocent youngsters into math professors and thousands of math professors into innocent youngsters\u2026\u201d<\/i><\/p>\n

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Though he influenced the field of recreational mathematics greatly, Gardner was not a professional mathematician.\u00a0 \u00a0 Starting with his monthly Scientific American <\/i>column in 1956, his iconic logic and math oriented puzzles still hold the standard in recreational math literature. \u00a0 \u00a0A simple example of this can be found in the following riddle Gardner wrote in his book Entertaining Mathematical Puzzles <\/i>entitled \u201cThe Two Tribes\u201d:<\/p>\n

\u201cAn island is inhabited by two tribes.\u00a0 Members of one tribe always tell the truth, members of the other always lie.<\/i><\/p>\n

A missionary met two of these natives, one tall, the other short. \u2018Are you a truth-teller?\u2019 he asked the taller one.<\/i><\/p>\n

\u2018Oopf,\u2019 the tall native answered.<\/i><\/p>\n

The missionary recognized this as a native word meaning either yes or no, but he couldn\u2019t recall which.\u00a0 The short native spoke English, so the missionary asked him what his companion had said.<\/i><\/p>\n

\u2018He say yes,\u2019 replied the short native, \u2018but him big liar!\u2019<\/i><\/p>\n

What tribe did each native belong to?\u201d<\/i><\/p>\n

<\/i>The answer is given by breaking the problem up into two cases. \u00a0Suppose the tall one is of the tribe that always tells the truth. \u00a0In this case, the tall native would have said yes. \u00a0Likewise, \u00a0If the tall native would have been of the tribe that always told lies, he still would have said yes. \u00a0Since the missionary understood this, he knew the short native was telling the truth when he translated what the tall native said. \u00a0Therefore, the missionary concluded that the short native was of the tribe that always told the truth and the tall native was of the tribe that always lied.<\/p>\n

The strategy and thought processes behind solving such riddles is very similar to how a mathematician proves theorems. \u00a0Gardner routinely placed mathematical methodologies into the solutions of his riddles and puzzles. \u00a0 The Pigeonhole Principle, theorems from Logic, Number Theory, Topology, etc. show up as regulars in Gardner’s playground of mathematical conundrums.<\/p>\n

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Mathematical conundrums are not always the product of academic rigor. \u00a0 Sometimes they come from a recreational playground built on the foundation of logic and unique understanding. \u00a0 Beyond numbers and sets, mathematical thinking is rooted in the ability to … Continue reading →<\/span><\/a><\/p>\n

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