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I am teaching English in Japan and I have a student who speaks English well, and to keep up his level, in our weekly lessons would like to learn some subjects related to my degree in mathematics.

I would like to get across reasons why mathematics can be beautiful and interesting in a way that is accessible to a non-mathematician. Ideally this would help him understand some ideas in mathematics, avoiding the rigour of undergraduate courses.

Are there any good resources, books, websites etc., which I could be pointed towards? Or additionally does anyone have any good topics which I could look into?

One topic I have found is the development of cryptography, based on The Code Book by Simon Singh

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  • $\begingroup$ Martin Gardner wrote a mathematical games column in Scientific American for many years. The articles have been anthologized. $\endgroup$ – Ben Crowell Sep 20 '15 at 18:02
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Perhaps (depending on background) give them one of the books by William Dunham, like "Journey through Genius" (Dover, 1990), "The Mathematical Universe" (Wiley, 1997), there are others. He won several awards for expository writing on mathematics, Aigner et al's "Proofs from THE BOOK" (Springer, 2003) is more demanding, but a lot of fun.

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I'm afraid that this might not be of use to you, because it's in Hebrew, but this question reminded me of the really nice Beauty of Mathematics course that Gil Kalai organized at the Hebrew University in Jerusalem for non-science undergraduates.

The slides from the course (in Hebrew) are available on his blog and they include various subjects such as: Euclid's proof of the infinitude of primes, a basic overview of the concept of symmetric tilings, examples of games that can be fully solved (e.g. Nim), n-dimensional geometry, Cantor's diagonal argument about the cardinality of infinite sets, and others. The topics were mostly presented in a non-rigorous, but very engaging, introductory manner.

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"Mathematics, a Human Endeavor: A Textbook for Those Who Think They Don't Like the Subject" is a great book. Lots of fun/interesting problems that are accessible to a wide range of math levels. The description says it's targeted at liberal arts math classes that focus on problem-solving:

"For instructors of liberal arts mathematics classes who focus on problem-solving, Harold Jacobs's remarkable textbook has long been the answer, helping teachers connect with of math-anxious students. Drawing on over thirty years of classroom experience, Jacobs shows students how to make observations, discover relationships, and solve problems in the context of ordinary experience."

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People are often keen to develop their knowledge of their current interests so, if you can find out what interests your student (sport, cars, travel, cookery, whatever), you can hang some elegant mathematics on that.

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