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When I was younger some of my favorite books were Raymond Smullyan's puzzle books. He strove to write books that covered topics more deep than more standard children's puzzles; To Mock a Mockingbird gets into serious thought about combinatory logic, and Forever Undecided, an introduction to Godel's incompleteness theorem, phrasing everything in terms of the classic children's puzzles about people who always lie and people who always tell the truth. (For a much shorter but higher-brow sample of what the writing looks like, he has a brief note about roughly the same subject.) I recently looked at them again, and they still hold up (or maybe are even made better) looking back at them with an actual math education.

What I would like are books that are accessible to a general audience - especially kids! - that nonetheless are able to introduce one to nontrivial, deep mathematics, like Smullyan's books above. I would prefer books that actually teach the mathematics (however disguised) as opposed to something like Simon Singh's expository book on Fermat's Last Theorem (which tells you the story of FLT, but nothing about the mathematics behind it.)

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Essentially, the first third of the book is devoted to two-dimensional symmetry, and is generally very layperson-friendly, with lots of good pictures and things that utilize intuition. Generally the first 100 pages should be doable for anybody with a genuine interest. The next 100 pages are a bit more difficult, and then abstract groups are encountered, at which point all bets are off (but to be fair, it's still a pretty gentle treatment!). So, maybe it's not a very good fit for young children, but I'd say ages 10-12+ should be able to get plenty out of it. It doesn't hide the fact that it's a book about math, whatever that means.

  • And while I realize this isn't a book, Satyan Devadoss has a series of video lectures about quality mathematics for a general audience, called The Shape of Nature. It's not cheap, but I managed to borrow a copy through an interlibrary loan and it was well worth it.
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In Russia, those 2 books were very popular:

not sure how their english translation compares to the original Russian though

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