# Gardner’s Two Tribes

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

Though he influenced the field of recreational mathematics greatly, Gardner was not a professional mathematician.    Starting with his monthly Scientific American column in 1956, his iconic logic and math oriented puzzles still hold the standard in recreational math literature.    A simple example of this can be found in the following riddle Gardner wrote in his book Entertaining Mathematical Puzzles entitled “The Two Tribes”:

“An island is inhabited by two tribes.  Members of one tribe always tell the truth, members of the other always lie.

A missionary met two of these natives, one tall, the other short. ‘Are you a truth-teller?’ he asked the taller one.

The missionary recognized this as a native word meaning either yes or no, but he couldn’t recall which.  The short native spoke English, so the missionary asked him what his companion had said.

‘He say yes,’ replied the short native, ‘but him big liar!’

What tribe did each native belong to?”

The answer is given by breaking the problem up into two cases.  Suppose the tall one is of the tribe that always tells the truth.  In this case, the tall native would have said yes.  Likewise,  If the tall native would have been of the tribe that always told lies, he still would have said yes.  Since the missionary understood this, he knew the short native was telling the truth when he translated what the tall native said.  Therefore, 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.

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